L1.3 L1.3 Department of Mathematics, Computer Science and Physics, University
of Udine, Italy<EMAIL_ADDRESS>Department of Mathematics, Computer
Science and Physics, University of Udine, Italy
<EMAIL_ADDRESS>by the Italian MIUR PRIN
2017FTXR7S _IT MATTERS_. Davide Castelnovo and Marino Miculan <ccs2012>
<concept> <concept_id>10003752.10003790</concept_id> <concept_desc>Theory of
computation Logic</concept_desc>
<concept_significance>500</concept_significance> </concept> </ccs2012>
[500]Theory of computation Logic A short version of this paper will appear in
the _CSL 2022_ conference proceedings. John Q. Open and Joan R. Access 2 42nd
Conference on Very Important Topics (CVIT 2016) CVIT 2016 CVIT 2016 December
24–27, 2016 Little Whinging, United Kingdom 42 xx

# Fuzzy algebraic theories

Davide Castelnovo  Marino Miculan

###### Abstract

In this work we propose a formal system for _fuzzy algebraic reasoning_. The
sequent calculus we define is based on two kinds of propositions, capturing
equality and existence of terms as members of a fuzzy set. We provide a sound
semantics for this calculus and show that there is a notion of free model for
any theory in this system, allowing us (with some restrictions) to recover
models as Eilenberg-Moore algebras for some monad. We will also prove a
completeness result: a formula is derivable from a given theory if and only if
it is satisfied by all models of the theory. Finally, leveraging results by
Milius and Urbat, we give HSP-like characterizations of subcategories of
algebras which are categories of models of particular kinds of theories.

###### keywords:

categorical logic, fuzzy sets, algebraic reasoning, equational
axiomatisations, monads, Eilenberg-Moore algebras.

††terms: L††terms: L††terms: L††terms: L††terms: L††terms: L††terms: L††terms:
L††terms: L††terms: L††terms: L††terms: L††terms: L††terms: L††terms:
L††terms: L††terms: L††terms: L††terms: L††terms: L††terms: L††terms:
L††terms: L

##  1 Introduction

One of the most fruitful and influential lines of research of Logic in
Computer Science is the algebraic study of computation. After Moggi’s seminal
work [16] showed that notions of computation can be represented as monads,
Plotkin and Power [19] approached the problem using operations and equations,
i.e., Lawvere theories. Since then, various extensions of the notion of
Lawvere theory have been introduced in order to accommodate an ever increasing
number of computational notions within this framework; see, e.g., [20, 9, 18],
and more recently [3, 4] for quantitative algebraic reasoning for
probabilistic computations.

Along this line of research, in this work we study algebraic reasoning on
_fuzzy sets_. Algebraic structures on fuzzy sets are well known since the
seventies (see e.g., [21, 14, 1, 17]). Fuzzy sets are very important in
computer science, with applications ranging from pattern recognition to
decision making, from system modeling to artificial intelligence. So, it is
natural to ask if it is possible to use an approach similar to above for
_fuzzy algebraic reasoning_.

In this paper we answer positively to this question. We propose a sequent
calculus based on two kind of propositions, one expressing equality of terms
and the other the existence of a term as a member of a fuzzy set. These
sequents have a natural interpretation in categories of fuzzy sets endowed
with operations. This calculus is sound and complete for such a semantics: a
formula is satisfied by all the models of a given theory if and only if it is
derivable from it.

It is possible to go further. Both in the classical and in the quantitative
settings there is a notion of free model for a theory; we show that is also
true for theories in our formal system for fuzzy sets. In general the category
of models of a given theory will not be equivalent to the category of
Eilenberg-Moore algebras for the induced monad, but we will show that this
equivalence holds for theories with sufficiently simple axioms. Finally we
will use the techniques developed in [15] to prove two results analogous to
Birkhoff’s theorem.

Synopsis. In Section 2 we recall the category ${\textbf{Fuz}}\quantity(H)$ of
fuzzy sets over a frame $H$. Section 3 introduces the syntax and the rules of
fuzzy theories. Then, in Section 4 we introduce the notions of algebras for a
signature and of models for a theory; in this section we will also show that
the calculus proposed is sound and complete. Section 5 is devoted to free
models and it is shown that if a theory is _basic_ then its category of models
arose as the category of Eilenberg-Moore algebras for a monad on
${\textbf{Fuz}}\quantity(H)$. In Section 6 we use the results of [15] to prove
two HSP-like theorems for our calculus. Finally, Section 7 draws some
conclusions and directions for future work. Appendix A contains two detailed
derivations used in the paper.

##  2 Fuzzy sets

In this section we will recall the definition and some well-known properties
of the category of fuzzy sets over a frame $H$ (i.e. a complete Heyting
algebra [10]).

######  Definition 2.1 ([23, 24]).

Let $H$ be a frame. A _$H$ -fuzzy set_ is a pair $(A,\mu_{A})$ consisting in a
set $A$ and a _membership function_ $\mu_{A}:A\rightarrow H$. The _support of
$\mu_{A}$_ is the set $\mathsf{supp}({A},\mu_{A})$ of elements $x\in A$ such
that $\mu_{A}(x)\neq\bot$. An arrow $f:(A,\mu_{A})\rightarrow(B,\mu_{B})$ is a
function $f:A\rightarrow B$ such that $\mu_{A}(x)\leq\mu_{B}(f(x))$ for all
$x\in A$.

We denote by ${\textbf{Fuz}}\quantity(H)$ the category of $H$-fuzzy sets and
their arrows. We will often drop the explicit reference to the frame $H$ when
there is no danger of confusion.

######  Proposition 2.2.

For any frame $H$, the forgetful functor
$\mathscr{V}:{\textbf{Fuz}}\quantity(H)\rightarrow{\textbf{Set}}$ has both a
left and a right adjoint $\nabla$ and $\Delta$ endowing a set $X$ with the
function constantly equal to the bottom and the top element of $H$,
respectively.

######  Proof 2.3.

If $\nabla(X)$ and $\Delta(X)$ are, respectively $(X,c_{\bot})$ and
$(X,c_{\top})$, where $c_{\bot}$ and $c_{\top}$ are the functions
$X\rightarrow H$ constant in $\bot$ and $\top$, then for any
$X\in{\textbf{Set}}$, $\mathrm{id}_{X}:\mathscr{V}(\Delta(X))=X\rightarrow
X=\mathscr{V}(\nabla(X))$ is the counit of $\mathscr{V}\dashv\Delta$ and the
unit of $\nabla\dashv\mathscr{V}$.

######  Definition 2.4.

Let $e:A\rightarrow B$ and $m:C\rightarrow D$ be two arrows in a category C,
we say that $m$ has _the left lifting property with respect to $e$_ if for any
two arrows $f:A\rightarrow C$ and $g:B\rightarrow D$ such that $m\circ
f=g\circ e$ there exists a unique $k:B\rightarrow C$ with $m\circ k=g$.

A _strong monomorphism_ is an arrow $m$ which has the left lifting property
with respect to all epimorphisms.

######  Proposition 2.5.

Let $f:(A,\mu_{A})\rightarrow(B,\mu_{B})$ be an arrow of
${\textbf{Fuz}}\quantity(H)$, then:

  1. 1.

$f$ is a monomorphism iff it is injective; $f$ is an epimorphism iff it is
surjective;

  2. 2.

$f$ is a strong monomorphism iff it is injective and
$\mu_{B}(f(x))=\mu_{A}(x)$ for all $x\in A$;

  3. 3.

$f$ is a split epimorphism iff for any $b\in B$ there exists $a_{b}\in
f^{-1}(b)$ s.t. $\mu_{B}(b)=\mu_{A}(a_{b})$.

Let us start with the following observation.

######  Remark 2.6.

In any category C, if $m:A\rightarrow B$ is a strong monomorphism then it is a
monomorphism. Suppose that $m\circ f=m\circ g$ for some $f$ and $g$, then we
have the square

$C$$C$$B$$A$$\mathrm{id}_{C}$$m\circ g$$f$$m$

thus there exists a unique $k$ such that $m\circ k=m\circ g$ but both $f$ and
$g$ satisfy this equation.

######  Remark 2.7.

Notice that the previous remark implies that for any square as in the
definition of strong monomorphism, $k\circ e=f$.

######  Proof 2.8 (Proof of Proposition 2.5).

  1. 1.

$\mathscr{V}:{\textbf{Fuz}}\quantity(H)\rightarrow{\textbf{Set}}$ is faithful,
so it reflects monomorphisms and epimorphisms, since it is both a left and a
right adjoint it preserves them too, hence $1$ follows.

  2. 2.

Let us show the two implications.

     * $\Rightarrow$

Injectivity follows from Remark 2.6, let
$(D,\mu_{D})\coloneqq(f(A),{\mu_{B}}_{|f(A)})$ be the set-theoretic image of
$f$ endowed with the restriction of $\mu_{B}$. We have a factorization of $f$
as $m\circ e$ where $m$ is a monomorphism and $e$ an epimorphism, so we have
the square

$(A,\mu_{A})$$(D,{\mu_{D}})$$(B,\mu_{B})$$(A,\mu_{A})$$e$$m$$\mathrm{id}_{(A,\mu_{A})}$$f$

and its diagonal filling $k:(D,\mu_{D})\rightarrow(A,\mu_{A})$. For any $x\in
A$:

$\displaystyle\mu_{B}(f(x))=\mu_{B}(m(e(x)))=\mu_{D}(e(x))\geq\mu_{A}(k(e(x)))=\mu_{A}(x)$

and we get the thesis.

     * $\Leftarrow$

Any square

$(C,\mu_{C})$$(D,{\mu_{D}})$$(B,\mu_{B})$$(A,\mu_{A})$$e$$m$$g$$f$

induces the square

$C$$D$$B$$A$$e$$m$$g$$f$

in Set, which, by point $1$ and $2$, has a diagonal filling $k:D\rightarrow
A$. Now:

$\displaystyle\mu_{D}(x)\leq\mu_{B}(m(x))=\mu_{B}(f(k(x)))=\mu_{A}(k(x))$

and we can conclude that $k$ is an arrow of ${\textbf{Fuz}}\quantity(H)$.

     * •

If $f$ is split then there exists a right inverse
$e:(B,\mu_{B})\rightarrow(A,\mu_{A})$ and we can put $a_{b}:=e(b)$. The other
direction follows noticing that $b\mapsto a_{b}$ is a splitting of $f$.

######  Definition 2.9 ([11]).

A _proper factorization system_ on a category C is a pair
$(\mathscr{E},\mathscr{M})$ given by two classes of arrows such that:

  * •

$\mathscr{E}$ and $\mathscr{M}$ are closed under composition;

  * •

every isomorphism belongs to both $\mathscr{E}$ and $\mathscr{M}$;

  * •

every $e\in\mathscr{E}$ is an epimorphism and every $m\in\mathscr{M}$ is a
monomorphism;

  * •

every $m\in\mathscr{M}$ has the left lifting property with respect to every
$e\in\mathscr{E}$;

  * •

every arrow of C is equal to $m\circ e$ for some $m\in\mathscr{M}$ and
$e\in\mathscr{E}$.

######  Lemma 2.10.

For any frame $H$, ${\textbf{Fuz}}\quantity(H)$ has all products. Moreover the
classes of epimorphisms and strong monomorphisms form a proper factorization
system on it.

######  Proof 2.11.

$\Delta(1)$ is the terminal object by Proposition 2.2. Given a family
$\quantity{\quantity(A_{i},\mu_{i})}$ of fuzzy sets, their product is
constructed taking their product in Set equipped with

$\displaystyle\mu:\prod_{i\in I}A_{i}\rightarrow H\qquad\quantity(a_{i})_{i\in
I}\mapsto\bigwedge_{i\in I}\mu_{i}(a_{i})$

The last part of the thesis follows at once noticing that every arrow
$f:(A,\mu_{A})\rightarrow(B,\mu_{B})$ factors as

$(A,\mu_{A})\xrightarrow{e}(f(A),{\mu_{B}}_{|f(A)})\xrightarrow{m}(B,\mu_{B})$

and applying Proposition 2.5.

######  Remark 2.12.

Completeness and the existence of both adjoints to $\mathscr{V}$ can be
deduced directly from the fact that ${\textbf{Fuz}}\quantity(H)$ is
topological over Set [23, Prop. 71.3].

##  3 Fuzzy Theories

In this section we introduce the syntax and logical rules of fuzzy theories.
The first step is to introduce an appropriate notion of signature. Differently
from usual signatures, in fuzzy theories constants cannot be seen simply as
0-arity operations, because , as we will see in Section 4, these are
interpreted as fuzzy morphisms from the terminal object, and these correspond
only to elements whose membership degree is $\top$.

######  Definition 3.1.

A _signature_ $\Sigma=(O,\mathsf{ar},C)$ is a set $O$ of _operations_ with an
_arity function_ $\mathsf{ar}:O\rightarrow\mathbb{N}_{+}$ and a set $C$ of
_constants_. Signatures form a category Sign in which an arrow
$\Sigma_{1}=(O_{1},\mathsf{ar}_{1},C_{1})\rightarrow\Sigma_{2}=(O_{2},\mathsf{ar}_{2},C_{2})$
is a pair $\mathbf{F}=(F_{1},F_{2})$ of functions: $F_{1}:O_{1}\rightarrow
O_{2}$ and $F_{2}:C_{1}\rightarrow C_{2}$ with the property that
$\mathsf{ar}_{2}\circ F_{1}=\mathsf{ar}_{1}$.

An _algebraic language_ $\mathcal{L}$ is a pair $(\Sigma,X)$ where $\Sigma$ is
a signature and $X$ a set. The category Lng of algebraic languages is just
${\textbf{Sign}}\times{\textbf{Set}}$.

######  Example 3.2.

The _signature of semigroups_ $\Sigma_{S}$ in which $O=\\{\cdot\\}$,
$\mathsf{ar}(\cdot)=2$ and $C=\emptyset$.

######  Example 3.3.

The _signature of groups_ $\Sigma_{G}$ is equal to $\Sigma_{S}$ plus an
operation $(-)^{-1}$ of arity $1$ and a constant $e$.

Given a language $\mathcal{L}$ we can inductively apply the operations to the
set of variables to construct terms, and once terms are built we can introduce
equations.

######  Definition 3.4.

Given a language $\mathcal{L}=(\Sigma,X)$, the set is the smallest set s.t.

  * •

$X\sqcup C\subset$;

  * •

if $f\in O$ and $t_{1},\dots,t_{\mathsf{ar}(f)}\in$ then
$f\quantity(t_{1},\dots,t_{\mathsf{ar}(f)})\in$.

######  Proposition 3.5.

There exists a functor
$\mathsf{Terms}:{\textbf{Lng}}\rightarrow{\textbf{Set}}$ sending $\mathcal{L}$
to .

######  Proof 3.6.

For any $F=((F_{1},F_{2}),h):(\Sigma_{1},X)\rightarrow(\Sigma_{2},Y)$ we can
define $\mathsf{Terms}\quantity({\mathbf{F}})$ by induction:

  * •

for any $x\in X$, $\mathsf{Terms}\quantity({F})(x)\coloneqq g(x)$;

  * •

for any $c\in C_{1}$, $\mathsf{Terms}\quantity({F})(c)\coloneqq F_{2}(c)$;

  * •

for any $f\in O_{1}$ and $t_{1},\dots,t_{\mathsf{ar}_{1}(f)}\in$ it is
$\mathsf{ar}_{2}(F_{1}(f))=\mathsf{ar}_{1}(f)$, so we can define

$\mathsf{Terms}\quantity({F})\quantity(f\quantity(t_{1},\dots,t_{\mathsf{ar}(f)}))\coloneqq
F_{1}(f)\quantity(\mathsf{Terms}\quantity({F})\quantity(t_{1}),\dots,\mathsf{Terms}\quantity({F})\quantity(t_{\mathsf{ar}(f)}))$

Identities are preserved and an easy induction shows that composition is
respected.

######  Definition 3.7 (Formulae).

For any language $\mathcal{L}$ we define the sets
$\mathsf{Eq}\quantity(\mathcal{L})$ of _equations_ as the product
$\mathsf{Eq}\quantity(\mathcal{L})\coloneqq\times$ and the set
$\mathsf{M}\quantity(\mathcal{L})$ of _membership propositions_ as
$\mathsf{M}\quantity(\mathcal{L})\coloneqq H\times$. We will write $s\equiv t$
for $(s,t)\in\mathsf{Eq}\quantity(\mathcal{L})$ and $\mathsf{E}\quantity(l,t)$
for $(l,t)\in\mathsf{M}\quantity(\mathcal{L})$. The set
$\mathsf{Form}\quantity(\mathcal{L})$ of formulae is then defined as
$\mathsf{Eq}\quantity(\mathcal{L})\sqcup\mathsf{M}\quantity(\mathcal{L})$.

Clearly, a proposition $s\equiv t$ means “$s$ and $t$ are equivalent and hence
interchangeable”; on the other hand, $\mathsf{E}\quantity(l,t)$ intuitively
means “the degree of existence of $t$ is at least $l$”.

######  Definition 3.8 (Sequent ant fuzzy theory).

A _sequent_ $\Gamma\vdash\psi$ is an element $(\Gamma,\psi)$ of
$\mathsf{Seq}\quantity(\mathcal{L})\coloneqq\mathcal{P}(\mathsf{Form}\quantity(\mathcal{L}))\times\mathsf{Form}\quantity(\mathcal{L})$,
where $\mathcal{P}$ is the (covariant) power set functor. A _fuzzy theory in
the language $\mathcal{L}$_ is a subset
$\Lambda\subset\mathsf{Seq}\quantity(\mathcal{L})$ and we will use
$\mathsf{Th}\quantity(\mathcal{L})$ for the set
$\mathcal{P}\quantity(\mathsf{Seq}\quantity(\mathcal{L}))$.

######  Notation.

We will write $\vdash\phi$ for $\emptyset\vdash\phi$.

For any function $\sigma:X\rightarrow$ and $t\in$ we denote $t[\sigma]$ the
term obtained substituting $\sigma(x)$ to any occurence of $x$ in $t$.
Moreover, for any formula $\phi\in\mathsf{Form}\quantity(\mathcal{L})$ we
define $\phi[\sigma]$ as $t[\sigma]\equiv s[\sigma]$ if $\phi$ is $t\equiv s$
or as $\mathsf{E}\quantity(l,t[\sigma])$ if $\phi$ is
$\mathsf{E}\quantity(l,t)$. Finally, given
$\Gamma\subset\mathcal{P}(\mathsf{Form}\quantity(\mathcal{L}))$ we put
$\Gamma[\sigma]\coloneqq\\{\phi[\sigma]\mid\phi\in\Gamma\\}$.

$\begin{array}[]{ccc}\inferrule*[right=A]{\phi\in\Gamma}{\Gamma\vdash\phi}\qquad\inferrule*[right=Weak]{\Gamma\vdash\phi}{\Gamma\cup\Delta\vdash\phi}&\inferrule*[right=Cut]{\quantity{\Gamma\vdash\phi\mid\phi\in\Phi}\\\
\Phi\vdash\psi}{\Gamma\vdash\psi}\\\ \inferrule*[right=Refl]{\hskip
1.0pt}{\Gamma\vdash s\equiv s}\qquad\inferrule*[right=Sym]{\Gamma\vdash
s\equiv t}{\Gamma\vdash t\equiv s}&\inferrule*[right=Trans]{\Gamma\vdash
s\equiv t\\\ \Gamma\vdash t\equiv u}{\Gamma\vdash t\equiv u}\\\
\inferrule*[right=Sub]{\sigma:X\rightarrow\\\
\Gamma\vdash\psi}{\Gamma[\sigma]\vdash\psi[\sigma]}&\inferrule*[right=Cong]{f\in
O\\\ \quantity{\Gamma\vdash t_{i}\equiv
s_{i}}_{i=1}^{\mathsf{ar}(f)}}{\Gamma\vdash
f\quantity(t_{1},\dots,t_{\mathsf{ar}(f)})\equiv
f\quantity(s_{1},\dots,s_{\mathsf{ar}(f)})}\\\ \inferrule*[right=Inf]{\hskip
1.0pt}{\Gamma\vdash\mathsf{E}\quantity(\bot,t)}\qquad\inferrule*[right=Mon]{\Gamma\vdash\mathsf{E}\quantity(l,t)}{\Gamma\vdash\mathsf{E}\quantity(l\wedge
l^{\prime},t)}&\inferrule*[right=Exp]{\quantity{\Gamma\vdash\mathsf{E}\quantity(l_{i},t_{i})}_{i=1}^{\mathsf{ar}(f)}}{\Gamma\vdash\mathsf{E}\quantity(\inf\quantity(\quantity{l_{i}}_{i=1}^{n}),f\quantity(t_{1},\dots,t_{\mathsf{ar}(f)}))}\\\
\inferrule*[right=Sup]{S\subset H\\\
\quantity{\Gamma\vdash\mathsf{E}\quantity(l,t)}_{l\in
S}}{\Gamma\vdash\mathsf{E}\quantity(\sup(S),t)}&\inferrule*[right=Fun]{\Gamma\vdash
t\equiv s\\\
\Gamma\vdash\mathsf{E}\quantity(l,t)}{\Gamma\vdash\mathsf{E}\quantity(l,s)}\end{array}$
Figure 1: Derivation rules for the fuzzy sequent calculus.

######  Definition 3.9.

For any $\mathcal{L}$, the _fuzzy sequent calculus_ is given by the rules in
Figure 1.

Given a fuzzy theory $\Lambda$, its _deductive closure_ $\Lambda^{\vdash}$ is
the smallest subset of $\mathsf{Seq}\quantity(\mathcal{L})$ which contains
$\Lambda$ and it is closed under the rules of fuzzy sequent calculus. A
sequent is _derivable from $\Lambda$_ (or simply derivable if
$\Lambda=\emptyset$) if it belongs to ${\Lambda}^{\vdash}$. We will write
$\vdash_{\Lambda}\phi$ if $\vdash\phi\in\Lambda^{\vdash}$.

Finally we say that two theories $\Lambda$ and $\Theta$ are _deductively
equivalent_ if $\Lambda^{\vdash}=\Theta^{\vdash}$.

The next result shows how maps between languages are lifted to theories.

######  Proposition 3.10.

For any $\mathbf{F}:\mathcal{L}_{1}\rightarrow\mathcal{L}_{2}$ arrow of Lng:

  1. 1.

there exists a Galois connection $\mathbf{F}_{*}\dashv\mathbf{F}^{*}$ between
$\quantity(\mathsf{Th}\quantity(\mathcal{L}_{1}),\subset)$ and
$\quantity(\mathsf{Th}\quantity(\mathcal{L}_{2}),\subset)$;

  2. 2.

$\mathbf{F}_{*}\quantity(\Lambda_{1}^{\vdash})\subset\quantity(\mathbf{F}_{*}\quantity(\Lambda_{1}))^{\vdash}$
and
$\quantity(\mathbf{F}^{*}\quantity(\Lambda_{2}))^{\vdash}\subset\mathbf{F}^{*}\quantity(\Lambda_{2}^{\vdash})$
for any $\Lambda_{1}\in\mathbf{Th}(\mathcal{L}_{1})$ and
$\Lambda_{2}\in\mathbf{Th}(\mathcal{L}_{2})$.

######  Proof 3.11.

  1. 1.

For any formula $\psi$ we can define (using Proposition 3.5):

$\displaystyle\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(t\equiv
s)\coloneqq\mathsf{Terms}\quantity({\mathbf{F}})(t)\equiv\mathsf{Terms}\quantity({\mathbf{F}})(s)$
$\displaystyle\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(\mathsf{E}\quantity(l,t))\coloneqq\mathsf{E}\quantity(l,\mathsf{Terms}\quantity({\mathbf{F}})(t))$

getting a function
$\mathsf{Form}\quantity(\mathcal{\mathbf{F}}):\mathsf{Form}\quantity(\mathcal{L}_{1})\rightarrow\mathsf{Form}\quantity(\mathcal{L}_{2})$.
We can extend it to sequents defining

$\mathsf{Seq}\quantity(\mathcal{\mathbf{F}})(\Phi\vdash\psi)\coloneqq\quantity{\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(\phi)\mid\phi\in\Phi}\vdash\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(\psi)$

Now it is enough to take $\mathbf{F}_{*}$ and $\mathbf{F}^{*}$ as the image
and the preimage of $\mathsf{Seq}\quantity(\mathcal{\mathbf{F}})$
respectively.

  2. 2.

First of all notice that the two inequalities are equivalent because
$\mathbf{F}_{*}\dashv\mathbf{F}^{*}$. Now, the first one holds since if a
sequent $\Gamma\vdash\phi$ follows from the set of sequents
$\quantity{\Gamma_{i}\vdash\phi_{i}}_{i\in I}$ by the application of one of
the rules of the fuzzy sequent calculus then the same rule can be applied to
$\quantity{\mathsf{Seq}\quantity(\mathcal{\mathbf{F}})\quantity(\Gamma_{i}\vdash\phi_{i})}_{i\in
I}$ to get
$\mathsf{Seq}\quantity(\mathcal{\mathbf{F}})\quantity(\Gamma\vdash\psi)$.

Usually, logics enjoy the so-called “deduction lemma”, which allows us to
treat elements of a theory on a par with assumptions in sequents. In fuzzy
theories, this holds in a slightly restricted form, as proved next.

######  Lemma 3.12 (Partial deduction lemma).

Let $\Lambda$ be in $\mathsf{Th}\quantity(\mathcal{L})$ and
$\Gamma\in\mathcal{P}\quantity(\mathsf{Form}\quantity(\mathcal{L}))$, let also
$\Lambda[\Gamma]$ be the theory
$\Lambda\cup\quantity{\emptyset\vdash\phi\mid\phi\in\Gamma}$. Then the
following are true:

  1. 1.

$\Gamma\cup\Delta\vdash\psi$ in $\Lambda^{\vdash}$ implies $\Delta\vdash\psi$
in $\quantity(\Lambda[\Gamma])^{\vdash}$;

  2. 2.

if $\Delta\vdash\psi$ is derivable from $\Lambda[\Gamma]$ without using rule
Sub then $\Gamma\cup\Delta\vdash\psi$ is in $\Lambda^{\vdash}$.

######  Proof 3.13.

  1. 1.

By hypothesis $\Gamma\cup\Delta\vdash\psi$ is in $\Lambda^{\vdash}$ then,
since $\Lambda\subset\Lambda[\Gamma]$, it is also in
$\quantity(\Lambda[\Gamma])^{\vdash}$. Now, for any $\phi\in\Gamma$ and
$\theta\in\Delta$ rules Weak and A give

$\inferrule*[right=Weak]{\vdash\phi}{\Delta\vdash\phi}\qquad\inferrule*[right=A]{\hskip
1.0pt}{\Delta\vdash\theta}$

so $\quantity{\Delta\vdash\phi\mid\phi\in\Gamma\cup\Delta}$ is contained in
$\quantity(\Lambda[\Gamma])^{\vdash}$ and we can apply Cut to get the thesis:

$\inferrule*[right=Cut]{\quantity{\Delta\vdash\phi\mid\phi\in\Gamma\cup\Delta}\\\
\Gamma\cup\Delta\vdash\psi}{\Delta\vdash\psi}$

  2. 2.

Let us proceed by induction on a derivation of $\Delta\vdash\psi$ from
$\Lambda[\Gamma]$.

     * •

If $\Delta\vdash\psi\in\Lambda[\Gamma]$ then or $\Delta\vdash\psi\in\Lambda$
and we are done, or $\psi\in\Gamma$ and we can conclude since
$\Gamma\cup\Delta\vdash\phi_{i}$ is in $\Lambda^{\vdash}$ by rules A and Weak

     * •

If $\Delta\vdash\psi$ follows from the application of one of the rules A, Inf
or Refl then it belongs to the closure of any theory, by Weak this is true
even for $\Gamma\cup\Delta\vdash\psi$ which, in particular, it belongs to
$\Lambda^{\vdash}$.

     * •

Suppose that $\Delta\vdash\psi$ comes from an application of Weak, then there
exists $\Psi$ and $\Phi$ such that $\Psi\cup\Phi=\Delta$ and $\Psi\vdash\phi$
is in $(\Lambda[\Gamma])^{\vdash}$. By inductive hypothesis we have that

$\inferrule*[right=Weak]{\Gamma\cup\Psi\vdash\psi}{\Gamma\cup\Psi\cup\Phi\vdash\psi}$

is a derivation of $\Gamma\cup\Delta\vdash\psi$ from $\Lambda$.

     * •

If $\Delta\vdash\psi$ is derived with an application of Cut as last rule then
there exists a set $\Theta$ such that
$\quantity{\Delta\vdash\theta\mid\theta\in\Theta}\cup\quantity{\Theta\vdash\psi}\subseteq\quantity(\Lambda[\Gamma])^{\vdash}$,
therefore, by the inductive hypothesis, we have that
$\quantity{\Gamma\cup\Delta\vdash\theta\mid\theta\in\Theta}\cup\quantity{\Gamma\cup\Theta\vdash\psi}$
is conteined in $\Lambda^{\vdash}$. Now, by rule A we have that
$\quantity{\Gamma\cup\Delta\vdash\phi\mid\phi\in\Gamma\cup\Theta}$ is
contained in $\Lambda^{\vdash}$ too and so we can apply Cut:

$\inferrule*[right=Cut]{\quantity{\Gamma\cup\Delta\vdash\phi\mid\phi\in\Gamma\cup\Theta}\\\
\Gamma\cup\Theta\vdash\psi}{\Gamma\cup\Delta\vdash\psi}$

which gives us the thesis.

     * •

Any other rule is of the form

$\inferrule*[right=R]{\quantity{\Psi\vdash\xi_{j}}_{j\in J}}{\Psi\vdash\xi}$

therefore, if $\Delta\vdash\psi$ is derived with an application of one of this
rules then the set of its premises must be an element of
$(\Lambda[\Gamma])^{\vdash}$ of type $\quantity{\Delta\vdash\xi_{j}}_{j\in
J}$, so by inductive hypothesis
$\quantity{\Gamma\cup\Delta\vdash\theta_{j}}_{j\in J}\subset\Lambda^{\vdash}$
and then

$\inferrule*[right=R]{\quantity{\Gamma\cup\Delta\vdash\xi_{j}}_{j\in
J}}{\Gamma\cup\Delta\vdash\psi}$

witnesses the thesis.

######  Example 3.14.

Our first set of running examples is inspired by [17]. Let $\Sigma_{S}$ be the
signature of semigroups and $X$ a countable set. The theory of _fuzzy
semigroups_ $\Lambda_{S}$ is simply the usual theory of semigroups, i.e given
by the sequent (using infix notation)

$\displaystyle\vdash(x\cdot y)\cdot z\equiv x\cdot(y\cdot z)$

We get the theory of _left ideal_ $\Lambda_{LI}$ if we add the axioms (one for
any $l\in L$):

$\mathsf{E}\quantity(l,y)\vdash\mathsf{E}\quantity(l,x\cdot y)$

Similarly the theory $\Lambda_{RI}$ of _right ideal_ is obtained from the
axioms:

$\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,x\cdot y)$

Finally we get the theory of _(bilateral) ideal_ $\Lambda_{I}$ taking the
union of the above theories.

######  Example 3.15 ([21, 1, 2]).

Let $\Sigma_{G}$ be the signature of groups and $X$ a countable set. The
theory $\Lambda_{G}$ of _fuzzy groups_ is simply the usual theory of groups,
i.e that given by the sequents

$\displaystyle\vdash x\cdot x^{-1}\equiv e\quad\vdash x^{-1}\cdot x\equiv
e\quad\vdash e\cdot x\equiv x\quad\vdash x\cdot x\equiv x\quad\vdash(x\cdot
y)\cdot z\equiv x\cdot(y\cdot z)$

We get the theory $\Lambda_{N}$ of _normal fuzzy groups_ ([14]) if we add the
axioms:

$\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,y\cdot(x\cdot y^{-1}))$

######  Proposition 3.16.

The sequent $\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,e)$ is
derivable from $\Lambda_{G}$ and $\mathsf{E}\quantity(l,y\cdot(x\cdot
y^{-1}))\vdash\mathsf{E}\quantity(l,x)$ is derivable from $\Lambda_{N}$.

######  Proof 3.17.

The two derivations are shown in Appendix A. In the second derivation we have
used the fact that $\vdash y\equiv\quantity(y^{-1})^{-1}$ and $\vdash x\equiv
y\cdot\quantity(\quantity(y^{-1}\cdot(x\cdot y))\cdot y^{-1})$ are derivable
in the theory of groups (this can be shown as in the case of ordinary group
theory) and we have substituted $y^{-1}\cdot(x\cdot y)$ for $x$ in the first
application of Sub and $y^{-1}$ for $y$ in the second one.

##  4 Fuzzy algebras and semantics

In this section we provide a sound and complete semantics to the syntax and
sequents introduced in Section 3. The first step is to define the semantic
counterpart of a signature.

######  Definition 4.1.

Given a signature $\Sigma$, a $\Sigma$-_fuzzy algebra_
$\mathcal{A}\coloneqq((A,\mu_{A}),\Sigma^{\mathcal{A}})$ is a fuzzy set
$(A,\mu_{A})$ and a collection $\Sigma^{\mathcal{A}}=\\{f^{\mathcal{A}}\mid
f\in O\\}\sqcup\\{c^{\mathcal{A}}\mid c\in C\\}$ of arrows:

$\displaystyle
f^{\mathcal{A}}:(A,\mu_{A})^{\mathsf{ar}(f)}\rightarrow(A,\mu_{A})\qquad
c^{\mathcal{A}}:(1,c_{\bot})\rightarrow(A,\mu_{A})$

where $c_{\bot}$ is the constant function in $\bot$. A _morphism of
$\Sigma$-fuzzy algebras_ $\mathcal{A}\rightarrow\mathcal{B}$ is an arrow
$g:(A,\mu_{A})\rightarrow(B,\mu_{B})$ such that $g\circ
c^{\mathcal{A}}=c^{\mathcal{B}}$ and $f^{\mathcal{B}}\circ
g^{\mathsf{ar}(f)}=g\circ f^{\mathcal{A}}$ for every $c\in C$ and $f\in O$. We
will write for the resulting category of $\Sigma$-fuzzy algebras.

######  Remark 4.2.

We will not distinguish between a function from the singleton and its value.

######  Definition 4.3.

Let $\mathcal{L}=(\Sigma,X)$ be a language and
$\mathcal{A}=\quantity((A,\mu_{A}),\Sigma^{\mathcal{A}})$ be a
$\Sigma$-algebra.

An _assignment_ is simply a function $\iota:X\rightarrow A$. We define the
_evaluation in $\mathcal{A}$ with respect to $\iota$_ as the function
$(\text{-})^{\mathcal{A},{\iota}}:\rightarrow A$ by induction:

  * •

$x^{\mathcal{A},{\iota}}\coloneqq\iota(x)$ if $x\in X$;

  * •

$c^{\mathcal{A},{\iota}}\coloneqq c^{\mathcal{A}}$ if $c\in C$;

  * •

$(f(t_{1},\dots,t_{\mathsf{ar}(f)}))^{\mathcal{A},{\iota}}\coloneqq
f^{\mathcal{A}}(t_{1}^{\mathcal{A},{\iota}},\dots,t_{\mathsf{ar}(f)}^{\mathcal{A},{\iota}})$
if $f\in O$.

######  Proposition 4.4.

Let $\mathcal{A}$ be a $\Sigma$-algebra. Given a function
$\sigma:X\rightarrow$ and an assignment $\iota:X\rightarrow A$ define
$\iota_{\sigma}:X\rightarrow A$ as the assignment sending $x$ to
$\quantity(\sigma\quantity(x))^{\mathcal{A},\iota}$. Then
$\mathcal{A}\vDash_{\iota}\phi[\sigma]$ if and only if
$\mathcal{A}\vDash_{\iota_{\sigma}}\phi$.

######  Proof 4.5.

This follows at once noticing that
$\big{(}t\quantity[\sigma]\big{)}^{\mathcal{A},\iota}=t^{\mathcal{A},\iota_{\sigma}}$
holds for every term $t$.

######  Definition 4.6.

$\mathcal{A}$ _satisfies $\phi\in\mathsf{Form}\quantity(\mathcal{L})$ with
respect to $\iota$_, and we write $\mathcal{A}\vDash_{\iota}\phi$, if $\phi$
is $\mathsf{E}\quantity(l,t)$ and $l\leq\mu_{A}(t^{\mathcal{A},{\iota}})$ or
if $\phi$ is $t\equiv s$ and
$t^{\mathcal{A},{\iota}}=s^{\mathcal{A},{\iota}}$.

$\mathcal{A}$ _satisfies $\phi$_ if $\mathcal{A}\vDash_{\iota}\phi$ for all
$\iota:X\rightarrow A$, and we write $\mathcal{A}\vDash\phi$.

Given $\Gamma\subset\mathsf{Form}\quantity(\mathcal{L})$,
$\mathcal{A}\vDash\Gamma$ ($\mathcal{A}\vDash_{\iota}\Gamma$) means
$\mathcal{A}\vDash\phi$ ($\mathcal{A}\vDash_{\iota}\phi$) for any
$\phi\in\Gamma$. Finally, given a sequent $\Gamma\vdash\phi$ we say that
$\mathcal{A}$ _satisfies it with respect to $\iota$_ and we will write
$\Gamma\vDash_{\mathcal{A},\iota}\phi$ if $\mathcal{A}\vDash_{\iota}\phi$
whenever $\mathcal{A}\vDash_{\iota}\Gamma$; if this happens for all
assignments $\iota$ we say that $\mathcal{A}$ _satisfies_ the sequent and we
will write $\Gamma\vDash_{\mathcal{A}}\phi$.

We say that a $\Sigma$-fuzzy algebra $\mathcal{A}$ is a _model_ of a fuzzy
theory $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ if it satisfies all the
sequents in it.

We will write ${\textbf{Mod}}\quantity(\Lambda)$ for the full subcategory of
given by the models of $\Lambda$.

Clearly . For any $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ there exist
two forgetful functors
$\mathscr{U}_{\Lambda}:{\textbf{Mod}}\quantity(\Lambda)\rightarrow{\textbf{Fuz}}\quantity(L)$
and
$\mathscr{V}_{\Lambda}:{\textbf{Mod}}\quantity(\Lambda)\rightarrow{\textbf{Set}}$.
We will write $\mathscr{U}_{\Sigma}$ and $\mathscr{V}_{\Sigma}$ for
$\mathscr{U}_{\emptyset}$ and $\mathscr{V}_{\emptyset}$.

######  Proposition 4.7.

For any signature $\Sigma$, $\mathscr{V}_{\Sigma}$ has a left adjoint
$\mathscr{F}^{{\textbf{Set}}}_{\Sigma}:{\textbf{Set}}\rightarrow{\textbf{Mod}}\quantity(\Lambda)$.

######  Proof 4.8.

For any set $X$ take the language $\mathcal{L}_{X}$ and define
$\mathscr{F}^{{\textbf{Set}}}_{\Sigma}(X)$ has
$(\nabla(\mathcal{L}_{X}\text{-}\mathbf{Terms}),\Sigma^{\mathscr{F}^{{\textbf{Set}}}_{\Sigma}(X)})$
where $c^{\mathscr{F}^{{\textbf{Set}}}_{\Sigma}(X)}:=c$ and

$\displaystyle
f^{\mathscr{F}^{{\textbf{Set}}}_{\Sigma}(X)}:\nabla(\mathcal{L}_{X}\text{-}\mathbf{Terms})^{\mathsf{ar}(f)}\rightarrow\nabla(\mathcal{L}_{X}\text{-}\mathbf{Terms})\qquad\quantity(t_{1},\dots,t_{\mathsf{ar}(f)})\mapsto
f(t_{1},\dots,t_{\mathsf{ar}(f)})$

for any $f\in O$. Now, it is immediate to see that for any
$\iota:X\rightarrow\mathscr{V}_{\Sigma}(\mathcal{A})$ the evaluation
$(\text{-})^{\mathcal{A},{\iota}}$ is the unique morphism of that composed
with the inclusion $X\rightarrow\mathcal{L}_{X}\text{-}\mathbf{Terms}$ gives
back $\iota.$

We now provide two technical results about interpretations. The first
describes how interpretations are moved along morphisms of algebras.

######  Proposition 4.9.

Let $\mathcal{L}=(\Sigma,X)$ be a language,
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ and
$\mathcal{A}=\quantity(\quantity(A,\mu_{A}),\Sigma^{\mathcal{A}})$,
$\mathcal{B}=\quantity(\quantity(B,\mu_{B}),\Sigma^{\mathcal{B}})$ be two
$\Sigma$-algebras. Let also $f:\mathcal{A}\rightarrow\mathcal{B}$ be a
morphism between them, then:

  1. 1.

$\mathcal{A}$ is a model of $\Lambda$ if and only if it is a model of
$\Lambda^{\vdash}$;

  2. 2.

$f\circ(-)^{\mathcal{A},\iota}=(-)^{\mathcal{B},f\circ\iota}$ for every
assignment $\iota:X\rightarrow A$;

  3. 3.

for any assignment $\iota:X\rightarrow A$, $\mathcal{A}\vDash_{\iota}\phi$
entails $\mathcal{B}\vDash_{f\circ\iota}\phi$;

  4. 4.

if $\mathscr{U}_{\Sigma}(f)$ is a strong monomorphism in
${\textbf{Fuz}}\quantity(H)$ and $\iota:X\rightarrow A$ is an assignment then,
for any formula $\phi$, $\mathcal{A}\vDash_{\iota}\phi$ if and only if
$\mathcal{B}\vDash_{f\circ\iota}\phi$;

  5. 5.

if $\mathscr{U}_{\Sigma}(f)$ is a strong monomorphism in
${\textbf{Fuz}}\quantity(H)$ and
$\mathcal{B}\in{\textbf{Mod}}\quantity(\Lambda)$ then
$\mathcal{A}\in{\textbf{Mod}}\quantity(\Lambda)$.

######  Proof 4.10.

  1. 1.

The implication from the right to the left is obvious since
$\Lambda\subseteq\Lambda^{\vdash}$. The other follows from soundness.

  2. 2.

This follows at once by structural induction.

  3. 3.

This follows from the previous point: if $\phi$ is $t\equiv s$ then

$\displaystyle
t^{\mathcal{B},f\circ\iota}=f\quantity(t^{\mathcal{A},\iota})=f\quantity(s^{\mathcal{A},\iota})=s^{\mathcal{B},f\circ\iota}$

otherwise

$\displaystyle
l\leq\mu_{A}\quantity(t^{\mathcal{A},\iota})\leq\mu_{B}\quantity(f\quantity(t^{\mathcal{A},\iota}))=\mu_{B}\quantity(t^{\mathcal{B},f\circ\iota})$

  4. 4.

We have to show only the implication from the right to the left. Let us
proceed by cases remembering that by Proposition 2.5 $\mathscr{U}_{\Sigma}(f)$
is a strong monomorphism if and only if it is injective and
$\mu_{A}(a)=\mu_{B}(f(a))$ for any $a\in A$.

     * •

$\phi=t\equiv s$. $\mathcal{A}\vDash_{f\circ\iota}t\equiv s$ if and only if
$t^{\mathcal{A},\iota}=s^{\mathcal{A},\iota}$, by injectivity of $f$ this is
equivalent to $f(t^{\mathcal{A,\iota}})=f(s^{\mathcal{A,\iota}})$ and by point
$2$ this is equivalent to
$t^{\mathcal{B},f\circ\iota}=s^{\mathcal{B},f\circ\iota}$, i.e.
$\mathcal{B}\vDash_{f\circ\iota}\phi$.

     * •

$\phi=\mathsf{E}\quantity(l,t)$. As before, $\mathcal{A}\vDash_{\iota}\phi$ if
and only if $\mu_{A}(t^{\mathcal{A},\iota})\geq l$ but

$\displaystyle\mu_{A}(t^{\mathcal{A},\iota})=\mu_{B}(f(t^{\mathcal{A},\iota}))=\mu_{B}(t^{\mathcal{B},f\circ\iota})$

and thus $\mathcal{A}\vDash_{\iota}\phi$ is equivalent to
$\mathcal{B}\vDash_{f\circ\iota}\phi$.

  5. 5.

Let $\Gamma\vdash\phi$ be in $\Lambda$ and suppose
$\mathcal{A}\vDash_{\iota}\Gamma$ for some assignment $\iota:X\rightarrow B$,
by the previous point this implies $\mathcal{B}\vDash_{f\circ\iota}\Gamma$,
so, by hypothesis $\mathcal{B}\vDash_{f\circ\iota}\phi$ and we can conclude
again using point $3$.

We can also move interpretations and theories along morphisms of signatures.

######  Definition 4.11.

For any $\mathbf{F}:\Sigma_{1}\rightarrow\Sigma_{2}$ arrow of Sign and any
$\mathcal{A}=\quantity(\quantity(A,\mu_{A}),\Sigma_{2}^{\mathcal{A}})\in$, we
define
$\mathsf{r}_{\mathbf{F}}(\mathcal{A})=\quantity(\quantity(A,\mu_{A}),\Sigma_{1}^{\mathsf{r}_{\mathbf{F}}(\mathcal{A})})\in$
putting, for any $f\in O_{1}$

$f^{\mathsf{r}_{\mathbf{F}}(\mathcal{A})}:(A,\mu_{A})^{\mathsf{ar}(f)}\rightarrow(A,\mu_{A})\qquad\quantity(a_{1},\dots,a_{\mathsf{ar}(f)})\mapsto
F_{2}(f)^{\mathcal{A}}\quantity(a_{1},\dots,a_{\mathsf{ar}(f)})$

and
$c^{\mathsf{r}_{\mathbf{F}}\quantity(\mathcal{A})}:=F_{3}(c)^{\mathcal{A}}$
for every $c\in C_{1}$.

######  Lemma 4.12.

Let $\mathcal{L}_{1}=(\Sigma_{1},X)$ and $\mathcal{L}_{2}=(\Sigma_{2},Y)$ and
$\mathbf{F}=\quantity(\quantity(F_{1},F_{2}),g):\mathcal{L}_{1}\rightarrow\mathcal{L}_{2}$,
then:

  1. 1.

there exists a functor $\mathsf{r}_{\mathbf{F}}:$ sending $\mathcal{A}$ to
$\mathsf{r}_{\mathbf{F}}(\mathcal{A})$;

  2. 2.

$t^{\mathsf{r}_{\mathbf{F}}(\mathcal{A}),\iota\circ
g}=\quantity(\mathsf{Terms}\quantity({\mathbf{F}})(t))^{\mathcal{A},\iota}$
for any assignment $\iota:Y\rightarrow A$ and
$t\in\mathcal{L}_{1}\text{-}\mathsf{Terms}$;

  3. 3.

for any assignment $\iota:Y\rightarrow A$,
$\mathsf{r}_{\mathbf{F}}(\mathcal{A})\vDash_{\iota\circ g}\phi$ if and only if
$\mathcal{A}\vDash_{\iota}\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(\phi)$;

  4. 4.

If $X=Y$ and $g=\mathrm{id}_{X}$ then $\mathsf{r}_{\mathbf{F}}$ restricts to a
functor $\mathsf{r}_{\mathbf{F},\hskip
1.0pt{\Lambda}}:{\textbf{Mod}}\quantity(\Lambda)\rightarrow{\textbf{Mod}}\quantity(\mathbf{F}^{*}(\Lambda))$.

######  Proof 4.13.

  1. 1.

Let $h:\mathcal{A}\rightarrow\mathcal{B}$ be a morphism of  then $h\circ
u^{\mathcal{A}}=u^{\mathcal{B}}\circ g^{\mathsf{ar}_{2}(h)}$ and
$d^{\mathcal{B}}=h\circ d^{\mathcal{A}}$ for all $u\in O_{2}$ and $d\in
C_{2}$. In particular this is true when $u=F_{2}(f)$ and $d=F_{3}(c)$ for
$f\in O_{1}$ and $c\in C_{1}$ but these are exactly the conditions making $h$
into an homomorphism
$\mathsf{r}_{\mathbf{F}}(\mathcal{A})\rightarrow\mathsf{r}_{\mathbf{F}}(\mathcal{B})$.

  2. 2.

This follows by induction:

     * •

for any $x\in X$

$\displaystyle x^{\mathsf{r}_{\mathbf{F}}(\mathcal{A}),\iota\circ
g}=\iota(g(x))=(\mathsf{Terms}\quantity({\mathbf{F}})(x))^{\mathcal{A},\iota}$

and for any constant $c\in C_{1}$

$\displaystyle c^{\mathsf{r}_{\mathbf{F}}(\mathcal{A}),\iota\circ
g}=c^{\mathsf{r}_{\mathbf{F}}(\mathcal{A})}=F_{3}(c)^{\mathcal{A}}=(\mathsf{Terms}\quantity({\mathbf{F}})(c))^{\mathcal{A}}=(\mathsf{Terms}\quantity({\mathbf{F}})(c))^{\mathcal{A},\iota}$

     * •

for any $f\in O_{1}$:

$\displaystyle\quantity(f\quantity(t_{1},\dots,t_{\mathsf{ar}_{1}(f)}))^{\mathcal{\mathsf{r}_{\mathbf{F}}(A)},\iota\circ
g}$
$\displaystyle=f^{\mathsf{r}_{\mathbf{F}}(\mathcal{A})}\quantity(t^{\mathcal{\mathsf{r}_{\mathbf{F}}(A)},\iota\circ
g}_{1},\dots,t^{\mathcal{\mathsf{r}_{\mathbf{F}}(A)},\iota\circ
g}_{\mathsf{ar}_{1}(f)})$
$\displaystyle=f^{\mathsf{r}_{\mathbf{F}}(\mathcal{A})}\quantity(\quantity(\mathsf{Terms}\quantity({\mathbf{F}})(t_{1}))^{\mathcal{A},\iota},\dots,\quantity(\mathsf{Terms}\quantity({\mathbf{F}})(t_{\mathsf{ar}_{1}(f)}))^{\mathcal{A},\iota})$
$\displaystyle=F_{1}(f)^{\mathcal{A}}\quantity(\quantity(\mathsf{Terms}\quantity({\mathbf{F}})\quantity(t_{1}))^{\mathcal{A},\iota},\dots,\quantity(\mathsf{Terms}\quantity({\mathbf{F}})\quantity(t_{\mathsf{ar}_{1}(f)}))^{\mathcal{A},\iota})$
$\displaystyle=\quantity(F_{1}(f)\quantity(\quantity(\mathsf{Terms}\quantity({\mathbf{F}})\quantity(t_{1})),\dots,\quantity(\mathsf{Terms}\quantity({\mathbf{F}})\quantity(t_{\mathsf{ar}_{1}(f)}))))^{\mathcal{A},\iota}$
$\displaystyle=\mathsf{Terms}\quantity({\mathbf{F}})\quantity(f\quantity(t_{1},\dots,t_{\mathsf{ar}_{1}(f)}))^{\mathcal{A},\iota}$

  3. 3.

This follows immediately by point $2$:

$\displaystyle\mathsf{r}_{\mathbf{F}}(\mathcal{A})\vDash_{\iota\circ
g}(t\equiv s)$ $\displaystyle\iff
t^{\mathsf{r}_{\mathbf{F}}(\mathcal{A}),\iota\circ
g}=s^{\mathsf{r}_{\mathbf{F}}(\mathcal{A}),\iota\circ g}$
$\displaystyle\iff\quantity(\mathsf{Terms}\quantity({\mathbf{F}})\quantity(t))^{\mathcal{A},\iota}=\quantity(\mathsf{Terms}\quantity({\mathbf{F}})\quantity(s))^{\mathcal{A},\iota}$
$\displaystyle\iff\mathcal{A}\vDash_{\iota}\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(t\equiv
s)$ $\displaystyle\mathsf{r}_{\mathbf{F}}(\mathcal{A})\vDash_{\iota\circ
g}\mathsf{E}\quantity(l,t)$ $\displaystyle\iff
l\leq\mu_{A}\quantity(t^{\mathsf{r}_{\mathbf{F}}(\mathcal{A}),\iota\circ g})$
$\displaystyle\iff
l\leq\mu_{A}\quantity(\mathsf{Terms}\quantity({\mathbf{F}})\quantity(t))^{\mathcal{A},\iota}$
$\displaystyle\iff\mathcal{A}\vDash_{\iota}\mathsf{Form}\quantity(\mathcal{\mathbf{F}})\quantity(\mathsf{E}\quantity(l,t))$

  4. 4.

Let $\Gamma\vdash\psi$ be in $\mathbf{F}^{*}(\Lambda)$, by the previous point,
for any assignment $\iota:X\rightarrow A$

$\mathsf{r}_{\mathbf{F}}(\mathcal{A})\vDash_{\iota\circ
g}\Gamma\iff\mathcal{A}\vDash_{\iota}\quantity{\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(\phi)}_{\phi\in\Gamma}$

Since
$\quantity{\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(\phi)}_{\phi\in\Gamma}\vdash\mathsf{Form}\quantity(\mathcal{\mathbf{F}})(\psi)$
is in $\Lambda$ we have
$\mathcal{A}\vDash_{\iota}\mathsf{Form}\quantity(\mathcal{\mathbf{F}})\quantity(\psi)$
but, using again point $3$, this implies
$\mathsf{r}_{\mathbf{F}}(\mathcal{A})\vDash_{\iota}\phi$.

######  Example 4.14.

The models for $\Lambda_{S}$, $\Lambda_{LI}$, $\Lambda_{RI}$ and $\Lambda_{I}$
(Example 3.14) are precisely the structures defined in [17], while the models
for $\Lambda_{G}$ (Example 3.15) are precisely the fuzzy groups as in [21] and
those of $\Lambda_{N}$ are the structures called _normal fuzzy subgroups_ in
[2, 1, 14].

#### Soundness

Now we can proceed proving the soundness of the rules in Figure 1.

######  Lemma 4.15.

Let $\mathcal{L}=(\Sigma,X)$ be a language and
$\mathcal{A}=((A,\mu_{A}),\Sigma^{\mathcal{A}})$ a $\Sigma$-algebra, then:

  1. 1.

for any assignment $\iota:X\rightarrow A$ and rule

$\inferrule*[right=R]{\quantity{\Psi_{i}\vdash\xi_{i}}_{i\in
I}}{\Psi\vdash\xi}$

different from Sub, if $\Psi_{i}\vDash_{\mathcal{A},\iota}\xi_{i}$ for all
$i\in I$ then $\Psi\vDash_{\mathcal{A},\iota}\xi$ too;

  2. 2.

for any $\sigma:X\rightarrow$, if $\Gamma\vDash_{\mathcal{A}}\psi$ then
$\Gamma[\sigma]\vDash_{\mathcal{A}}\psi[\sigma]$.

######  Proof 4.16.

  1. 1.

The thesis is vacuous for rule A and it is trivial for Refl, Sym and Trans.
For rule Weak if $\mathcal{A}\vDash_{\iota}\Gamma\cup\Delta$ then
$\mathcal{A}\vDash_{\iota}\Gamma$ and so, since
$\Gamma\vDash_{\mathcal{A},\iota}\phi$ this implies
$\mathcal{A}\vDash_{\iota}\phi$. The thesis for Inf it follows from the fact
that $\bot$ is the bottom element, for Cong and Fun it follows since $\mu_{A}$
and $f^{\mathcal{A}}\in\Sigma^{\mathcal{A}}$ are functions for any $f\in O$.
We’re left with four rules.

     * Cut. For any assignment $\iota$, if $\mathcal{A}\vDash_{\iota}\Gamma$ then, since $\Gamma\vDash_{\mathcal{A},\iota}\phi_{i}$ for any $i\in I$ we have $\mathcal{A}\vDash_{\iota}\quantity{\phi_{i}}_{i\in I}$, but $\quantity{\phi_{i}}_{i\in I}\vDash_{\mathcal{A},\iota}\psi$ and this implies $\mathcal{A}\vDash_{\iota}\psi$.

     * Sup. Let $\iota:X\rightarrow A$ be such that $\mathcal{A}\vDash_{\iota}\Gamma$, then $\Gamma\vDash_{\mathcal{A},\iota}\mathsf{E}\quantity(l,t)$ for all $l\in\mathcal{L}$ implies that $\mu_{A}(t^{\mathcal{A},\iota})$ is an upper bound of $\mathcal{L}$, so the thesis follows.

     * Mon. As before let $\iota$ be an assignment for which $\mathcal{A}$ satisfies all the formulae in $\Gamma$, then, since $\Gamma\vDash_{\mathcal{A},\iota}\mathsf{E}\quantity(l,t)$ we have that $l\leq\mu_{A}(t^{\mathcal{A},\iota})$, so, for any $l^{\prime}\in H$, $l^{\prime}\wedge l\leq\mu_{A}(t^{\mathcal{A},\iota})$ holds too.

     * Exp. For any $\iota$ such that $\mathcal{A}\vDash_{\iota}\Gamma$, $\Gamma\vDash_{\mathcal{A},\iota}\mathsf{E}\quantity(l_{i},t_{i})$ entails $l_{i}\leq\mu_{A}(t^{\mathcal{A},\iota}_{i})$ for any $1\leq i\leq\mathsf{ar}(f)$, therefore:

$\displaystyle\bigwedge_{i=1}^{\mathsf{ar}(f)}l_{i}$
$\displaystyle\leq\bigwedge_{i=1}^{\mathsf{ar}(f)}\mu_{A}(t^{\mathcal{A},\iota}_{i})=\mu_{A^{n}}(t^{\mathcal{A},\iota}_{1},\dots,t^{\mathcal{A},\iota}_{\mathsf{ar}(f)})\leq\mu_{A}(f^{\mathcal{A}}(t^{\mathcal{A},\iota}_{1},\dots,t^{\mathcal{A},\iota}_{\mathsf{ar}(f)}))$

and we are done.

  2. 2.

Let $\iota:X\rightarrow A$ an assignment such that
$\mathcal{A}\vDash_{\iota}\phi_{i}[\sigma]$ for any $\phi_{i}\in\Gamma$. By
Proposition 4.4 this means that $\mathcal{A}\vDash_{\iota_{\sigma}}\Gamma$ and
so, by hypothesis, $\mathcal{A}\vDash_{\iota_{\sigma}}\psi$ which, again by
Proposition 4.4, entails $\mathcal{A}\vDash_{\iota}\psi[\sigma]$.

######  Corollary 4.17 (Soundness).

If a $\Sigma$-algebra satisfies all the premises of a rule of the fuzzy
sequent calculus then it satisfies also its conclusion.

######  Remark 4.18.

Let us see why the deduction lemma (Lemma 3.12) cannot be extended to rule
Sub. Take $\Sigma$ to be the empty set, $X=\\{x,y,z\\}$ and $H=\\{0,1\\}$.
Notice that . We have the derivation

$\inferrule*[Right=Sub]{\vdash x\equiv y}{\vdash x\equiv z}$

If the deduction lemma held for Sub, $x\equiv y\vdash x\equiv z$ would be in
$\emptyset^{\vdash}$, hence satisfied by any fuzzy set, but
$(H,\mathrm{id}_{H})$ with $\iota:X\rightarrow H$ sending $x$ and $y$ to $0$
and $z$ to $1$ is a counterexample.

######  Remark 4.19.

Let us take $\Sigma=\emptyset$ and $H=\\{0,1\\}$ and $X=\\{x,y,z\\}$ and the
derivation as in Remark 4.18. Now, a fuzzy set $(A,\mu_{A})$ satisfies
$\vdash_{\iota}x\equiv y$ if and only if $\iota(x)=\iota(y)$, thus, if we take
$(H,\mathrm{id}_{H})$ and the assignment $\iota$ of the previous example, then
$(H,\mathrm{id}_{H})\vdash_{\iota}x\equiv y$ but it does not satisfy $x\equiv
z$ with respect to $\iota$.

#### Completeness

Now we prove that the calculus we have provided in Section 3 is complete. Let
us start with the following observation.

######  Remark 4.20.

For any $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ the relation
$\sim_{\Lambda}$ given by all $t$ and $s$ such that $\vdash_{\Lambda}t\equiv
s$, is an equivalence relation on .

Using this equivalence, we can define the model of terms, as done next.

######  Definition 4.21.

Let $\mathcal{L}=(\Sigma,X)$ be a language and
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$, we define
$\mathsf{Terms}\quantity({\Lambda})$ to be the quotient of by
$\sim_{\Lambda}$, moreover, by rule Fun, the function

$\displaystyle\hat{\mu}:\rightarrow H\qquad t\mapsto\sup\quantity{l\in
H\mid\hskip 2.0pt\vdash_{\Lambda}\mathsf{E}\quantity(l,t)}$

induces a function
$\mu_{\Lambda}:\mathsf{Terms}\quantity({\Lambda})\rightarrow H$. For any $f\in
O$ and $c\in C$ putting $c^{\mathcal{T}_{\Lambda}}\coloneqq[c]$ and

$\displaystyle
f^{\mathcal{T}_{\Lambda}}:\mathsf{Terms}\quantity({\Lambda})^{\mathsf{ar}(f)}\rightarrow\mathsf{Terms}\quantity({\Lambda})\qquad\quantity([t_{1}],\dots,[t_{\mathsf{ar}(f)}])\mapsto\quantity[f\quantity(t_{1},\dots,t_{\mathsf{ar}(f)})]$

gives us a $\Sigma$-algebra
$\mathcal{T}_{\Lambda}=\quantity(\quantity(\mathsf{Terms}\quantity({\Lambda}),\mu_{\Lambda}),\Sigma^{\mathcal{T}_{\Lambda}})$,
called _the $\Sigma$-algebra of terms in $\Lambda$_. The _canonical
assignment_ is the function
$\iota_{can}:X\rightarrow\mathsf{Terms}\quantity({\Lambda})$ sending $x$ to
its class $[x]$.

######  Remark 4.22.

Rule Cong assures us that $f^{\mathcal{T}_{\Lambda}}$ is well defined while
Exp implies that it is an arrow of ${\textbf{Fuz}}\quantity(H)$.

######  Lemma 4.23.

Let $\mathcal{L}=(\Sigma,X)$ be a language and
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$, then:

  1. 1.

for any $\phi\in\mathsf{Form}\quantity(\mathcal{L})$ the following are
equivalent:

    1. (a)

$\mathcal{T}_{\Lambda}\vDash\phi$,

    2. (b)

$\mathcal{T}_{\Lambda}\vDash_{\iota_{can}}\phi$,

    3. (c)

$\vdash_{\Lambda}\phi$;

  2. 2.

for any assignment $\iota:X\rightarrow\mathsf{Terms}\quantity({\Lambda})$ and
formula $\phi$, $\mathscr{T}_{\Lambda}\vDash_{\iota}\phi$ if and only if
$\vdash_{\Lambda}\phi[\sigma\circ\iota]$ for one (and thus any) section
$\sigma$ of the quotient $\rightarrow\mathsf{Terms}\quantity({\Lambda})$;

  3. 3.

$\mathcal{T}_{\Lambda}=\quantity(\quantity(\mathsf{Terms}\quantity({\Lambda}),\mu_{\Lambda}),\Sigma^{\mathcal{T}_{\Lambda}})$
is a model of $\Lambda$.

######  Proof 4.24.

Let us start with a technical result.

######  Proposition 4.25.

Let $\mathcal{L}=(\Sigma,X)$ be a language,
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$, and
$\sigma:\mathsf{Terms}\quantity({\Lambda})\rightarrow$ a section of the
quotient $\rightarrow\mathsf{Terms}\quantity({\Lambda})$. The equation
$t^{\mathcal{T}_{\Lambda},\iota}=\quantity[t\quantity[\sigma\circ\iota]]$
holds for any assignment
$\iota:X\rightarrow\mathsf{Terms}\quantity({\Lambda})$ and $t\in$. In
particular $t^{\mathcal{T}_{\Lambda},\iota_{can}}=[t]$.

######  Proof 4.26.

For constants it is obvious, let us proceed by induction.

  * •

For any $x\in X$, $\sigma\quantity(\iota(x))$ is a representative of
$\iota(x)$, so:

$\displaystyle
x^{\mathcal{T}_{\Lambda},\iota}=\iota(x)=\quantity[\sigma(\iota(x))]=\quantity[\sigma\quantity(\iota\quantity(x))/x]=\quantity[x\quantity[\sigma\circ\iota]]$

  * •

For any $f\in O$ and $t_{1},\dots,t_{\mathsf{ar}(f)}\in$:

$\displaystyle\quantity(f\quantity(t_{1},\dots,t_{\mathsf{ar}(f)}))^{\mathcal{T}_{\Lambda},\iota}=f^{\mathcal{T}_{\Lambda}}\quantity(t^{\mathcal{T}_{\Lambda},\iota}_{1},\dots,t^{\mathcal{T}_{\Lambda},\iota}_{\mathsf{ar}(f)})=f^{\mathcal{T}_{\Lambda}}\quantity(t^{\mathcal{T}_{\Lambda},\iota}_{1},\dots,t^{\mathcal{T}_{\Lambda},\iota}_{\mathsf{ar}(f)})$
$\displaystyle=f^{\mathcal{T}_{\Lambda}}\quantity(\big{[}t_{1}\quantity[\sigma\circ\iota]\big{]},\dots,\quantity[t_{\mathsf{ar}(f)}\quantity[\sigma\circ\iota]])=\quantity[f\quantity(t_{1}[\sigma\circ\iota],\dots,t_{\mathsf{ar}(f)}[\sigma\circ\iota])]$
$\displaystyle=\quantity[\quantity(f\quantity(t_{1},\dots,t_{\mathsf{ar}(f)}))[\sigma\circ\iota]]$

Now we can proceed with the proof of Lemma 4.23.

  1. 1.

Let us show the three implications.

    1. (a)$\Rightarrow$(b). By definition.

    2. (b)$\Rightarrow$(c). We split the cases.

       * •

$\phi$ is $t\equiv s$. Then $\mathcal{T}_{\Lambda}\vDash_{\iota_{can}}\phi$
means

$\displaystyle[t]=t^{\mathscr{T}_{\Lambda},\iota_{can}}=s^{\mathscr{T}_{\Lambda},\iota_{can}}=[s]$

thus $t\sim_{\Lambda}s$ i.e. $\vdash_{\Lambda}t\equiv s$.

       * •

$\phi$ is $\mathsf{E}\quantity(l,t)$. Let $S$ be $\quantity{l^{\prime}\in
H\mid\Lambda\vdash\mathsf{E}\quantity(l^{\prime},t)}$, by hypothesis
$\mathcal{T}_{\Lambda}\vDash_{\iota_{can}}\phi$, so

$\displaystyle
l\leq\mu_{\Lambda}\quantity(t^{\mathscr{T}_{\Lambda},\iota_{can}})=\mu_{\Lambda}\quantity([t])=\sup(S)$

hence $l=l\wedge\sup(S)$ and, since $H$ is a frame, it follows that
$l=\sup\quantity(\quantity{l\wedge l^{\prime}\mid l^{\prime}\in S})$, by rule
Mon we know that that $\vdash_{\Lambda}\mathsf{E}\quantity(l\wedge
l^{\prime},t)$ for all $l^{\prime}\in S$ and so rule Sup gives us
$\vdash_{\Lambda}\mathsf{E}\quantity(l,t)$.

    3. (c)$\Rightarrow$(a). Let $\iota:X\rightarrow\mathsf{Terms}\quantity({\Lambda})$ be an assignment and $\sigma$ a section as in the hypothesis, then we can apply Sub to get $\vdash_{\Lambda}\phi[\sigma\circ\iota]$, by Proposition 4.25 we get the thesis.

  2. 2.

By Proposition 4.25 we have

$\displaystyle
t^{\mathscr{T}_{\Lambda},\iota}=[t[\sigma\circ\iota]]=\quantity(t[\sigma\circ\iota])^{\mathscr{T}_{\Lambda},\iota_{can}}$

we can conclude using the previous point.

  3. 3.

Let $\Gamma\vdash\psi$ be a sequent in $\Lambda$ with
$\Gamma=\quantity{\phi_{i}}_{i=1}^{n}$ and
$\iota:X\rightarrow\mathsf{Terms}\quantity({\Lambda})$ an assignment such that
$\mathcal{T}_{\Lambda}\vDash_{\iota}\Gamma$. By point $1$ above this means
that $\vdash_{\Lambda}\Gamma[\sigma\circ\iota]$ and applying Sub and Cut we
can conclude that $\vdash_{\Lambda}\psi[\sigma\circ\iota]$. By the previous
point this is equivalent to $\mathscr{T}_{\Lambda}\vDash_{\iota}\psi$.

######  Corollary 4.27 (Completeness for formulae).

For any theory $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$,
$\mathcal{A}\vDash\phi$ for all
$\mathcal{A}\in{\textbf{Mod}}\quantity(\Lambda)$ if and only if
$\vdash_{\Lambda}\phi$.

##  5 From theories to monads

Given a language $\mathcal{L}=\quantity(\Sigma,X)$ and a fuzzy theory
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ we have a forgetful functor:
$\mathscr{U}_{\Lambda}:{\textbf{Mod}}\quantity(\Lambda)\rightarrow{\textbf{Fuz}}\quantity(L)$.
In this section we first show that it has a left adjoint (Section 5.1) and
that for a specific class of theories, models correspond to Eilenberg-Moore
algebras for the monad induced by this adjunction (Section 5.2).

###  5.1 The free fuzzy algebra on a fuzzy set

To give the definition of free models (Definition 5.11) we need some
preliminary constructions.

######  Definition 5.1.

Let $\mathcal{A}$ be a $\Sigma$-algebra and
$f:(B,\mu_{B})\rightarrow\mathscr{U}_{\Sigma}(\mathcal{A})$ an arrow in
${\textbf{Fuz}}\quantity(H)$, a $\Sigma$-algebra _generated by $f$ in
$\mathcal{A}$_ is a morphism
$\epsilon:\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}\rightarrow\mathcal{A}$
such that:

  * •

$\mathscr{U}_{\Sigma}(\epsilon)$ is strong mono;

  * •

there exists
$\bar{f}:(B,\mu_{B})\rightarrow\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}$
such that $\mathscr{U}_{\Sigma}(\epsilon)\circ\bar{f}=f$;

  * •

if $g:\mathcal{C}\rightarrow\mathcal{A}$ is a morphism such that
$\mathscr{U}_{\Sigma}(g)$ is strong monomorphism and
$\mathscr{U}_{\Sigma}(g)\circ h=f$ for some $h$ then there exists a unique
$k:\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}\rightarrow\mathcal{C}$ such
that $g\circ k=\epsilon$.

######  Proposition 5.2.

For any signature $\Sigma$, $\Sigma$-algebra $\mathcal{A}$ and
$f:(B,\mu_{B})\rightarrow\mathscr{U}_{\Sigma}(\mathcal{A})$,
$\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}$ exists and it is unique up to
isomorphism.

######  Proof 5.3.

This is straightforward: if $\mathcal{A}=((A,\mu_{A}),\Sigma^{\mathcal{A}})$
it is enough to consider as carrier set
$\mathscr{U}_{\Sigma}(\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}})$ the
smallest subset of $A$ such that:

  * •

$f(B)\subset\mathscr{U}_{\Sigma}(\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}})$;

  * •

$\quantity{c^{\mathcal{A}}}_{c\in
C}\subset\mathscr{U}_{\Sigma}(\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}})$;

  * •

if $g\in O$ and
$b_{1},\dots,b_{\mathsf{ar}(g)}\in\mathscr{U}(\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}})$
then
$g^{\mathcal{A}}(b_{1},\dots,b_{\mathsf{ar}(g)})\in\mathscr{U}_{\Sigma}(\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}})$.

with the restriction of $\mu_{A}$ as membership degree function. As
$\Sigma$-algebra structure we can take the restriction of
$\Sigma^{\mathcal{A}}$. By Proposition 2.5 the inclusion $i$ is a strong
monomorphism and by construction an arrow of . Let
$g:\mathcal{C}\rightarrow\mathcal{A}$ and $h$ be as in the definition, if we
consider the inclusion $i:(f(B),{\mu_{A}}_{|f(B)})\rightarrow(A,\mu_{A})$, by
the left lifting property we obtain

$(B,\mu_{B})$$(f(B),{\mu_{A}}_{|f(B)})$$(A,\mu_{A})$$\mathscr{U}_{\Sigma}(\mathcal{C})$$\bar{f}$$i$$h$$\bar{k}$$\mathscr{U}_{\Sigma}(g)$

and we can define $k$ by induction as:

  * •

$k(b)=\bar{k}(b)$ if $b\in f(B)$;

  * •

$k(c^{\mathcal{A}})=c^{\mathcal{C}}$ if $c\in C$;

  * •

$k(t^{\mathcal{A}}(b_{1},\dots,b_{\mathsf{ar}(g)}))=t^{\mathcal{C}}(k(b_{1}),\dots,k(b_{\mathsf{ar}(t)}))$
if $t\in O$ and
$b_{1},\dots,b_{\mathsf{ar}(t)}\in\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}$.

Notice that injectivity of $g$ implies that $k$ is actually well-defined, in
fact, proceeding by induction:

  * •

if $x\in B$ is such that $f(x)=c^{\mathcal{A}}$ for some $c\in C$ then

$\displaystyle
g(h(x))=\epsilon(\bar{f}(x))=c^{\mathcal{A}}=g(c^{\mathcal{C}})$

so

$\displaystyle k(f(x))=h(x)=c^{\mathcal{C}}=k(c^{\mathcal{A}})$

  * •

if $x\in B$ is such that
$f(x)=t^{\mathcal{A}}(b_{1},\dots,b_{\mathsf{ar}(t)})$ for some $t\in O$ and
$b_{1},\dots,b_{\mathsf{ar}(t)}\in\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}$
then

$\displaystyle g(h(x))$
$\displaystyle=\epsilon(\bar{f}(x))=t^{\mathcal{A}}(b_{1},\dots,b_{\mathsf{ar}(t)})$
$\displaystyle=t^{\mathcal{A}}(g(k(b_{1})),\dots,g(k(b_{\mathsf{ar}(t)})))=g(t^{\mathcal{C}}(b_{1},\dots,b_{\mathsf{ar}(t)}))$

so $h(x)=t^{\mathcal{C}}(b_{1},\dots,b_{\mathsf{ar}(t)})$ and we are done;

  * •

if $c\in C$ is such that
$c^{\mathcal{A}}=t^{\mathcal{A}}(b_{1},\dots,b_{\mathsf{ar}(t)})$ for some
$t\in O$ and
$b_{1},\dots,b_{\mathsf{ar}(t)}\in\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}$
then

$\displaystyle g(c^{\mathcal{C}})$
$\displaystyle=c^{\mathcal{A}}=t^{\mathcal{A}}(b_{1},\dots,b_{\mathsf{ar}(t)})=t^{\mathcal{A}}(g(k(b_{1})),\dots,g(k(b_{\mathsf{ar}(t)})))=g(t^{\mathcal{C}}(b_{1},\dots,b_{\mathsf{ar}(t)}))$

so $c^{\mathcal{C}}=t^{\mathcal{C}}(b_{1},\dots,b_{\mathsf{ar}(t)})$ and we
are done again;

  * •

if
$s^{\mathcal{A}}(b^{\prime}_{1},\dots,b^{\prime}_{\mathsf{ar}(s)})=t^{\mathcal{A}}(b_{1},\dots,b_{\mathsf{ar}(t)})$
for some $t,s\in O$ and
$b^{\prime}_{1},\dots,b^{\prime}_{\mathsf{ar}(s)},b_{1},\dots,b_{\mathsf{ar}(t)},\in\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}$
then

$\displaystyle
g(s^{\mathcal{C}}(k(b^{\prime}_{1}),\dots,k(b^{\prime}_{\mathsf{ar}(s)}))$
$\displaystyle=s^{\mathcal{A}}(b^{\prime}_{1},\dots,b^{\prime}_{\mathsf{ar}(s)})=t^{\mathcal{A}}(b_{1},\dots,b_{\mathsf{ar}(t)})$
$\displaystyle=t^{\mathcal{A}}(g(k(b_{1})),\dots,g(k(b_{\mathsf{ar}(t)})))=g(t^{\mathcal{C}}(b_{1},\dots,b_{\mathsf{ar}(t)}))$

so
$s^{\mathcal{C}}(k(b^{\prime}_{1}),\dots,k(b^{\prime}_{\mathsf{ar}(s)}))=t^{\mathcal{C}}(b_{1},\dots,b_{\mathsf{ar}(t)})$
and even in this case $k$ is well defined.

Moreover, for any $t\in\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}$, if
$\mathscr{U}(\mathcal{C})=(D,\mu_{D})$ then

$\displaystyle\mu_{D}(k(t))=\mu_{A}(g(k(t)))=\mu_{A}(i(t))$

Uniqueness follows at once by induction.

######  Remark 5.4.

Proposition 4.9 implies that, given a model
$\mathcal{A}=\quantity(\quantity(A,\mu_{A}),\Sigma^{\mathcal{A}})$ of a theory
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$, and a morphism
$f:(B,\mu_{B})\rightarrow(A,\mu_{A})$, the $\Sigma$-algebra
$\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}$ is in
${\textbf{Mod}}\quantity(\Lambda)$.

######  Proposition 5.5.

Let $\mathcal{A}$ be a $\Sigma$-algebra and
$f:(B,\mu_{B})\rightarrow\mathscr{U}_{\Sigma}(\mathcal{A})$, then, for any
other $\Sigma$-algebra $\mathcal{C}$ and
$h:(B,\mu_{B})\rightarrow\mathscr{U}_{\Sigma}(\mathcal{C})$ there exists at
most one
$k:\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}\rightarrow\mathcal{C}$ such
that $k\circ\bar{f}=h$.

######  Proof 5.6.

Let $k$ and $k^{\prime}$ as in the thesis, we can proceed by induction:

  * •

$k(f(x))=k^{\prime}(f(x))$ for any $x\in B$ by hypothesis;

  * •

$k\quantity(c^{\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}})=k^{\prime}\quantity(\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}})$
since $k$ and $k^{\prime}$ are morphism of ;

  * •

$k\quantity(g^{\langle{B},\mu_{B}\rangle_{{\mathcal{A}},{f}}}\quantity(b_{1},\dots,b_{\mathsf{ar}(g)}))=g^{\mathcal{C}}\quantity(k\quantity(b_{1}),\dots,k(b_{\mathsf{ar}(g)}))$,
by inductive hypothesis this is equal to
$g^{\mathcal{C}}(k^{\prime}(b_{1}),\dots,k^{\prime}(b_{\mathsf{ar}(g)}))$ and
we conclude using again the fact that $k^{\prime}$ is an arrow of .

The next definition explains how to extend a theory in a given language with
the data of a fuzzy set.

######  Definition 5.7.

Let $(M,\mu_{M})$ be a fuzzy set, $\mathcal{L}=(\Sigma,X)$ a language with
$\Sigma=(O,\mathsf{ar},C)$. We define $\Sigma[M,\mu_{M}]$ to be
$\quantity(O,\mathsf{ar},C\sqcup M)$ and $\mathcal{L}_{(M,\mu_{M})}$ to be
$\quantity(\Sigma[M,\mu_{M}],X)$. We have an obvious morphism
$\mathbf{I}:\mathcal{L}\rightarrow\mathcal{L}_{(M,\mu_{M})}$ given by the
identities and the inclusion $i_{C}:C\rightarrow C\sqcup M$.

For any $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ we define
$\Lambda[M,\mu_{M}]\in\mathcal{L}_{(M,\mu_{M})}$ as
$\mathbf{I}_{*}\quantity(\Lambda)\cup\overline{\quantity(M,\mu_{M})}$ where
$\overline{\quantity(M,\mu_{M})}=\quantity{\vdash\mathsf{E}\quantity(l,m)\mid
m\in M,l\in L\text{ and }\mu_{M}(m)\geq l}$.

######  Remark 5.8.

It is immediate to see that
$\mathbf{I^{*}}\quantity(\Lambda[M,\mu_{M}])=\Lambda$.

In the next proposition we show that, for any theory $\Lambda$, a fuzzy set
can be mapped into the term model of the theory $\Lambda$ extended with it.
Hence, the natural candidate to be the free model is the algebra generated by
such map.

######  Proposition 5.9.

For any fuzzy set $(M,\mu_{M})$ and any theory
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$:

  1. 1.

the function $\bar{\eta}_{(M,\mu_{M})}$ sending $m$ to the class $[m]$ of the
corresponding constant defines an arrow of fuzzy sets
$\bar{\eta}_{(M,\mu_{M})}:(M,\mu_{M})\rightarrow\mathsf{Terms}\quantity({\Lambda\quantity[M,\mu_{M}]})$;

  2. 2.

any element in $\langle
M,\mu_{M}\rangle_{\mathcal{T}_{\Lambda\quantity[M,\mu_{M}]},\bar{\eta}_{(M,\mu_{M})}}$
has a representative without variables;

  3. 3.

$\langle
M,\mu_{M}\rangle_{\mathcal{T}_{\Lambda\quantity[M,\mu_{M}]},\bar{\eta}_{\quantity(M,\mu_{M})}}$
is the initial object of ${\textbf{Mod}}\quantity(\Lambda[M,\mu_{M}])$;

######  Proof 5.10.

  1. 1.

The only thing to do is to show that
$\mu_{M}(m)\leq\mu_{{\Lambda[M,\mu_{M}]}}\quantity([m])$ but we are easily
done since
$\vdash\mathsf{E}\quantity(\mu_{M},m)\in\Lambda\quantity[M,\mu_{M}]$.

  2. 2.

This follows by the explicit construction given in Proposition 5.2.

  3. 3.

$\langle
M,\mu_{M}\rangle_{\mathcal{T}_{\Lambda\quantity[M,\mu_{M}]},\bar{\eta}_{\quantity(M,\mu_{M})}}$
is in ${\textbf{Mod}}\quantity(\Lambda[M,\mu_{M}])$ by Remark 5.4. For every
model $\mathcal{A}$ of $\Lambda[M,\mu_{M}]$, mapping $m\in M$ to
$m^{\mathcal{A}}$ provides a morphism of fuzzy sets
$(M,\mu_{M})\rightarrow\mathscr{U}_{\Sigma[M,\mu_{M}]}(\mathcal{A})$. By
Proposition 5.5 there exists at most one morphism of
$\Sigma[M,\mu_{M}]$-algebras from $\langle
M,\mu_{M}\rangle_{\mathcal{T}_{\Lambda\quantity[M,\mu_{M}]},\bar{\eta}_{\quantity(M,\mu_{M})}}$
to $\mathcal{A}$. On the other hand, by point 2 for every $[t]\in\langle
M,\mu_{M}\rangle_{\mathcal{T}_{\Lambda\quantity[M,\mu_{M}]},\bar{\eta}_{(M,\mu_{M})}}$
there exists a term $s$ with no variables such that $\Lambda[M,\mu_{M}]\vdash
t\equiv s$, therefore $t^{\mathcal{A},\iota}=t^{\mathcal{A},\iota^{\prime}}$
for any two assignments $\iota,\iota^{\prime}:X\rightarrow A$. Hence we get a
morphism of $\Sigma[M,\mu_{M}]$-algebras sending the class $[t]$ of a term $t$
to its interpretation $t^{\mathcal{A},\iota}$ for any chosen assignment
$\iota$.

######  Definition 5.11.

For any language $\mathcal{L}$, $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$
and $(M,\mu_{M})\in{\textbf{Fuz}}\quantity(H)$ we define the _free model
$\mathcal{F}_{\Lambda}\quantity(M,\mu_{M})$ of $\Lambda$ generated by
$\quantity(M,\mu_{M})$_ to be $\mathsf{r}_{\mathbf{I},\hskip
1.0pt{\Lambda\quantity[M,\mu_{M}]}}\quantity(\langle
M,\mu_{M}\rangle_{\mathcal{T}_{\Lambda\quantity[M,\mu_{M}]},\bar{\eta}_{\quantity(M,\mu_{M})}})$.
We set $\mathsf{T}_{\Lambda}\quantity(M,\mu_{M})$ to be
$\mathscr{U}_{\Lambda}\quantity(\mathcal{F}_{\Lambda}(M,\mu_{M}))$.

Now it is enough to check that the free model just defined actually provides
the left adjoint.

######  Theorem 5.12.

For any language $\mathcal{L}$ and
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ the functor
$\mathscr{U}_{\Lambda}:{\textbf{Mod}}\quantity(\Lambda)\rightarrow{\textbf{Fuz}}\quantity(L)$
has a left adjoint $\mathscr{F}_{\Lambda}$.

######  Proof 5.13.

By construction $\bar{\eta}_{(M,\mu_{M})}$ factors through
$\eta_{(M,\mu_{M})}:(M,\mu_{M})\rightarrow\mathsf{T}_{\Lambda}(M,\mu_{M})$
which sends $m$ to $[m]$. Let now
$g:(M,\mu_{M})\rightarrow\mathscr{U}_{\Lambda}(\mathcal{B})$ be another arrow
in ${\textbf{Fuz}}\quantity(H)$, we can turn $\mathcal{B}$ into a
$\Sigma[M,\mu_{M}]$-algebra $\mathcal{B}^{g}$ setting $m^{\mathcal{B}^{g}}$ to
be $g(m)$ for any $m\in M$.

Let us show that $\mathcal{B}^{g}$ is a model of $\Lambda[M,\mu_{M}]$. Surely
it is a model of $\Lambda$ since $\mathcal{B}$ is, let
$\vdash\mathsf{E}\quantity(l,m)$ be a sequent in $\overline{(M,\mu_{M})}$,
then for any assignment $\iota:V\rightarrow B$:

$\displaystyle\mathcal{B}^{g}\vDash_{\iota}\mathsf{E}\quantity(l,m)$
$\displaystyle\iff
l\leq\mu_{B}(m^{\mathcal{B}^{g},\iota})l\leq\mu_{\Lambda}\quantity(t^{\mathscr{F}_{\Lambda}(M,\mu_{M}),\eta_{(M,\mu_{M})}\circ\iota})\iff
l\leq\mu_{B}(g(m))$

but $g$ is an arrow of ${\textbf{Fuz}}\quantity(H)$ so
$\mu_{B}(g(m))\geq\mu_{M}(m)$ and we are done.

Now, since $\mathcal{B}^{g}$ is a model of $\Lambda[M,\mu_{M}]$, we can take
$\bar{g}$ to be the image through $\mathsf{r}_{\mathbf{\mathbf{I}},\hskip
1.0pt{\Lambda[M,\mu_{M}]}}$ of the unique arrow $\langle
M,\mu_{M}\rangle_{\mathcal{T}_{\Lambda[(M,\mu_{M})]},\bar{\eta}_{(M,\mu_{M})}}\rightarrow\mathcal{B}^{g}$,
by construction

$\displaystyle\bar{g}(\eta_{(M,\mu_{M})}(m))=\bar{g}([m])=m^{\mathcal{B}^{g}}=g(m)$

so $\mathscr{U}_{\Lambda}(\bar{g})\circ\eta_{(M,\mu_{M})}=g$. Uniqueness
follows from Proposition 5.9.

######  Definition 5.14.

Given a theory $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$, its _associated
monad_
$\mathsf{T}_{\Lambda}:{\textbf{Fuz}}\quantity(H)\rightarrow{\textbf{Fuz}}\quantity(H)$
is the composite $\mathscr{U}_{\Lambda}\circ\mathscr{F}_{\Lambda}$.

######  Remark 5.15.

If $\Lambda$ is the empty theory (in any language), then, by Proposition 4.7,
the composittion $\mathscr{F}_{\emptyset}\circ\nabla$ gives us a functor
isomorphic to $\mathscr{F}^{{\textbf{Set}}}_{\Sigma}$.

######  Notation.

We will denote by $\mathscr{F}_{\emptyset}$ with $\mathscr{F}_{\Sigma}$ and
with $\mathsf{T}_{\Sigma}$ the monad
$\mathsf{T}_{\emptyset}=\mathscr{U}_{\Sigma}\circ\mathscr{F}_{\Sigma}$.

In this setting we can provide a result similar to Lemma 4.23.

######  Lemma 5.16.

For any language $\mathcal{L}=(\Sigma,X)$ we define the _natural assignment_
$\iota_{nat}$ as the adjoint to the unit
$\nabla(X)\rightarrow\mathsf{T}_{\Lambda}(\nabla(X))$. Then
$\mathscr{F}_{\Lambda}(\nabla(X))\vDash_{\iota_{nat}}\phi$ if and only if
$\vdash_{\Lambda}\phi$.

######  Proof 5.17.

The implication from the right to the left follows immediately since
$\mathscr{F}_{\Lambda}(\nabla(X))$ is a model for $\Lambda$. By adjointness he
canonical assignment $\iota_{can}$ induces an arrow
$\nabla(X)\rightarrow\mathcal{U}_{\Lambda[\nabla(X)]}\quantity(\mathcal{T}_{\Lambda[\nabla(X)]})$,
which, in turn, induces a morphism
$e:\mathscr{F}_{\Lambda}(\nabla(X))\rightarrow\mathcal{T}_{\Lambda[\nabla(X)]}$
of $\Sigma$-algebras such that, as function between sets,
$e\circ\iota_{nat}=\iota_{can}$. Recalling that $\mathbf{I}$ is the arrow
$(\Sigma,X)\rightarrow(\Sigma[\nabla(X)],X)$ and using Proposition 4.9, Lemma
4.12 and Lemma 4.23:

$\displaystyle\mathscr{F}_{\Lambda}(\nabla X)\vDash_{\iota_{nat}}\phi$
$\displaystyle\iff\mathsf{r}_{\mathbf{I},\hskip
1.0pt{\Lambda[\nabla(X)]}}\quantity({\langle\nabla(X)\rangle_{{\mathcal{T}_{\Lambda[\nabla(X)]}},{\bar{\eta}_{\nabla(X)}}}})\vDash_{\iota_{nat}}\phi$
$\displaystyle\iff\mathsf{r}_{\mathbf{I}}\quantity({\langle\nabla(X)\rangle_{{\mathcal{T}_{\Lambda[\nabla(X)]}},{\bar{\eta}_{\nabla(X)}}}})\vDash_{\iota_{nat}}\phi$
$\displaystyle\iff{\langle\nabla(X)\rangle_{{\mathcal{T}_{\Lambda[\nabla(X)]}},{\bar{\eta}_{\nabla(X)}}}}\vDash_{\iota_{nat}}\phi$
$\displaystyle\hskip
5.69046pt\Longrightarrow\mathcal{T}_{\Lambda[\nabla(X)]}\vDash_{e\circ\iota_{nat}}\phi$
$\displaystyle\iff\mathcal{T}_{\Lambda[\nabla(X)]}\vDash_{\iota_{can}}\phi$
$\displaystyle\iff\vdash_{\Lambda[\nabla(X)]}\phi$

Now, by definition $\overline{\nabla(X)}$ is equal to
$\quantity{\vdash\mathsf{E}\quantity(\bot,x)\mid x\in X}$, therefore
$\quantity(\Lambda[\nabla(X)])^{\vdash}=\Lambda^{\vdash}$ and we get the
thesis.

###  5.2 Eilenberg-Moore algebras and models

In this section we will compare the category
${\textbf{Mod}}\quantity(\Lambda)$ of models of some
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ and
${\textbf{Alg}}(\mathsf{T}_{\Lambda})$ of Eilenberg-Moore algebras for the
corresponding monad $\mathsf{T}_{\Lambda}$. First of all we recall the
following classic lemma ([7, Prop. 4.2.1] and [12, Theorem VI.3.1]).

######  Lemma 5.18.

Let $\mathscr{L}:{\textbf{C}}\rightarrow{\textbf{D}}$ be a functor with right
adjoint $\mathscr{R}$ and let $\mathsf{T}=\mathscr{R}\circ\mathscr{L}$ be its
associated monad, then there exists a _comparison functor_
$\mathscr{K}:{\textbf{D}}\rightarrow{\textbf{Alg}}(\mathsf{T})$ such that
$\mathscr{U}_{\mathsf{T}}\circ\mathscr{K}=\mathscr{R}$, where
$\mathscr{U}_{\mathsf{T}}:{\textbf{Alg}}(\mathsf{T})\rightarrow{\textbf{C}}$
is the forgetful functor. $\mathscr{K}$ sends $D$ in
$(\mathscr{R}(D),\mathscr{R}(\epsilon_{D}))$, where $\epsilon$ is the counit
of the adjunction.

As a consequence, for any theory $\Lambda$ we have a functor from
${\textbf{Mod}}\quantity(\Lambda)$ to ${\textbf{Alg}}(\mathsf{T}_{\Lambda})$.
We want to construct an inverse of such functor.

######  Definition 5.19.

Let $\Lambda$ be in $\mathsf{Th}\quantity(\mathcal{L})$ and
$\xi:\mathsf{T}_{\Lambda}(M,\mu_{M})\rightarrow(M,\mu_{M})$ an object of
${\textbf{Alg}}(\mathsf{T}_{\Lambda})$, we define its _associated algebra_
$\mathscr{H}(\xi)=\quantity(\quantity(M,\mu_{M}),\Sigma^{\mathscr{H}(\xi)})$
putting

$c^{\mathscr{H}(\xi)}\coloneqq\xi\quantity(c^{\mathscr{F}_{\Lambda}(X,\mu_{X})})\qquad
f^{\mathscr{H}\quantity(\xi)}\coloneqq\xi\circ
f^{\mathscr{F}_{\Lambda}(X,\mu_{X})}\circ\eta_{\quantity(M,\mu_{M})}^{\mathsf{ar}(f)}$

for every $c\in C$ and $f\in O$.

######  Lemma 5.20.

For any Eilenberg-Moore algebra
$\xi:\mathsf{T}_{\Lambda}(M,\mu_{M})\rightarrow(M,\mu_{M})$, $\xi$ itself is
an arrow $\mathscr{F}_{\Lambda}(X,\mu_{X})\rightarrow\mathscr{H}(\xi)$ of .

######  Proof 5.21.

Let us start with the following observation.

######  Claim 1.

Let $\Lambda$ be a theory in the language $\mathcal{L}$ and $\hat{\mu}$ the
multiplication of $\mathsf{T}_{\Lambda}$, then, for any
$g:(A,\mu_{A})\rightarrow(B,\mu_{B})$ and operation $f$ the following diagrams
commutes:

$\mathsf{T}_{\Lambda}(B,\mu_{B})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(B,\mu_{B})$$\mathsf{T}_{\Lambda}(M,\mu_{M})$$\mathsf{T}_{\Lambda}(M,\mu_{M})^{\mathsf{ar}(f)}$$f^{\mathscr{F}_{\Lambda}(B,\mu_{B})}$$\mathsf{T}_{\Lambda}(g)$$\mathsf{T}_{\Lambda}(g)^{\mathsf{ar}(f)}$$f^{\mathscr{F}_{\Lambda}(M,\mu_{M})}$$\mathsf{T}_{\Lambda}(M,\mu_{M})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(M,\mu_{M})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(\mathsf{T}_{\Lambda}(M,\mu_{M}))$$\mathsf{T}_{\Lambda}(\mathsf{T}_{\Lambda}(M,\mu_{M}))^{\mathsf{ar}(f)}$$f^{\mathscr{F}_{\Lambda}(M,\mu_{M})}$$\hat{\mu}_{(M,\mu_{M})}$$\hat{\mu}_{(M,\mu_{M})}^{\mathsf{ar}(f)}$$f^{\mathscr{F}_{\Lambda}\quantity(\mathsf{T}_{\Lambda}(M,\mu_{M}))}$$(1,c_{\bot})$$\mathsf{T}_{\Lambda}\quantity(\mathsf{T}_{\Lambda}(M,\mu_{M}))$$\mathsf{T}_{\Lambda}(M,\mu_{M})$$c^{\mathscr{F}_{\Lambda}\quantity(\mathsf{T}_{\Lambda}\quantity(M,\mu_{M}))}$$\hat{\mu}_{(M,\mu_{M})}$$c^{\mathscr{F}_{\Lambda}(M,\mu_{M})}$$(1,c_{\bot})$$\mathsf{T}_{\Lambda}(M,\mu_{M})$$\mathsf{T}_{\Lambda}(M,\mu_{M})$$c^{\mathscr{F}_{\Lambda}(M,\mu_{M})}$$\mathsf{T}_{\Lambda}(g)$$c^{\mathscr{F}_{\Lambda}(M,\mu_{M})}$

######  Proof 5.22.

$\mathsf{T}_{\Lambda}(g)=\mathscr{U}_{\Lambda}(\mathscr{F}_{\Lambda}(g))$ and
$\hat{\mu}_{(A,\mu_{A})}=\mathscr{U}_{\Lambda}(\epsilon_{\mathscr{F}_{\Lambda}(A,\mu_{A})})$
where
$\epsilon:\mathscr{F}_{\Lambda}\circ\mathscr{U}_{\Lambda}\rightarrow\mathrm{id}_{{\textbf{Mod}}\quantity(\Lambda)}$
is the counit ([7] proposition $4.2.1$ or [12], chapter VI). Now the thesis
follows since both $\mathscr{F}_{\Lambda}(g)$ and
$\epsilon_{\mathscr{F}_{\Lambda}(A,\mu_{A})}$ are arrows of .

We can now come back to the proof of our lemma: $\xi\circ
c^{\mathscr{F}_{\Lambda}(M,\mu_{M})}=c^{\mathscr{H}(\xi)}$ while the other
condition is equivalent to commutativity of the outer rectangle in the
diagram:

$\mathsf{T}_{\Lambda}\quantity(\mathsf{T}_{\Lambda}(M,\mu_{M}))^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(M,\mu_{M})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(M,\mu_{M})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(M,\mu_{M})^{\mathsf{ar}(f)}$$(M,\mu_{M})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(M,\mu_{M})$$(M,\mu_{M})$$\mathsf{T}_{\Lambda}\quantity(\mathsf{T}_{\Lambda}(M,\mu_{M}))$$\mathsf{T}_{\Lambda}(M,\mu_{M})$$\eta_{(M,\mu_{M})}^{\mathsf{ar}(f)}$$\eta_{\mathsf{T}_{\Lambda}(M,\mu_{M})}^{\mathsf{ar}(f)}$$\xi^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(\xi)^{\mathsf{ar}(f)}$$f^{\mathscr{F}_{\Lambda}\quantity(\mathscr{F}_{\Lambda}\quantity(M,\mu_{M}))}$$\hat{\mu}_{(M,\mu_{M})}^{\mathsf{ar}(f)}$$f^{\mathscr{F}_{\Lambda}\quantity(M,\mu_{M})}$$\hat{\mu}_{(M,\mu_{M})}$$\xi$$\xi$$\mathsf{T}_{\Lambda}(\xi)$$f^{\mathscr{F}_{\Lambda}\quantity(M,\mu_{M})}$$\mathrm{id}_{\mathsf{T}_{\Lambda}(M,\mu_{M})}$$1$$3$$4$$5$$2$

but 1 commutes by naturality of $\eta$, 2 and 4 by 1, 3 since
$\mathsf{T}_{\Lambda}$ is a monad and 5 from the fact that $\xi$ is an
Eilenberg-Moore algebra.

In general $\mathscr{H}(\xi)$ is not a model of $\Lambda$, but we can restrict
to a class of theories such this holds. As in [4, 13], we consider theories
whose sequents’ premises contain only variables.

######  Definition 5.23.

A theory $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ is _basic_ 111In [3]
such theories are called _simple_. if, for any sequent $\Gamma\vdash\phi$ in
it, all the formulae in $\Gamma$ contain only variables.

######  Example 5.24.

Fuzzy groups, fuzzy normal groups, fuzzy semigroups and left, right, bilateral
ideals (Examples 3.15 and 3.14) are all examples of basic theories.

######  Lemma 5.25.

$\mathscr{H}(\xi)$ is a model of $\Lambda$ for any basic theory
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ and Eilenberg-Moore algebra
$\xi:\mathsf{T}_{\Lambda}(M,\mu_{M})\rightarrow(M,\mu_{M})$.

######  Proof 5.26.

We can start by observing that if $\Gamma\vdash\phi$ is in $\Lambda$ and
$\iota:X\rightarrow M$ is an assignment such that
$\mathscr{H}(\xi)\vDash_{\iota}\Gamma$ then
$\mathscr{F}_{\Lambda}(M,\mu_{M})\vDash_{\eta_{(M,\mu_{M})}\circ\iota}\Gamma$.
Indeed, for every $\psi$ in $\Gamma$, we have two cases:

  * •

$\psi$ is $x\equiv y$. Since $\iota(x)=\iota(y)$ we can easily conclude.

  * •

$\psi$ is $\mathsf{E}\quantity(l,x)$. The thesis follows at once since the
membership degree of $\eta_{(M,\mu_{M})}(\iota(x))$ in
$\mathsf{T}_{\Lambda}(M,\mu_{M})$ is greater than $\mu_{M}(\iota(x))$.

Therefore, we know that
$\mathscr{F}_{\Lambda}(M,\mu_{M})\vDash_{\eta_{(M,\mu_{M})}\circ\iota}\phi$.
Let us split again the two cases.

  * •

$\phi$ is $t\equiv s$. In this case,
$t^{\mathscr{F}_{\Lambda}(M,\mu_{M}),\eta_{(M,\mu_{M})}\circ\iota}=s^{\mathscr{F}_{\Lambda}(M,\mu_{M}),\eta_{(M,\mu_{M})}\circ\iota}$,
point $2$ of Proposition 4.9 and the fact that $\xi$ is an Eilenberg-Moore
algebra thus imply:

$\displaystyle t^{\mathscr{H}(\xi),\iota}$
$\displaystyle=t^{\mathscr{H}(\xi),\xi\circ\eta_{(M,\mu_{M})}\circ\iota}=\xi\quantity(t^{\mathscr{F}_{\Lambda}(M,\mu_{M}),\eta_{(M,\mu_{M})}\circ\iota})$
$\displaystyle=\xi\quantity(s^{\mathscr{F}_{\Lambda}(M,\mu_{M}),\eta_{(M,\mu_{M})}\circ\iota})=s^{\mathscr{H}(\xi),\xi\circ\eta_{(M,\mu_{M})}\circ\iota}=s^{\mathscr{H}(\xi),\iota}$

  * •

$\phi$ is $\mathsf{E}\quantity(l,t)$. This means that
$l\leq\mu_{\Lambda}\quantity(t^{\mathscr{F}_{\Lambda}(M,\mu_{M}),\eta_{(M,\mu_{M})}\circ\iota})$,
hence, using again Lemma 5.20 and Proposition 4.9:

$\displaystyle l$
$\displaystyle\leq\mu_{\Lambda}\quantity(t^{\mathscr{F}_{\Lambda}(M,\mu_{M}),\eta_{(M,\mu_{M})}\circ\iota})\leq\mu_{M}\quantity(\xi\quantity(t^{\mathscr{F}_{\Lambda}(M,\mu_{M}),\eta_{(M,\mu_{M})}\circ\iota}))$
$\displaystyle=\mu_{M}\quantity(t^{\mathscr{H}(\xi),\xi\circ\eta_{(M,\mu_{M})}\circ\iota})=\mu_{M}\quantity(t^{\mathscr{H}(\xi),\iota})$

and we can conclude.

######  Theorem 5.27.

For any basic theory $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$, the
functor
$\mathscr{K}:{\textbf{Mod}}\quantity(\Lambda)\rightarrow{\textbf{Alg}}(\mathsf{T}_{\Lambda})$
has an inverse
$\mathscr{H}:{\textbf{Alg}}(\mathsf{T}_{\Lambda})\rightarrow{\textbf{Mod}}\quantity(\Lambda)$
sending $\xi:\mathsf{T}_{\Lambda}(M,\mu_{M})\rightarrow(M,\mu_{M})$ to
$\mathscr{H}(\xi)$.

######  Proof 5.28.

Let $\xi:\mathsf{T}_{\Lambda}(M,\mu_{M})\rightarrow(M,\mu_{M})$,
$\xi^{\prime}:\mathsf{T}_{\Lambda}(N,\mu_{N})\rightarrow(N,\mu_{N})$ be two
Eilenberg-Moore algebras and $g:(M,\mu_{M})\rightarrow(N,\mu_{N})$ an arrow
between them, we claim that $g$ itself is a morphism of
${\textbf{Mod}}\quantity(\Lambda)$. Let $f\in O$ and $c\in C$, we have
diagrams:

$(M,\mu_{M})$$(N,\mu_{N})$$\mathsf{T}_{\Lambda}(N,\mu_{N})$$\mathsf{T}_{\Lambda}(M,\mu_{M})$$g$$\xi^{\prime}$$\xi$$\mathsf{T}_{\Lambda}(g)$$2$$1$$(1,c_{\bot})$$c^{\mathscr{F}_{\Lambda}(M,\mu_{M})}$$c^{\mathscr{F}_{\Lambda}(N,\mu_{N})}$$(M,\mu_{M})^{\mathsf{ar}(f)}$$(N,\mu_{N})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(M,\mu_{M})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(N,\mu_{N})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(M,\mu_{M})$$\mathsf{T}_{\Lambda}(N,\mu_{N})$$(M,\mu_{M})$$(N,\mu_{N})$$3$$4$$5$$g^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(g)^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(g)$$g$$\eta_{(M,\mu_{M})}^{\mathsf{ar}(f)}$$\eta_{(N,\mu_{N})}^{\mathsf{ar}(f)}$$f^{\mathscr{F}_{\Lambda}(M,\mu_{M})}$$f^{\mathscr{F}_{\Lambda}(N,\mu_{N})}$$\xi$$\xi^{\prime}$

Commutativity of 1 and 4 follows from 1, that of 3 from naturality of $\eta$,
2 and 5 from the fact that $g$ is an arrow of
${\textbf{Alg}}(\mathsf{T}_{\Lambda})$. So $\mathscr{H}$ is a functor. Since
both $\mathscr{H}$ and $\mathscr{K}$ are the identity on arrows, it is enough
to show that they are the inverse of each other on objects. Let
$\xi:\mathsf{T}_{\Lambda}(M,\mu_{M})\rightarrow(M,\mu_{M})$ be in
${\textbf{Alg}}(\mathsf{T}_{\Lambda})$, then Lemma 5.20 implies that $\xi$ is
a morphism of ${\textbf{Mod}}\quantity(\Lambda)$ such that
$\mathscr{U}_{\Lambda}(\xi)\circ\eta_{(M,\mu_{M})}=\mathrm{id}_{\quantity(M,\mu_{M})}$,
but there is only one such morphism, namely the component of the counit in
$\mathscr{H}(\xi)$, so
$\mathscr{U}_{\Lambda}\quantity(\epsilon_{\mathscr{H}(\xi)})=\xi$ and
$\mathscr{K}\circ\mathscr{H}=\mathrm{id}_{{\textbf{Alg}}(\mathsf{T}_{\Lambda})}$.
On the other hand, let
$\mathcal{A}=\quantity(\quantity(A,\mu_{A}),\Sigma^{\mathcal{A}})\in{\textbf{Mod}}\quantity(\Lambda)$,
and consider, for any $f\in O$ and $c\in C$, the diagrams

$\mathsf{T}_{\Lambda}(A,\mu_{A})^{\mathsf{ar}(f)}$$(A,\mu_{A})^{\mathsf{ar}(f)}$$\mathsf{T}_{\Lambda}(A,\mu_{A})$$(A,\mu_{A})$$(A,\mu_{A})^{\mathsf{ar}(f)}$$\eta_{(A,\mu_{A})}^{\mathsf{ar}(f)}$$\mathrm{id}_{{(A,\mu_{A})}^{\mathsf{ar}(f)}}$$\mathscr{U}_{\Lambda}(\epsilon_{\mathcal{A}})^{\mathsf{ar}(f)}$$\mathscr{U}_{\Lambda}(\epsilon_{\mathcal{A}})$$f^{\mathcal{A}}$$f^{\mathcal{A}}$$2$$3$$(1,c_{\bot})$$\mathsf{T}_{\Lambda}(A,\mu_{A})$$(A,\mu_{A})$$\mathscr{U}_{\Lambda}(\epsilon_{\mathcal{A}})$$c^{{\mathscr{F}_{\Lambda}(A,\mu_{A})}}$$c^{\mathcal{A}}$$1$

Commutativity of 1 and 3 follows since each component of $\epsilon$ is an
arrow of ${\textbf{Mod}}\quantity(\Lambda)$, that of 2 since $\epsilon$ is the
counit. So we can deduce now that
$f^{\mathscr{H}(\mathscr{U}_{\Lambda}(\epsilon_{\mathcal{A}}))}=f^{\mathcal{A}}$
and
$c^{\mathscr{H}(\mathscr{U}_{\Lambda}(\epsilon_{\mathcal{A}}))}=c^{\mathcal{A}}$
from which we can deduce that
$\mathscr{H}\circ\mathscr{K}=\mathrm{id}_{{\textbf{Mod}}\quantity(\Lambda)}$.

######  Corollary 5.29.

For any basic theory $\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$,
${\textbf{Alg}}(\mathsf{T}_{\Lambda})$ and ${\textbf{Mod}}\quantity(\Lambda)$
are isomorphic, and thus equivalent, categories.

##  6 Equational axiomatizations

In this section we prove two results for our calculus analogous to the classic
HSP theorem [5], using the results by Milius and Urbat [15].

#### The abstract framework

Let us start recalling the tools introduced in [15], adapted to our situation.
In the following we will fix a tuple222In their work Milius and Urbat
additionaly require a full subcategory of C and a fixed class of cardinals,
but we will not need this level of generality.
$\quantity({\textbf{C}},\quantity(\mathscr{E},\mathscr{M}),\mathscr{X})$ where
C is a category, $\quantity(\mathscr{E},\mathscr{M})$ is a proper
factorization system on C and $\mathscr{X}$ is a class of objects of C.

######  Definition 6.1.

An object $X$ of C is _projective with respect to an arrow $f:A\rightarrow B$_
if for any $h:X\rightarrow B$ there exists a $k:X\rightarrow A$ such that
$f\circ k=h$. We define $\mathscr{E}_{\mathscr{X}}$ as the class of
$e\in\mathscr{E}$ such that for every $X\in\mathscr{X}$, $X$ is projective
with respect to $e$. A _$\mathscr{E}_{\mathscr{X}}$ -quotient_ is just an
arrow in $\mathscr{E}_{\mathscr{X}}$.

In the rest of the section, we assume that
$\quantity({\textbf{C}},\quantity(\mathscr{E},\mathscr{M}),\mathscr{X})$
satisfies the following requirements:

  * •

C has all (small) products;

  * •

for any $X\in\mathscr{X}$, the class
${X}{\mathrel{\rotatebox[origin={c}]{270.0}{$\twoheadrightarrow$}}}{{\textbf{C}}}$
of all $e\in\mathscr{E}$ with domain $X$ is essentially small, i.e. there is a
set
$\mathcal{J}\subset{X}{\mathrel{\rotatebox[origin={c}]{270.0}{$\twoheadrightarrow$}}}{{\textbf{C}}}$
such that for any $e:X\rightarrow
C\in{X}{\mathrel{\rotatebox[origin={c}]{270.0}{$\twoheadrightarrow$}}}{{\textbf{C}}}$
there exists $e^{\prime}:X\rightarrow D\in\mathcal{J}$ and an isomorphism
$\phi$ such that $\phi\circ e=e^{\prime}$;

  * •

for every object $C$ of C there exists $e:X\rightarrow C$ in
$\mathscr{E}_{\mathscr{X}}$ with $X\in\mathscr{X}$.

######  Definition 6.2.

An _$\mathscr{X}$ -equation_ is an arrow
$e\in{X}{\mathrel{\rotatebox[origin={c}]{270.0}{$\twoheadrightarrow$}}}{{\textbf{C}}}$
with $X\in\mathscr{X}$. We say that an object $A$ of C _satisfies_
$e:X\rightarrow C$, and we write $A\vDash_{\mathscr{X}}e$, if for every
$h:X\rightarrow A$ there exists $q:C\rightarrow A$ such that $q\circ e=h$.
Given a class $\mathbb{E}$ of $\mathscr{X}$-equations, we define
$\mathcal{V}(\mathbb{E})$ as the full subcategory of C given by objects that
satisfy $e$ for every $e\in\mathbb{E}$. A full subcategory V is
$\mathscr{X}$-_equationally presentable_ if there exists $\mathbb{E}$ such
that ${\textbf{V}}=\mathcal{V}(\mathbb{E})$.

######  Remark 6.3.

The definition of equation in [15, Def. 3.3] is given in terms of suitable
subclasses of
${X}{\mathrel{\rotatebox[origin={c}]{270.0}{$\twoheadrightarrow$}}}{{\textbf{C}}}$.
However in our setting Milius and Urbat’s definition reduces to ours (cfr.
[15, Remark 3.4]).

######  Theorem 6.4 ([15, Th. 3.15, 3.16]).

A full subcategory V of C is $\mathscr{X}$-equationally presentable if and
only if it is closed under $\mathscr{E}_{\mathscr{X}}$-quotients,
$\mathscr{M}$-subobjects and (small) products.

#### Application to fuzzy algebras

In order to apply the results above to , we need to define the required
inputs, i.e., to specify a factorization system and a class of
$\Sigma$-algebras.

######  Lemma 6.5.

For any signature $\Sigma$ the classes
$\mathscr{E}_{\Sigma}\coloneqq\quantity{e\text{ arrow of
}\alg{\Sigma}\mid\mathscr{U}_{\Sigma}(e)\text{ is epi}}$ and
$\mathscr{M}_{\Sigma}\coloneqq\quantity{m\text{ arrow of
}\alg{\Sigma}\mid\mathscr{U}_{\Sigma}(m)\text{ is strong mono}}$ form a proper
factorization system on .

######  Proof 6.6.

Let $\mathcal{A}=\quantity((A,\mu_{A}),\Sigma^{\mathcal{A}})$ and
$\mathcal{B}=\quantity((B,\mu_{B}),\Sigma^{\mathcal{B}})$ two
$\Sigma$-algebras with a morphism $g:\mathcal{A}\rightarrow\mathcal{B}$
between them. $\mathcal{U}_{\Sigma}(g)$ factors as $m\circ e$ where
$e:(A,\mu_{A})\rightarrow(g(A),{\mu_{B}}_{|g(A)})$ and
$m:(g(A),{\mu_{B}}_{|g(A)})\rightarrow(B,\mu_{B})$ is the usual epi-
monomorphism factorization of $f$ on ${\textbf{Fuz}}\quantity(H)$ (cfr. Lemma
2.10). Notice that $c^{\mathcal{B}}=g(c^{\mathcal{A}})\in g(A)$ for all $c\in
C$ and
$f^{\mathcal{B}}\quantity(g(a_{1}),\dots,g(a_{\mathsf{ar}(g)}))=g\quantity(f^{\mathcal{A}}(a_{1},\dots,a_{\mathsf{ar}(g)}))\in
g(A)$ for every $f\in O$ and $g(a_{1}),\dots,g(a_{\mathsf{ar}(g)})\in g(A)$ so
$\Sigma^{\mathcal{B}}$ restricts to a $\Sigma$-algebra structure on
$(g(A),{\mu_{B}}_{|g(A)})$ and it is clear that with this choice $e$ and $m$
becomes morphisms of . We have now to show the left lifting property. Given
$e\in\mathscr{E}_{\Sigma}$, $m\in\mathscr{M}_{\Sigma}$ and $g$ and $h$ such
that $m\circ g=h\circ e$ we can apply $\mathscr{U}_{\Sigma}$ and get a square

$\mathscr{U}_{\Sigma}(\mathcal{A})$$\mathscr{U}_{\Sigma}(\mathcal{B})$$\mathscr{U}_{\Sigma}\quantity(\mathcal{B^{\prime}})$$\mathscr{U}_{\Sigma}\quantity(\mathcal{A^{\prime}})$$\mathscr{U}_{\Sigma}(g)$$\mathscr{U}_{\Sigma}(m)$$\mathscr{U}_{\Sigma}(e)$$\mathscr{U}_{\Sigma}(h)$

which, by Lemma 2.10, has a diagonal filling
$k:\mathscr{U}_{\Sigma}\quantity(\mathcal{A}^{\prime})\rightarrow\mathscr{U}_{\Sigma}(\mathcal{B})$.
Let us show that $k$ is a morphism of . For any $c\in C$ we have
$c^{\mathcal{A^{\prime}}}=e(c^{\mathcal{A}})$ and for every $f\in O$ and
$x_{1},\dots,x_{\mathsf{ar}(f)}\in\mathscr{U}_{\Sigma}\quantity(\mathcal{A}^{\prime})$
there exist
$a_{1},\dots,a_{\mathsf{ar}(f)}\in\mathscr{U}_{\Sigma}\quantity(\mathcal{A})$
such that $e(a_{i})=x_{i}$ for $1\leq i\leq\mathsf{ar}(f)$, so
$k(c^{\mathcal{A^{\prime}}})=k\quantity(e\quantity(c^{\mathcal{A}}))=g\quantity(c^{\mathcal{A}})=c^{\mathcal{B}}$
and

$\displaystyle
k\quantity(f^{\mathcal{A^{\prime}}}\quantity(x_{1},\dots,x_{\mathsf{ar}(f)}))$
$\displaystyle=k\quantity(f^{\mathcal{A^{\prime}}}\quantity(e\quantity(a_{1}),\dots,e\quantity(a_{\mathsf{ar}(f)})))=k\quantity(e\quantity(f^{\mathcal{A}}\quantity(a_{1},\dots,a_{\mathsf{ar}(f)})))$
$\displaystyle=g\quantity(f^{\mathcal{A}}\quantity(a_{1},\dots,a_{\mathsf{ar}(f)}))=f^{\mathcal{B}}\quantity(g\quantity(a_{1}),\dots,g\quantity(a_{\mathsf{ar}(f)}))$
$\displaystyle=f^{\mathcal{B}}\quantity(k\quantity(e\quantity(a_{1})),\dots,k\quantity(e\quantity(a_{\mathsf{ar}(f)})))=f^{\mathcal{B}}\quantity(k\quantity(x_{1}),\dots,k\quantity(x_{\mathsf{ar}(f)}))$

Finally, properness follows at once since $\mathscr{U}_{\Sigma}$ is faithful
and so reflects epis and monos.

######  Definition 6.7.

We define the following two classes of $\Sigma$-algebras:

$\displaystyle\mathscr{X}_{0}\coloneqq\quantity{\mathscr{F}^{{\textbf{Set}}}_{\Sigma}\quantity(X)\mid
X\in{\textbf{Set}}}\qquad\mathscr{X}_{\mathsf{E}}\coloneqq\quantity{\mathscr{F}_{\Sigma}(X,\mu_{X})\mid(X,\mu_{X})\in{\textbf{Fuz}}\quantity(H)}$

We will use $\mathscr{E}_{\Sigma,\mathscr{X}_{0}}$ (resp.,
$\mathscr{E}_{\Sigma,\mathscr{X}_{\mathsf{E}}}$) for the class of
$e\in\mathscr{E}$ such that every $X\in\mathscr{X}_{0}$ (resp.
$X\in\mathscr{X}_{\mathsf{E}}$) is projective with respect to $e$.

######  Remark 6.8.

$\mathscr{X}_{0}=\quantity{\mathscr{F}_{\Sigma}\quantity(X,\mu_{X})\mid\mathsf{supp}({X},\mu_{X})=\emptyset}$.

We have now all the ingredients needed to use the results recalled above.

######  Lemma 6.9.

With the above definitions:

  1. 1.

$\mathscr{E}_{\Sigma,\mathscr{X}_{0}}=\mathscr{E}_{\Sigma}$;

  2. 2.

$\mathscr{E}_{\Sigma,\mathscr{X}_{\mathsf{E}}}=\quantity{e\in\mathscr{E}_{\Sigma}\mid\mathscr{U}_{\Sigma}(e)\text{
is split}}$;

  3. 3.

$($ and $($ satisfy the conditions of our settings.

######  Proof 6.10.

Let us start adapting the usual notion of congruence to our set environment.

######  Definition 6.11.

Given a $\Sigma$-algebra
$\mathcal{A}=\quantity(\quantity(A,\mu_{A}),\Sigma^{\mathcal{A}})$, a _fuzzy
congruence_ on $\mathcal{A}$ is a pair $(\sim,\mu)$ where

  * •

$\sim$ is a congruence: i.e. an equivalence relation such that, for any $f\in
O$, if $a_{i}\sim b_{i}$ fo $1\leq i\leq\mathsf{ar}(f)$ then
$f^{\mathcal{A}}\quantity(a_{1},\dots a_{\mathsf{ar}(f)})\sim
f^{\mathcal{A}}\quantity(b_{1},\dots,b_{\mathsf{ar}(f)})$;

  * •

$\mu:A\rightarrow H$ is a function such $\mu(a)=\mu(b)$ whenever $a\sim b$;

  * •

for any $f\in O$,

$\bigwedge_{i=1}^{\mathsf{ar}(f)}\mu(a_{i})\leq\mu\quantity(f^{\mathcal{A}}\quantity(a_{1},\dots
a_{\mathsf{ar}(f)}))$

  * •

$\mu_{A}(a)\leq\mu(a)$ for every $a\in A$.

######  Proposition 6.12.

Let $\mathcal{A}=\quantity(\quantity(A,\mu_{A}),\Sigma^{\mathcal{A}})$ be a
$\Sigma$-algebra, then:

  1. (a)

if $\quantity{\quantity(\sim_{i},\mu_{i})}_{i\in I}$ is a family of fuzzy
congruence then $\quantity(\bigcap_{i\in I}\sim_{i},\mu)$ with
$\mu:A\rightarrow H$ the pointwise infimum of $\quantity{\mu_{i}}_{i\in I}$ is
a fuzzy congruence;

  2. (b)

for every fuzzy congruence $(\sim,\mu)$ there exists an epimorphism:
$e_{(\sim,\mu)}:\mathcal{A}\rightarrow\mathcal{B}$ such that
$\mu_{B}(b)=\mu(a)$ for any $a\in e^{-1}(b)$ and
$e_{(\sim,\mu)}(a)=e_{(\sim,\mu)}(b)$ if and only if $a\sim b$;

  3. (c)

for every epimorphism $e:\mathcal{A}\rightarrow\mathcal{B}$ there exists a
fuzzy congruence $(\sim_{e},\mu_{e})$ on $\mathcal{A}$ such that $e\leq
e_{(\sim,\mu)}$ and $e_{(\sim,\mu)}\leq e$ in
$\mathcal{A}\mathrel{\rotatebox[origin={c}]{270.0}{$\twoheadrightarrow$}}$.

######  Proof 6.13.

  1. (a)

This is straightforward.

  2. (b)

Define $\mathcal{B}=\quantity(\quantity(B,\mu_{B}),\Sigma^{\mathcal{B}})$
setting $B:=A/\sim$, $\mu_{B}([a]):=\mu(a)$ and, for any $f\in O$

$f^{\mathcal{A}}\quantity(\quantity[a_{1}],\dots,[a_{\mathsf{ar}(f)}]):=\quantity[f^{\mathcal{B}}\quantity(\quantity[a_{1}],\dots,[a_{\mathsf{ar}(f)}])]$

Since $(\sim,\mu)$ is a fuzzy congruence all these objects are well defined,
the fact that $f^{\mathcal{B}}$ is an arrow of fuzzy sets follows from the
second condition on $\mu$, while the last condition entails that the
projection on the quotient is an arrow of ${\textbf{Fuz}}\quantity(H)$.

  3. (c)

Put $a\sim_{e}b$ if and only if $e(a)=e(b)$ and $\mu_{e}(a):=\mu_{B}(e(a))$.
Since $e$ is a morphism of  we get the first and the last condition in the
definition of a fuzzy congruence, while the second one follows since

$\displaystyle\mu_{e}\quantity(f^{\mathcal{A}}\quantity(a_{1},\dots,a_{\mathsf{ar}(f)}))$
$\displaystyle=\mu_{B}\quantity(e\quantity(f^{\mathcal{A}}\quantity(a_{1},\dots,a_{\mathsf{ar}(f)})))=\mu_{B}\quantity(f^{\mathcal{B}}\quantity(e\quantity(a_{1}),\dots,e\quantity(a_{\mathsf{ar}(f)})))$
$\displaystyle\geq\bigwedge_{i=1}^{\mathsf{ar}(f)}\mu_{B}\quantity(e\quantity(a_{i}))=\bigwedge_{i=1}^{\mathsf{ar}(f)}\mu_{e}\quantity(a_{i})$

Now, it is immediate to see that the function sending the equivalence class
$[a]$ of $a\in A$ to $e(b)$ induces an isomorphism of  witnessing the thesis.

So equipped we can turn back to the proof of Lemma 6.9.

  1. 1.

Let $e:\mathcal{A}\rightarrow\mathcal{B}$ be an arrow in
$\mathscr{E}_{\Sigma}$ and let
$h:\mathscr{F}^{{\textbf{Set}}}_{\Sigma}(X)\rightarrow\mathcal{B}$ be any
morphism of . By point $2$ of Proposition 2.5 $e$ is surjective so for any
$x\in X$ we can take a $a_{x}\in
e^{-1}\quantity(h\quantity(\eta_{X}\quantity(x)))$, where $\eta$ is the unit
of the adjunction
$\mathscr{F}^{{\textbf{Set}}}_{\Sigma}\dashv\mathscr{V}_{\Sigma}$ of
Proposition 4.7, and define $\bar{k}:X\rightarrow A$ mapping $x$ to $a_{x}$,
where $\mathcal{A}=\quantity(\quantity(A,\mu_{A}),\Sigma^{\mathcal{A}})$. By
adjointness, from $\bar{k}$ we get
$k:\mathscr{F}^{{\textbf{Set}}}_{\Sigma}(X)\rightarrow\mathcal{A}$ and

$\displaystyle\quantity(e\circ
k)\circ\eta_{X}=e\circ\quantity(k\circ\eta_{X})=e\circ\bar{k}=h\circ\eta_{X}$

so $e\circ k=h$.

$\mathscr{V}_{\Sigma}(\mathcal{A})$$\mathscr{V}_{\Sigma}(\mathcal{B})$$\mathscr{V}_{\Sigma}\quantity(\mathscr{F}^{{\textbf{Set}}}_{\Sigma}\quantity(X))$$X$$\mathscr{V}_{\Sigma}(e)$$\eta_{\nabla(X)}$$\mathscr{V}_{\Sigma}(h)$$\mathscr{V}_{\Sigma}(k)$$\bar{k}$

  2. 2.

Let $e:\mathcal{A}\rightarrow\mathcal{B}$ be in $\mathscr{E}_{\Sigma}$ such
that $\mathscr{U}_{\Sigma}(e)$ is split and let $s$ be a section in
${\textbf{Fuz}}\quantity(H)$, then, for any
$h:\mathscr{F}_{\Sigma}(X,\mu_{X})\rightarrow\mathcal{B}$ we can consider the
arrow $s\circ h\circ\eta_{(X,\mu_{X})}$, which, by adjointness provides a
$k:\mathscr{F}_{\Sigma}(X,\mu_{X})\rightarrow\mathcal{A}$, moreover:

$\displaystyle\quantity(e\circ k)\circ\eta_{(X,\mu_{X})}$
$\displaystyle=e\circ\quantity(k\circ\eta_{(X,\mu_{X})})=e\circ\quantity(s\circ
h\circ\eta_{(X,\mu_{X})})$ $\displaystyle=\quantity(e\circ
s)\circ\quantity(h\circ\eta_{(X,\mu_{X})})=h\circ\eta_{(X,\mu_{X})}$

so $k$ is the wanted lifting. On the other hand, if $e$ is in
$\mathscr{E}_{\Sigma,\mathscr{X}_{1}}$ we can take the diagram:

$\mathscr{U}_{\Sigma}(\mathcal{A})$$\mathscr{U}_{\Sigma}(\mathcal{B})$$\mathscr{U}_{\Sigma}\quantity(\mathscr{F}_{\Sigma}(\mathscr{U}_{\Sigma}(\mathcal{B})))$$\mathscr{U}_{\Sigma}(\mathcal{B})$$\mathscr{U}_{\Sigma}(e)$$\eta_{\mathscr{U}_{\Sigma}(\mathcal{B})}$$\mathscr{U}_{\Sigma}(\epsilon_{\mathcal{B}})$$\mathscr{U}_{\Sigma}(k)$$\mathrm{id}_{\mathscr{U}_{\Sigma}(\mathcal{B})}$

where $\epsilon_{\mathcal{B}}$ is the component of the counit
$\epsilon:\mathscr{F}_{\Sigma}\circ\mathscr{U}_{\Sigma}\rightarrow\mathrm{id}$
and $k$ its lifting. Taking
$\mathscr{U}_{\Sigma}(k)\circ\eta_{\mathscr{U}_{\Sigma}(\mathcal{B})}$ we get
the desired section of $\mathscr{U}_{\Sigma}(e)$.

  3. 3.

Let us proceed by steps.

     * •

${\textbf{Fuz}}\quantity(H)$ has all products by Lemma 2.10.

     * •

${X}{\mathrel{\rotatebox[origin={c}]{270.0}{$\twoheadrightarrow$}}}{{\textbf{C}}}$
is essentially small by point $3$ of Proposition 6.12.

     * •

For any fuzzy set $(X,\mu_{X})$ we can consider the identity
$\mathrm{id}_{(X,\mu_{X})}:(X,\mu_{X})\rightarrow(X,\mu_{X})$ and the counit
$\epsilon_{(X,\mu_{X})}:\nabla(X)\rightarrow(X,\mu_{X})$ of the adjunction
$\nabla\dashv\mathscr{U}$ of Proposition 2.2. They induce arrows
$e_{0}:\mathscr{F}^{{\textbf{Set}}}_{\Sigma}(X)\rightarrow(X,\mu_{X})$ and
$e_{\mathsf{E}}:\mathscr{F}_{\Sigma}(X,\mu_{X})\rightarrow(X,\mu_{X})$ such
that
$\mathscr{U}_{\Sigma}\quantity(e_{0})\circ\eta_{\nabla(X)}=\epsilon_{(X,\mu_{X})}$
and
$\mathscr{U}_{\Sigma}\quantity(e_{H})\circ\eta_{(X,\mu_{X})}=\mathrm{id}_{(X,\mu_{X})}$.
So $\mathscr{U}_{\Sigma}\quantity(e_{H})$ is split and, since
$\epsilon_{(X,\mu_{X})}$ is surjective, point $2$ of Proposition 2.5 allows us
to conclude that $\mathscr{U}_{\Sigma}\quantity(e_{0})$ is an epimorphism.

We want now to translate formulae of our sequent calculus into
$\mathscr{X}_{0}$\- and $\mathscr{X}_{\mathsf{E}}$-equations. To this end, we
have to restrict to two classes of theories, which we introduce next.

######  Definition 6.14.

Let $\mathcal{L}=(\Sigma,X)$ be a language, a
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ is said to be:

  * •

_unconditional_ ([15, App. B.5]) if any sequent in $\Lambda$ is of the form
$\vdash\phi$ for some formula $\phi$;

  * •

_of type $\mathsf{E}$_ if any sequent in $\Lambda$ is of the form
$\quantity{\mathsf{E}\quantity(l_{i},x_{i})}_{i\in I}\vdash\phi$ for some
formula $\phi$, $\\{x_{i}\\}_{i\in I}\subset X$ and $\\{l_{i}\\}_{i\in
I}\subset H$.

######  Lemma 6.15.

For any signature $\Sigma$ and $\mathscr{X}_{\mathsf{E}}$-equation
$e:\mathscr{F}_{\Sigma}(X,\mu_{X})\rightarrow\mathcal{B}$ there exists a type
${\mathsf{E}}$ theory $\Lambda_{e}$ such that, for every $\Sigma$-algebra
$\mathcal{A}$, $\mathcal{A}\vDash_{\mathscr{X}_{1}}e$ if and only if
$\mathcal{A}\in{\textbf{Mod}}(\Lambda_{e})$. Moreover
$\absolutevalue{\Gamma}\leq\absolutevalue{\mathsf{supp}({X},\mu_{X})}$ for any
$\Gamma\vdash\phi\in\Lambda_{e}$.

######  Proof 6.16.

Let $\mathcal{L}_{e}$ be $(\Sigma,X)$. We define
$\Gamma_{X}:=\\{\mathsf{E}\quantity(\mu_{X}(x),x)\mid
x\in\mathsf{supp}({X},\mu_{X})\\}$ and
$\Lambda_{e}\in\mathsf{Th}\quantity(\mathcal{L})$ as
$\Lambda_{e}^{1}\cup\Lambda_{e}^{2}$ where

$\displaystyle\Lambda_{e}^{1}\coloneqq\quantity{\Gamma_{X}\vdash\mathsf{E}\quantity(l,t)\mid
l\leq\mu_{B}\quantity(e\quantity([t]))}\qquad\Lambda_{e}^{2}\coloneqq\quantity{\Gamma_{X}\vdash[s]\equiv[t]\mid
e\quantity([t])=e\quantity([s])}$

and $(B,\mu_{B})$ is $\mathscr{U}_{\Sigma}(\mathcal{B})$. Let us show the two
implications.

  * $\Rightarrow$

Any $\iota:X\rightarrow A$ such that $\mathcal{A}\vDash_{\iota}\Gamma_{X}$
defines an arrow
$\bar{\iota}(X,\mu_{X})\rightarrow\mathscr{U}_{\Sigma}(\mathcal{A})$. By
adjointness we have a homomorphism
$h_{\iota}:\mathscr{F}_{\Sigma}(X,\mu_{X})\rightarrow\mathcal{A}$ hence, by
hypothesis, there exists $q_{\iota}:\mathcal{B}\rightarrow\mathcal{A}$ such
that $q_{\iota}\circ e=h_{\iota}$. Now, notice that (see Theorem 5.12, and
Proposition 5.9(4)) $h_{\iota}([t])=t^{\mathcal{A,\iota}}$ Take a sequent
$\Gamma_{X}\vdash\psi$ in $\Lambda_{e}$, we have two cases, depending on
$\psi$.

    * –

If $\psi=\mathsf{E}\quantity(l,t)\in\Lambda^{me}_{e}$ we have

$\displaystyle
l\leq\mu_{B}\quantity(e\quantity([t]))\leq\mu_{A}\quantity(q_{\iota}\quantity(e\quantity([t])))=\mu_{A}\quantity(h_{\iota}\quantity([t]))={t}^{\mathcal{A},\iota}$

therefore $\mathcal{A}\vDash_{\iota}\psi$.

    * –

If $\phi=[s]\equiv[t]\in\Lambda^{eq}_{e}$ then

$\displaystyle{t}^{\mathcal{A},\iota}=h_{\iota}\quantity([t])=q_{\iota}\quantity(e([t]))=q_{\iota}\quantity(e([s]))=h_{\iota}\quantity([t])={s}^{\mathcal{A},\iota}$

and even in this case $\mathcal{A}\vDash_{\iota}\psi$.

  * $\Leftarrow$

Take $h:\mathscr{F}_{\Sigma}\quantity(X,\mu_{X})\rightarrow\mathcal{A}$,
$\mathscr{U}_{\Sigma}(h)\circ\eta_{\nabla(X)}$ is an arrow
$(X,\mu_{X})\rightarrow\mathscr{U}_{\Sigma}(\mathcal{A})$, so forgetting the
fuzzy set structure too gives us an assignment $\iota_{h}:X\rightarrow A$ such
that $\mathcal{A}\vDash_{\iota_{h}}\Gamma_{X}$. As before
$h([t])=t^{\mathcal{A},\iota_{h}}$ for every
$[t]\in\mathscr{F}_{\Sigma}\quantity(X,\mu_{X})$. Since
$\mathcal{A}\in{\textbf{Mod}}\quantity(\Lambda_{e})$ we have

    * –

$t^{\mathcal{A},\iota_{h}}=s^{\mathcal{A},\iota_{h}}$ for all terms $t$ and
$s$ such that $e([t])=e([s])$;

    * –

$l\leq\mu_{A}(t^{\mathcal{A},\iota_{h}})$ for all terms $t$ such that
$l\leq\mu_{B}(e([t]))$.

So, the function $q:B\rightarrow A$ which sends $b\in B$ to $h([t])$ for some
$[t]\in e^{-1}(b)$, provides us with an arrow
$\mathscr{U}_{\Sigma}(\mathcal{B})\rightarrow\mathscr{U}_{\Sigma}(\mathcal{A})$
such that $q\circ e=h$. Now:

$\begin{split}&q\quantity(c^{\mathcal{B}})=q\quantity(e\quantity(c^{\mathscr{F}_{\Sigma}\quantity(X,\mu_{X})}))\\\
&=h\quantity(c^{\mathscr{F}_{\Sigma}\quantity(X,\mu_{X})})=c^{\mathcal{A}}\\\
\\\
\end{split}\quad\begin{split}&q\quantity(f^{\mathcal{B}}\quantity(b_{1},\dots,b_{\mathsf{ar}(f)}))=q\quantity(f^{\mathcal{B}}\quantity(e\quantity(c_{1}),\dots,\quantity(e\quantity(c_{\mathsf{ar}(f)}))))\\\
&=q\quantity(e\quantity(f^{\mathscr{F}_{\Sigma}\quantity(X,\mu_{X})}\quantity(c_{1},\dots,c_{\mathsf{ar}\quantity(f)})))\\\
&=h\quantity(f^{\mathscr{F}_{\Sigma}\quantity(X,\mu_{X})}\quantity(c_{1},\dots,c_{\mathsf{ar}\quantity(f)}))=f^{\mathcal{A}}\quantity(h\quantity(c_{1}),\dots,h\quantity(c_{\mathsf{ar}(f)}))\\\
&=f^{\mathcal{A}}\quantity(q\quantity(e\quantity(c_{1})),\dots,q\quantity(e\quantity(c_{\mathsf{ar}(f)})))=f^{\mathcal{A}}\quantity(q\quantity(b_{1}),\dots,q\quantity(b_{\mathsf{ar}(f)}))\end{split}$

so we can conclude that $q$ is an arrow of  and we are done.

######  Corollary 6.17.

For any signature $\Sigma$ and $\mathscr{X}_{0}$-equation
$e:\mathscr{F}^{{\textbf{Set}}}_{\Sigma}\quantity(X)\rightarrow\mathcal{B}$
there exists an unconditional theory $\Lambda_{e}$ such that, for any
$\Sigma$-algebra $\mathcal{A}$, $\mathcal{A}\vDash_{\mathscr{X}_{0}}e$ if and
only if $\mathcal{A}\in{\textbf{Mod}}(\Lambda_{e})$.

Finally, from the results above we can easily conclude HSP-like results for .

######  Theorem 6.18.

Let V be a full subcategory of , then

  1. 1.

V is closed under epimorphisms, (small) products and strong monomorphisms if
and only if there exists a class of unconditional theories
$\quantity{\Lambda_{e}}_{e\in\mathbb{E}}$ such that
$\mathcal{A}\in{\textbf{V}}$ if and only if
$\mathcal{A}\in{\textbf{Mod}}\quantity(\Lambda_{e})$ for all $e\in\mathbb{E}$.

  2. 2.

V is closed under split epimorphisms, (small) products and strong
monomorphisms if and only if there exists a class of type $\mathsf{E}$
theories $\quantity{\Lambda_{e}}_{e\in\mathbb{E}}$ such that
$\mathcal{A}\in{\textbf{V}}$ if and only if
$\mathcal{A}\in{\textbf{Mod}}\quantity(\Lambda_{e})$ for all $e\in\mathbb{E}$.

######  Proof 6.19.

This is straightforward in light of Theorem 6.4, Lemma 6.15 and Corollary
6.17.

######  Remark 6.20.

We cannot arrange the collection $\\{\Lambda_{e}\\}_{e\in\mathbb{E}}$ into a
unique theory since a proper class of variables is needed to write down all
the necessary sequents. A possible way to deal with this issue is to fix two
Grothendieck universes ([22]) ${\textbf{U}}_{1}\subset{\textbf{U}}_{2}$ and
modify the definition of language allowing for a possible class (i.e., an
element of ${\textbf{U}}_{2}$) of variables. All the proof of this paper can
be repeated verbatim in this context carefully distinguishing between fuzzy
_sets_ (i.e., those defined on an element of ${\textbf{U}}_{1}$) and fuzzy
_classes_ (i.e., those defined on an element of ${\textbf{U}}_{2}$). Then the
algebras of terms will be a fuzzy class in general but it is possible to show,
using the explicit construction, that $\mathsf{T}_{\Lambda}(X,\mu_{X})$ is a
fuzzy set if $X\in{\textbf{U}}_{1}$ and so we can retain all the results of
Section 5.

The issue mentioned in the previous remark can be avoided if the family
$\\{\Lambda_{e}\\}_{e\in\mathbb{E}}$ satisfies a boundedness property about
the premises of the sequents belonging to each $\Lambda_{e}$.

######  Definition 6.21.

Given a cardinal $\kappa$ we say that a $\mathscr{X}_{\mathsf{E}}$-equation
$e:\mathscr{F}_{\Sigma}(X,\mu_{X})\rightarrow\mathcal{B}$ is _$\kappa$
-supported_ if $\absolutevalue{\mathsf{supp}({X},\mu_{X})}<\kappa$.

######  Proposition 6.22.

Let ${\textbf{V}}=\mathcal{V}(\mathbb{E})$ be an
$\mathcal{X}_{\mathsf{E}}$-equational defined subcategory of and suppose every
$e\in\mathbb{E}$ is $\kappa$-supported, then there exists a theory
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$, where
$\mathcal{L}=(\Sigma,\kappa)$, such that
${\textbf{V}}={\textbf{Mod}}\quantity(\Lambda)$.

######  Proof 6.23.

For any $e:\mathscr{F}_{\Sigma}(X_{e},\mu_{X_{e}})\rightarrow\mathcal{B}_{e}$
in $\mathbb{E}$ we can fix an injection
$i_{e}:\mathsf{supp}({X_{e}},\mu_{X_{e}})\rightarrow\kappa$ and an extension
let $\bar{i}_{e}:X\rightarrow\kappa$ of it, fix also morphisms
$\mathbf{I}^{e}:\mathcal{L}_{e}\rightarrow\mathcal{L}$ given by
$(\mathrm{id}_{\Sigma},\bar{i}_{e})$. Let now
$\quantity{\Lambda_{e}}_{e\in\mathbb{E}}$ be the collection of theories given
by Corollary 6.17 and Theorem 6.18, since each
$\Lambda_{e}\in\mathbf{Form}\quantity(\mathcal{L}_{e})$ we can define
$\Lambda$ as
$\bigcup_{e\in\mathbb{E}}\mathbf{I}^{e}_{*}\quantity(\Lambda_{e})$. We have to
show that $\mathcal{A}\in{\textbf{V}}$ if and only if
$\mathcal{A}\in{\textbf{Mod}}\quantity(\Lambda)$.

  * $\Rightarrow$

Let
$\mathsf{Form}\quantity(\mathbf{I}^{e})\quantity(\Gamma_{X_{e}})\vdash\mathsf{Form}\quantity(\mathbf{I}^{e})\quantity(\psi)$
be a sequent in $\Lambda$ and let $\iota:\kappa\rightarrow A$ an assignment
such that
$\mathcal{A}\vDash_{\iota}\mathsf{Form}\quantity(\mathbf{I}^{e})\quantity(\Gamma_{X_{e}})$.
By point $3$ of Lemma 4.12 this implies
$\mathcal{A}\vDash_{\iota\circ\bar{i}_{e}}\Gamma_{X_{e}}$, therefore
$\mathcal{A}\vDash_{\iota\circ\bar{i}_{e}}\psi$ and we conclude applying lemma
4.12 again.

  * $\Leftarrow$

If
$\mathscr{U}_{\Sigma}(\mathcal{A})=(\emptyset,\rotatebox[origin={c}]{180.0}{!}_{H})$,
($\rotatebox[origin={c}]{180.0}{!}_{H}$ being the empty map
$\emptyset\rightarrow H$) then there are no assignment $\kappa\rightarrow A$
and so $\mathcal{A}$ is in ${\textbf{Mod}}\quantity(\Lambda)$. In the other
cases let $\Gamma_{X_{e}}\vdash\psi$ be in $\Lambda_{e}$ and
$\iota:X_{e}\rightarrow A$ such that
$\mathcal{A}\vDash_{\iota}\Gamma_{X_{e}}$, since $A\neq\emptyset$ there exists
$\hat{\iota}:\kappa\rightarrow A$ such that
$\hat{\iota}\circ\bar{i}_{e}=\iota$ as in the previous point Lemma 4.12
implies
$\mathcal{A}\vDash_{\hat{\iota}}\mathsf{Form}\quantity(\mathbf{I}^{e})\quantity(\Gamma_{X_{e}})$,
so
$\mathcal{A}\vDash_{\hat{\iota}}\mathsf{Form}\quantity(\mathbf{I}^{e})\quantity(\psi)$
and again this is equivalent to $\mathcal{A}\vDash_{\iota}\psi$.

######  Corollary 6.24.

V is closed under epimorphisms, (small) products and strong monomorphisms if
and only if there exists a language $\mathcal{L}$ and an unconditional theory
$\Lambda\in\mathsf{Th}\quantity(\mathcal{L})$ such that
${\textbf{V}}={\textbf{Mod}}\quantity(\Lambda)$.

##  7 Conclusions and future work

In this paper we have introduced a _fuzzy sequent calculus_ to capture
equational aspects of fuzzy sets. While equalities are captured by usual
equations, information contained in the membership function is captured by
_membership proposition_ of the form $\mathsf{E}\quantity(l,t)$, to be
interpreted as “the membership degree of $t$ is at least $l$”. We have used a
natural concept of _fuzzy algebras_ to provide a sound and complete semantics
for such calculus in the sense that a formula is satisfied by all the models
of a given theory if and only if it is derivable from it using the rules of
our calculus.

As in the classical and quantitative contexts, there is a notion of _free
model_ of a theory $\Lambda$ and thus an associated monad
$\mathsf{T}_{\Lambda}$ on the category ${\textbf{Fuz}}\quantity(H)$. However,
in general Eilenberg-Moore algebras for such monad are not equivalent to
models of $\Lambda$, but we have shown that this equivalence holds if
$\Lambda$ is _basic_. In this direction it would be interesting to better
understand the categorical status of our approach, investigating possible
links between our notion of fuzzy theory and
${\textbf{Fuz}}\quantity(H)$-Lawvere theories as introduced in full generality
by Nishizawa and Power in [18]. A difference between the two approaches is
that for us arities are simply finite sets, while following [18] a
${\textbf{Fuz}}\quantity(H)$-Lawvere theory arities would be given by finite
fuzzy sets.

Finally, using the results provided in [15] we have proved that, given a
signature $\Sigma$, subcategories of which are closed under products, strong
monomorphisms and epimorphic images correspond precisely with categories of
models for _unconditional theories_ , i.e. theories axiomatised by sequents
without premises. Moreover, using the same results, we have also proved that
the categories of models of _theories of type $\mathsf{E}$_, i.e. those whose
axioms’ premises contain only membership propositions involving variables, are
exactly those subcategories closed under products, strong monomorphisms and
split epimorphisms.

Our category ${\textbf{Fuz}}\quantity(H)$ of fuzzy sets has crisp arrows and
crisp equality: arrows are ordinary functions between the underlying sets and
equalities can be judged to be either true or false. A way to further
“fuzzifying” concepts is to use the topos of _$H$ -sets_ over the frame $H$
introduced in [8]: this is equivalent to the topos of sheaves over $H$ and
contains ${\textbf{Fuz}}\quantity(H)$ as a (non full) subcategory. By
construction, equalities and functions are “fuzzy”. It would be interesting to
study an application of our approach to this context. A promising feature is
that in an $H$-set the membership degree function is built-in as simply the
equality relation, so it would not be necessary to distinguish between
equations and membership propositions. Even more generally, we can replace $H$
with an arbitrary quantale $\mathcal{V}$ and consider the category of sets
endowed with a “$\mathcal{V}$-valued equivalence relation” [6].

## References

  * [1] N. Ajmal.  Homomorphism of fuzzy groups, correspondence theorem and fuzzy quotient groups.  Fuzzy sets and systems, 61(3):329–339, 1994.
  * [2] N. Ajmal and A. S. Prajapati.  Fuzzy cosets and fuzzy normal subgroups.  Information sciences, 64(1-2):17–25, 1992.
  * [3] G. Bacci, R. Mardare, P. Panangaden, and G. Plotkin.  An algebraic theory of Markov processes.  In 33rd Symposium on Logic in Computer Science (LICS), pages 679–688, 2018.
  * [4] G. Bacci, R. Mardare, P. Panangaden, and G. Plotkin.  Quantitative equational reasoning.  Foundations of Probabilistic Programming, page 333, 2020.
  * [5] G. Birkhoff.  On the structure of abstract algebras.  Proceedings of the Cambridge Philosophical Society, 10:433–454, 1935.
  * [6] F. Bonchi, B. König, and D. Petrisan.  Up-to techniques for behavioural metrics via fibrations.  In CONCUR, volume 118 of LIPIcs, pages 17:1–17:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018.
  * [7] F. Borceux.  Handbook of Categorical Algebra: Volume 2, Categories and Structures, volume 2.  Cambridge University Press, 1994.
  * [8] M. P. Fourman and D. S. Scott.  Sheaves and logic.  In Applications of sheaves, pages 302–401. Springer, 1979.
  * [9] M. Hyland and J. Power.  The category theoretic understanding of universal algebra: Lawvere theories and monads.  Electron. Notes Theor. Comput. Sci., 172:437–458, 2007.
  * [10] P. T. Johnstone.  Stone spaces, volume 3.  Cambridge University Press, 1982.
  * [11] G. M. Kelly.  A note on relations relative to a factorization system.  In Category Theory, pages 249–261. Springer, 1991.
  * [12] S. MacLane.  Categories for the working mathematician, volume 5.  Springer Science & Business Media, 2013.
  * [13] R. Mardare, P. Panangaden, and G. Plotkin.  On the axiomatizability of quantitative algebras.  In 32nd Symposium on Logic in Computer Science (LICS), pages 1–12. IEEE, 2017.
  * [14] A. S. Mashour, M. H. Ghanim, and F. I. Sidky.  Normal fuzzy subgroups.  Information Sciences, 20:53–59, 1990.
  * [15] S. Milius and H. Urbat.  Equational axiomatization of algebras with structure.  In International Conference on Foundations of Software Science and Computation Structures, pages 400–417. Springer, 2019.
  * [16] E. Moggi.  Notions of computation and monads.  Inf. Comput., 93(1):55–92, 1991.
  * [17] J. N. Mordeson, D. S. Malik, and N. Kuroki.  Fuzzy semigroups, volume 131.  Springer, 2012.
  * [18] K. Nishizawa and J. Power.  Lawvere theories enriched over a general base.  Journal of Pure and Applied Algebra, 213(3):377–386, 2009.
  * [19] G. D. Plotkin and J. Power.  Notions of computation determine monads.  In FoSSaCS, volume 2303 of Lecture Notes in Computer Science, pages 342–356. Springer, 2002.
  * [20] G. D. Plotkin and J. Power.  Algebraic operations and generic effects.  Appl. Categorical Struct., 11(1):69–94, 2003.
  * [21] A. Rosenfeld.  Fuzzy groups.  Journal of mathematical analysis and applications, 35(3):512–517, 1971.
  * [22] N. H. Williams.  On Grothendieck universes.  Compositio Mathematica, 21(1):1–3, 1969.
  * [23] O. Wyler.  Lecture notes on topoi and quasitopoi.  World Scientific, 1991.
  * [24] O. Wyler.  Fuzzy logic and categories of fuzzy sets.  In Non-Classical Logics and Their Applications to Fuzzy Subsets, pages 235–268. Springer, 1995.

##  Appendix A Derivations for Proposition 3.16

$\inferrule*[right=Fun]{\vdash e\equiv x\cdot x^{-1}\\\
\inferrule*[right=Exp]{\inferrule*[right=A]{\hskip
1.0pt}{\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,x)}\\\
\inferrule*[right=Exp]{\inferrule*[right=A]{\hskip
1.0pt}{\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,x)}}{\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,x^{-1})}}{\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,x\cdot
x^{-1})}}{\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,e)}$
$\inferrule*[right=Fun]{\vdash y\equiv\quantity(y^{-1})^{-1}\\\
\inferrule*[right=Sub]{\inferrule*[right=Fun]{\vdash x\equiv
y\cdot\quantity(\quantity(y^{-1}\cdot(x\cdot y))\cdot y^{-1})\\\
\inferrule*[right=Sub]{\mathsf{E}\quantity(l,x)\vdash\mathsf{E}\quantity(l,y\cdot(x\cdot
y^{-1}))}{\mathsf{E}\quantity(l,y^{-1}\cdot(x\cdot
y))\vdash\mathsf{E}\quantity(l,y\cdot\quantity(\quantity(y^{-1}\cdot(x\cdot
y))\cdot y^{-1}))}}{\mathsf{E}\quantity(l,y^{-1}\cdot(x\cdot
y))\vdash\mathsf{E}\quantity(l,x)}}{\mathsf{E}\quantity(l,\quantity(y^{-1})^{-1}\cdot\quantity(x\cdot
y^{-1}))\vdash\mathsf{E}\quantity(l,x)}}{\mathsf{E}\quantity(l,y\cdot\quantity(x\cdot
y^{-1}))\vdash\mathsf{E}\quantity(l,x)}$