# Quantum monadic algebras

J. Harding 111The author gratefully acknowledges support of US Army grant
W911NF-21-1-0247.  New Mexico State University, Las Cruces, NM 88003, USA
<EMAIL_ADDRESS>
###### Abstract

We introduce quantum monadic and quantum cylindric algebras. These are
adaptations to the quantum setting of the monadic algebras of Halmos, and
cylindric algebras of Henkin, Monk and Tarski, that are used in algebraic
treatments of classical and intuitionistic predicate logic. Primary examples
in the quantum setting come from von Neumann algebras and subfactors. Here we
develop the basic properties of these quantum monadic and cylindric algebras
and relate them to quantum predicate logic.

  * February 2022

Keywords: cylindric algebra, monadic algebra, quantum logic, subfactor, von
Neumann algebra

##  1 Introduction

Beginning with the work of Boole in the mid 1800’s, algebraic techniques have
played a significant role in logic. The Boolean algebras that grew from
Boole’s work serve as algebraic models of classical propositional calculus.
However, the significance of Boolean algebras has grown past this original
motivation. Boolean algebras are a recognized feature in such diverse areas as
digital electronics, set theory, probability and measure theory, commutative
algebra, and operator theory. In addition to the role played by Boolean
algebras within each subject is the important role they play in moving ideas
between these subjects.

The algebraic techniques used to model classical propositional logic were
extended to algebraic treatments of intuitionistic propositional logic by
Heyting and to modal propositional logic by MacColl, Lewis, and others. In the
first case, the relevant algebras are Heyting algebras, and in the second
Boolean algebras with additional operations.

The algebraic treatment of classical propositional logic was extended in
another direction to give an algebraic view of classical first order logic,
also known as predicate calculus. Roughly, predicate calculus extends
propositional logic by allowing quantification over individuals using $\forall
x$ and $\exists x$.

There were two closely related streams in this direction, one originated by
Halmos [17], and the other by Tarski [22, 23]. Halmos first treated the
1-variable fragment of predicate calculus via what he termed a monadic
algebra, that is, a Boolean algebra with an additional unary operation
$\exists$. Halmos’ extension to an arbitrary number of variables came under
the name of a polyadic algebra, a Boolean algebra with a family of unary
operations satisfying certain properties. Tarski’s cylindric algebras were
closely related to Halmos’ polyadic algebras and again consisted of a Boolean
algebra with a family of additional operations.

It their 1937 paper [3], Birkhoff and von Neumann proposed that the projection
lattice $P(\mathcal{H})$ of a Hilbert space $\mathcal{H}$ be used as an
algebraic semantics for a type of propositional logic arising in quantum
mechanics just as Boolean algebras are used as an algebraic semantics for
classical propositional logic. They termed this the “logic” of quantum
mechanics. This process of replacing a Boolean algebra with a projection
lattice, or more generally an orthomodular lattice [29], can be viewed as the
first of many “quantizations” of classical mathematical notions. Others
include such notions as quantum probability, non-commutative geometry and
quantum groups.

Formal propositional logical systems can be built from projection lattices
$P(\mathcal{H})$, or general orthomodular lattices, rather than Boolean
algebras. This gives an obvious meaning to the term “quantum logic”. There is
a more subtle meaning of the term “quantum logic”, and likely the more
prevalent one. This refers to viewing the algebraic structure $P(\mathcal{H})$
as a vehicle to to carry ideas between areas, much as one looks at Boolean
algebras in this way.

As cornerstones of this general view of quantum logic, the spectral theorem,
Gleason’s theorem, Wigner’s theorem, and Stone’s theorem describe self-adjoint
operators, states, unitary and anti-unitary operators, representations, and
dynamics of a quantum system in terms of certain natural mappings involving
the projection lattice $P(\mathcal{H})$ (see [27, 12, 38, 21] for further
details). In addition, the structure of $P(\mathcal{H})$ has strong ties to
projective geometry that form the basis of the “geometric” approach to quantum
mechanics of Varadarajan [39].

In this note, we introduce quantum analogs of the monadic and cylindric
algebras that are used to study classical predicate calculus. These quantum
monadic algebras and quantum cylindric algebras are projection lattices
$P(\mathcal{H})$, or general orthomodular lattices, equipped with additional
operations satisfying conditions similar to their classical counterparts. We
develop the basic theory of such algebras, including a treatment of their dual
spaces in the spirit of Kripke frames [5], and detail the use of these
algebras for an algebraic treatment of quantum predicate calculus. This
treatment of quantum predicate calculus involves a detailed view of the
projection lattice $P(\mathcal{H}^{\otimes n})$ of a tensor power of a Hilbert
space $\mathcal{H}$.

Further examples of quantum monadic algebras arise from von Neumann algebras.
Any concrete representation of a von Neumann algebra $\mathcal{M}$ in the
bounded operators $B(\mathcal{H})$ of a Hilbert space $\mathcal{H}$ provides a
quantum monadic algebra structure on the projection lattice $P(\mathcal{H})$,
and any subfactor $\mathcal{M}\leq\mathcal{N}$ provides a quantum monadic
algebra structure on the projection lattice $P(\mathcal{N})$. Additionally, it
appears there are ties between quantum cylindric algebras and the commuting
squares [35, 41] of projections that provide a quantum analog of independent
$\sigma$-fields. It is our hope that quantum monadic and cylindric algebras
can eventually serve as a vehicle to import logical perspective into the study
of these operator algebras.

This paper is arranged in the following way. Sections 2–4 provide background
on classical monadic and cylindric algebras, von Neumann algebras, and quantum
logic. The fifth section introduces quantum monadic and quantum cylindric
algebras and develops their basic properties. The sixth section gives a
treatment of quantum predicate calculus in terms of quantum cylindric
algebras. The seventh section deals with an analog of Kripke frame
representations of these algebras. The final section concludes with some
comments and suggestions for further directions.

##  2 Classical monadic and cylindric algebras

For a detailed account of monadic and polyadic algebras see [17] and for an
account of cylindric algebras see [22, 23]. Throughout we use
$\wedge,\vee,\perp,0,1$ for the meet, join, complementation, and bounds of a
Boolean algebra (abbrev. ba).

######  Definition 2.1.

A quantifier $\exists$ on a ba $B$ is a unary operation that satisfies

  * (Q1)

$\exists 0=0$,

  * (Q2)

$p\leq\exists p$,

  * (Q3)

$\exists(p\vee q)=\exists p\vee\exists q$,

  * (Q4)

$\exists\exists p=\exists p$,

  * (Q5)

$\exists(\exists p)^{\perp}=(\exists p)^{\perp}$.

A monadic algebra $(B,\exists)$ is a ba $B$ with a quantifier $\exists$.

######  Lemma 2.2.

[17] The conditions (Q1) – (Q5) are equivalent to (Q1), (Q2), (Q6) where

  * (Q6)

$\exists(p\wedge\exists q)=\exists p\wedge\exists q$.

One can similarly give a definition of a universal quantifier $\forall$. This
definition is dual to that given above, replacing $0$ with 1, $\leq$ with
$\geq$, and join with meet. The relation between universal and existential
quantifiers is the expected, if $\exists$ is an existential quantifier, then
$\forall p=(\exists p^{\perp})^{\perp}$ is a universal quantifier and
conversely.

######  Example 2.3.

Halmos’ motivating example is that of a functional monadic algebra. Suppose
individuals for the single variable we wish to quantify over range in a set
$X$. Then a proposition depending on this variable has a truth value for each
member of $X$, and we assume these truth values belong to a complete ba $B$.
So propositions give functions $f:X\to B$. The quantifier $\exists f$ applied
to the propositional function $f$ is defined to take constant value
$\bigvee\\{f(x):x\in X\\}$ and the universal quantifier $\forall f$ applied to
$f$ takes constant value $\bigwedge\\{f(x):x\in X\\}$. A key step in Halmos’
development of the theory of monadic algebras is to show that each monadic
algebra is a subalgebra of a functional one.

Halmo extends monadic algebras to allow quantification over more than one
variable with his polyadic algebras [17]. This notion is closely related to
the cylindric algebras initiated by Tarski, and developed by Henkin, Monk and
Tarski [22, 23]. Roughly, cylindric algebras amount to polyadic algebras with
equality [14]. We prefer to present things from the more common perspective of
cylindric algebras. Their presentation here is modified somewhat to take
advantage of our current development of quantifiers.

######  Definition 2.4.

A cylindric algebra $(B,c_{i},d_{i,j})$ consists of a ba $B$ with a family
$c_{i}$ of unary operations and $d_{i,j}$ of constants for $i,j\in I$ such
that for each $i,j,k\in I$

  * (C1)

$c_{i}$ is a quantifier,

  * (C2)

$c_{i}c_{j}x=c_{j}c_{i}x$,

  * (C3)

$d_{i,j}=d_{j,i}$ and $d_{i,i}=1$,

  * (C4)

if $j\neq i,k$ then $d_{i,k}=c_{j}(d_{i,j}\wedge d_{j,k})$,

  * (C5)

if $i\neq j$ then $c_{i}(d_{i,j}\wedge x)\wedge c_{i}(d_{i,j}\wedge
x^{\perp})=0$.

The operations $c_{i}$ are called cylindrifications, and the $d_{i,j}$ are
diagonals.

Usually the indexing set $I$ is assumed to be an ordinal and called the
dimension of the cylindric algebra. A diagonal-free cylindric algebra is
simply a ba with a family of pairwise commuting quantifiers.

######  Example 2.5.

The motivating example of an $I$-dimensional cylindric algebra comes from
considering the the ba $B$ of all subsets of $X^{I}$ for some set $X$.
Elements $f\in X^{I}$ are $I$-indexed families in $X$. For a subset
$A\subseteq B$, the cylindrification $c_{i}A$ is the set of all such indexed
families $g$ that agree with some $f\in A$ except possibly in the coordinate
$i$. The diagonal $d_{i,j}=\\{f\mid f(i)=f(j)\\}$.

######  Example 2.6.

One obtains an $\omega$-dimensional cylindric algebra, where $\omega$ is the
natural numbers, by considering a first-order structure based on a set $X$.
For each formula $\phi$ consider the subset of $X^{\omega}$ of all
$\omega$-indexed families $f$ that satisfy $\phi$ when $f(n)$ is used for all
free occurrences of the variable $x_{n}$ in $\phi$. Cylindrification and
diagonals are as in the previous example.

######  Example 2.7.

An $\omega$-dimensional cylindric algebra is obtained by taking a set $\Sigma$
of sentences in a first-order language and considering equivalence classes of
formulas under the relation $\Sigma\vdash\phi\leftrightarrow\psi$ and usual
ordering given by implication. This is a ba with cylindrification $c_{n}$ of
the equivalence class of $\phi$ being the equivalence class $\exists_{n}\phi$
and the diagonal $d_{m,n}$ being the equivalence class of the formula
$x_{m}=x_{n}$.

For further details of these examples, see [22, p. 6-10] and [23, Ch. 4].

######  Definition 2.8.

For an $I$-dimensional cylindric algebra, define for each $i,j\in I$ the
substitution operator $S_{j}^{\,i}x=c_{i}(d_{i,j}\wedge x)$ for $i\neq j$ and
$S_{i}^{\,i}x=x$.

When interpreted in the cylindric algebra of Example 2.7, the substitution
operator $S_{j}^{\,i}$ applied to the equivalence class of a formula $\phi$ is
the equivalence class of the formula obtained from $\phi$ by substituting the
variable $x_{j}$ in place of all free occurrences of $x_{i}$ in the formula
$\phi$. Substitution operators are Boolean endomorphisms of the cylindric
algebra, as one would expect of substitution. They play an important role in
the theory, and are used to develop the connection between relation algebras
and cylindric algebras [23, Ch. 4]. In fact, the notion of cylindric algebras
can be developed by taking these substitution operators as primitive [10].

##  3 von Neumann algebras

Our purpose is treat several details of von Neumann algebras that feature in
later sections. We assume a basic familiarity with the topic as is found in
[27, 28]. In particular, a von Neumann algebra $\mathcal{M}$ is a unital
$*$-subalgebra of the bounded operators $B(\mathcal{H})$ of a Hilbert space
$\mathcal{H}$ that is closed in the weak operator topology. Equivalently, it
is a subalgebra of $B(\mathcal{H})$ that is equal to its double commutant
$\mathcal{M}^{cc}$ where the commutant $S^{c}$ of $S\subseteq B(\mathcal{H})$
is the collection of all operators commuting with each operator in $S$.

######  Definition 3.1.

A self-adjoint projection of a von Neumann algebra $\mathcal{M}$ is an element
$p\in\mathcal{M}$ with $p=p^{*}=p^{2}$.

We often refer to these simply as projections since we do not consider others,
and use $P(\mathcal{M})$ for the projections of $\mathcal{M}$. If
$\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$, then projections of
$\mathcal{M}$ are projection operators of $\mathcal{H}$, so correspond to
closed subspaces of $\mathcal{H}$. An essential point of difference between
von Neumann algebras and general C∗-algebras is that von Neumann algebras have
a rich supply of projections that largely determine their behavior.

######  Definition 3.2.

An ortholattice (abbrev. ol) is a bounded lattice $L$ with a unary operation
$\perp$ that is order-inverting, period two, and satisfies $x\wedge
x^{\perp}=0$ and $x\vee x^{\perp}=1$. An orthomodular lattice (abbrev. oml) is
an ol that satisfies $x\leq y\Rightarrow x\vee(x^{\perp}\wedge y)=y$.

For details of omls, see [29]. There is a close tie between von Neumann
algebras and omls. To describe this, we recall that $C$ is a complete
sublattice of a complete lattice $L$ if $C$ is closed under all (even
infinite) joins and meets of $L$.

######  Theorem 3.3.

If $\mathcal{M}$ is a von Neumann subalgebra of the bounded operators
$B(\mathcal{H})$ of a Hilbert space $\mathcal{H}$, then $P(\mathcal{M})$ is a
complete sub-ortholattice of the complete oml $P(\mathcal{H})$ of projections
of $\mathcal{H}$. Further, $\mathcal{M}$ is generated as a von Neumann algebra
by its projections.

This theorem is found in [27, 28, 18]. In fact, much more is true, although we
won’t need it. Modulo certain small pathologies, Dye’s theorem shows that the
Jordan structure (structure under the symmetrized product $x\circ
y=\frac{1}{2}(xy+yx)$) is determined by the projections $P(\mathcal{M})$. See
[18] for details.

######  Definition 3.4.

The tensor product $\mathcal{H}\otimes\mathcal{K}$ of Hilbert spaces
$\mathcal{H}$ and $\mathcal{K}$ is the metric space completion of their
algebraic tensor product.

If $(a_{i})_{I}$ is an orthonormal basis (abbrev. onb) of $\mathcal{H}$ and
$(b_{j})_{J}$ is an onb of $\mathcal{K}$, it is well-known that $(a_{i}\otimes
b_{j})_{I\times J}$ is an onb of $\mathcal{H}\otimes\mathcal{K}$. This is the
case whether the spaces are separable or not. Also, if $A$ is a bounded
operator of $\mathcal{H}$ and $B$ is a bounded operator of $\mathcal{K}$, then
there is a bounded operator $A\otimes B$ on the tensor product extending the
algebraic tensor product of $A$ and $B$.

######  Definition 3.5.

If $\mathcal{M}$ is a von Neumann subalgebra of $B(\mathcal{H})$ and
$\mathcal{N}$ is a von Neumann subalgebra of $B(\mathcal{K})$, then
$\mathcal{M}\otimes\mathcal{N}$ is the von Neumann subalgebra of
$B(\mathcal{H}\otimes\mathcal{K})$ generated by all $A\otimes B$ where
$A\in\mathcal{M}$ and $B\in\mathcal{N}$.

We consider an infinite tensor product of Hilbert spaces. There are two paths,
often called the complete and the incomplete (or Guichardet) tensor product.
The name refers to the construction, both methods produce Hilbert spaces which
are complete metric spaces. For details, see [40, 37, 33].

The incomplete tensor product assumes a unit vector $\xi_{i}$ is chosen for
each $\mathcal{H}_{i}$ and takes the metric space completion of all pure
tensors $\otimes_{I}v_{i}$ where $v_{i}=\xi_{i}$ for all but finitely many
$i\in I$. The incomplete tensor is written
$\bigotimes^{\xi}_{I}\mathcal{H}_{i}$. When applied to a countable family of
separable Hilbert spaces it produces a separable Hilbert space and the
construction is associative.

The complete tensor product $\bigotimes_{I}\mathcal{H}_{i}$ does not require a
choice of unit vectors. It is much larger. Even when applied to a countable
family of separable Hilbert spaces, it is not separable. The details of the
construction are involved, and won’t be necessary for us. Roughly, one
considers all tensors $\otimes_{I}v_{n}$ for which $\prod_{I}\|v_{i}\|^{2}$ is
finite, then takes a metric space completion with respect to a certain inner
product. While the complete tensor product is in general not associative, the
following holds [40, p. 43].

######  Proposition 3.6.

For $i\in I$ we have $\mathcal{H}_{i}\otimes\bigotimes_{j\neq
i}\mathcal{H}_{j}\simeq\bigotimes_{I}\mathcal{H}_{i}$.

The above ideas can be used to give notions of the tensor product of an
infinite family of von Neumann algebras $\mathcal{M}_{i}$ where
$\mathcal{M}_{i}$ acts on the Hilbert space $\mathcal{H}_{i}$ for $i\in I$. We
won’t require this, and leave the interested reader to see for example [32].

##  4 Quantum logic

There are many versions of propositional “quantum logic” based in the original
idea of Birkhoff and von Neumann [3] of interpreting terms in a projection
lattice $P(\mathcal{H})$ or some generalization (see for example [11, 29]). To
give the bare bones of one approach, take an infinite set $x_{1},x_{2},\ldots$
of propositional variables and connectives $\wedge,\vee,\perp,0,1$. Terms $t$
formed in the usual recursive way are called propositions.

######  Definition 4.1.

Orthologic consists of the propositions $t$ such that the equation $t\approx
1$ is valid in every ol; orthomodular logic consists of such $t$ with
$t\approx 1$ valid in every oml; and quantum logic is such $t$ valid in each
projection lattice $P(\mathcal{H})$.

One issue is the lack of an implication in this treatment, and it appears that
there is no adequate solution to this issue. The common approach is to view
the Sasaki hook as implication.

######  Definition 4.2.

The Sasaki product is given by $x\cdot_{s}y=x\wedge(x^{\perp}\vee y)$ and the
Sasaki hook by $x\to_{s}y=x^{\perp}\vee(x\wedge y)$.

While the Sasaki hook does not share all properties one would like of
implication [29], it and the Sasaki product are residuated [6, 13], meaning
that in any oml we have

$x\cdot_{s}y\leq z\Leftrightarrow y\leq x\to_{s}z.$

There are versions of predicate orthologic, orthomodular logic, and quantum
logic developed using infinite joins and meets to interpret quantifiers. We
refer to [11]. A different approach to predicate quantum logic, more tightly
tied to projection lattices, is given in [42]. In Section 6 we view our
quantum cylindric algebras through this lens.

##  5 Quantum monadic and cylindric algebras

Adaptations of classical notions are straightforward.

######  Definition 5.1.

A monadic ol $(L,\exists)$ is an ol $L$ with unary operation $\exists$ where

  * (Q1)

$\exists 0=0$,

  * (Q2)

$p\leq\exists p$,

  * (Q3)

$\exists(p\vee q)=\exists p\vee\exists q$,

  * (Q4)

$\exists\exists p=\exists p$,

  * (Q5)

$\exists(\exists p)^{\perp}=(\exists p)^{\perp}$.

This is a quantum monadic algebra if $L$ is additionally an oml.

As in the classical case, there is an alternate way to view quantifiers and
monadic ols. A subalgebra $S$ of an ol $L$ is an approximating subalgebra if
for each $a\in L$ there is a largest element $a^{-}\in S$ below $a$ and a
least element $a^{+}\in S$ above $a$. When one of these exists, so does the
other, and they are related by $a^{-}=((a^{\perp})^{+})^{\perp}$. A subalgebra
$S$ of a complete ol $L$ is a complete subalgebra if it is closed under
arbitrary joins and meets as taken $L$. A complete subalgebra $S$ is
approximating with $a^{+}=\bigwedge\\{s\in S:a\leq s\\}$.

######  Proposition 5.2.

Let $L$ be an ol. If $S\leq L$ is approximating, then $\exists a=a^{+}$ and
$\forall a=a^{-}$ are existential and universal quantifiers. Conversely, if
$\exists$ is an existential quantifier, then $S=\\{\exists a:a\in L\\}$ is an
approximating subalgebra.

###### Proof.

Suppose $S$ is approximating and $\exists a=a^{+}$ is the least element in $S$
above $a$. By definition (Q2) holds, and (Q1), (Q4), (Q5) hold since
$0,\exists a,(\exists a)^{\perp}\in S$. Clearly $\exists a\vee\exists b\in S$
and lies above $a\vee b$. But any element of $S$ that lies above $a\vee b$
lies above $\exists a$ and $\exists b$, so lies above $\exists a\vee\exists
b$. So $\exists(a\vee b)=\exists a\vee\exists b$, giving (Q3). So $\exists$ is
a quantifier and by definition $\exists a$ is least in $S$ above $a$. From
basic properties of orthocomplementation $\forall a=(\exists
a^{\perp})^{\perp}$ is largest in $S$ beneath $a$. Conversely, if $\exists$ is
a quantifier on an ol $L$, then (Q1), (Q3), (Q5) give that $S=\\{\exists
a:a\in L\\}$ is a subalgebra, and (Q2), (Q4) imply that $\exists a$ is least
in $S$ above $a$, so $S$ is approximating. ∎

The following observation is trivial, but useful. It requires only the
definition of a pair of residuated maps and their well-known properties [6].
The point is that $\exists a\leq b$ and $a\leq\forall b$ are each equivalent
to saying that there is an element $s$ of the associated approximating
subalgebra with $a\leq s\leq b$.

######  Proposition 5.3.

In a monadic ol, $\exists a\leq b\Leftrightarrow a\leq\forall b$. Therefore
$\exists$ preserves all existing joins and $\forall$ preserves all existing
meets.

Monadic ols and quantum monadic algebras fall under the scope of universal
algebra in the well-known sense [8]. Thus there are standard notions of
homomorphisms, subalgebras and products for these structures. It is easy to
slightly reformulate their definitions to be purely equational, and thus both
form varieties. We won’t look closely at universal algebraic properties, but
provide some basics. Here the difference between monadic ols and quantum
monadic algebras is essential.

######  Proposition 5.4.

Every quantum monadic algebra $(L,\exists)$ is congruence distributive and
congruence permutable; and its congruences correspond to ideals $I$ of $L$
such that for all $a\in I$ and $b\in L$ we have $\exists a\in I$ and
$b\wedge(a\vee b^{\perp})\in I$.

###### Proof.

Since omls are congruence distributive and permutable [7] the same is true of
quantum monadic algebras. Further, [7] there is a correspondence between
congruences $\theta$ of $L$ and ideals $I$ of $L$ that satisfy $a\in I$ and
$b\in L$ imply $b\wedge(a\vee b^{\perp})\in I$. Such ideals are called
$p$-ideals. This associates to $\theta$ the $p$-ideal $0/\theta$ and
associates to the $p$-ideal $I$ the congruence $\\{(x,y):x\vee a=y\vee a\mbox{
for some }a\in I\\}$. If $\theta$ is compatible with $\exists$, then since
$\exists 0=0$ we have $I$ is closed under $\exists$. Conversely, if $I$ is
closed under $\exists$ and $x\vee a=y\vee a$ for some $a\in I$, then $\exists
x\vee\exists a=\exists(x\vee a)=\exists(y\vee a)=\exists y\vee\exists a$, so
$(\exists x,\exists y)\in\theta$. ∎

Elements $x,y$ of an oml $L$ commute if $x=(x\wedge y)\vee(x\wedge
y^{\prime})$. Commutativity is symmetric in an oml. An element $c$ is central
if it commutes with all $x\in L$. An element $c$ is central iff
${\downarrow}\,c$ is a p-ideal. Central elements of an oml correspond to its
direct product decompositions. Since $\exists c=c$ iff $\exists
c^{\perp}=c^{\perp}$, the direct product decompositions of a quantum monadic
algebra correspond to central elements $c$ with $\exists c=c$. See [7] for
details about commuting elements in an oml.

######  Proposition 5.5.

If $(L,\exists)$ is a quantum monadic algebra and $a\in L$ with $a=\exists a$,
then $C(a)=\\{x\in L:a\mbox{ commutes with }x\\}$ is a subalgebra.

###### Proof.

For any $a\in L$, it is known [7, 29] that $C(a)$ is a sub-oml of $L$. It
remains to show that if $a$ commutes with $x$, then $a$ commutes with $\exists
x$. Using that $a,x$ commute and that a quantifier preserves joins, $\exists
x=\exists((x\wedge a)\vee(x\wedge a^{\perp}))=\exists(x\wedge
a)\vee\exists(x\wedge a^{\perp})$. Since $\exists(x\wedge a)\leq\exists a=a$
and $\exists(x\wedge a^{\perp})\leq\exists a^{\perp}=a^{\perp}$, it follows
that from any three of the elements $a,a^{\perp},\exists(x\wedge a)$ and
$\exists(x\wedge a^{\perp})$, one commutes with the other two, so by the
Foulis-Holland theorem [7], the sublattice generated by these four elements is
distributive. Then $(\exists x\vee a)\wedge(\exists x\vee
a^{\perp})=[\exists(x\wedge a^{\perp})\vee a]\wedge[\exists(x\wedge a)\vee
a^{\perp}]$ and using distributivity this is $\exists(x\wedge
a)\vee\exists(x\wedge a^{\perp})=\exists x$. So $\exists x$ commutes with $a$.
∎

######  Proposition 5.6.

If $(L,\exists)$ is a quantum monadic algebra and $a\in L$ with $\exists a=a$,
then the interval $([0,a],\exists)$ is a quantum monadic algebra with
orthocomplement $x^{\\#}=a\wedge x^{\perp}$ and this quantum monadic algebra
is a factor of the subalgebra $(C(a),\exists)$ of $(L,\exists)$.

###### Proof.

Since $a$ is central in $C(a)$, it is well-known that $[0,a]$ and
$[0,a^{\perp}]$ are oml’s under the indicated orthocomplementations and
$L\simeq[0,a]\times[0,a^{\perp}]$ via $x\mapsto(x\wedge a,x\wedge a^{\perp})$.
To see that $\exists$ is a quantifier on $[0,a]$ only (Q5) is non-trivial. But
If $x\leq a$, then $(\exists x)^{\\#}=a\wedge(\exists x)^{\perp}$ is a meet of
elements in the approximating subalgebra, so is in the approximating
subalgebra, giving (Q5). But $\exists x$ commuting with $a$ gives $\exists
x=\exists(x\wedge a)\vee\exists(x\wedge a^{\perp})$, so $\exists x\wedge
a=\exists(x\wedge a)$ and $\exists x\wedge a^{\perp}=\exists(x\wedge
a^{\perp})$, showing $x\mapsto(x\wedge a,x\wedge a^{\perp})$ is a quantum
monadic algebra isomorphism. ∎

######  Example 5.7.

If $L$ is an ol and $S$ is a finite subalgebra of $L$, then $S$ is an
approximating subalgebra, so yields a monadic ol. The diagram below shows a
finite oml and a subalgebra indicated by filled in dots. This happens to be a
Boolean subalgebra.

$p$$q$

For each element $x$, the smallest filled in dot above $x$ is $\exists x$. In
this case, $\exists(p\wedge\exists q)=0$ while $\exists p\wedge\exists q$ is
an atom. This provides an example of a quantum monadic algebra that does not
satisfy (Q6) of Definition 2.1.

######  Example 5.8.

As mentioned, a complete subalgebra of a complete ol produces a monadic ol. A
block of an oml is a maximal Boolean subalgebra. It is known that a block $B$
of $L$ is closed under all existing joins and meets in $L$. So if $L$ is
complete, a block is a complete subalgebra, hence produces a quantum monadic
algebra. Completeness here is required. There is a method to construct an oml
$K(L)$ from a bounded lattice $L$ [19, 29]. Let $L$ be the bounded lattice
$([0,1)\times 2)\oplus(0,1]$, where $[0,1)$ and $(0,1]$ are intervals in the
reals and $\oplus$ is ordinal sum. For $x$ the element $\langle
0<(0,1)\rangle$ in $K(L)$ and $B$ the block of $K(L)$ corresponding to the
maximal chain $C=([0,1)\times\\{0\\})\oplus(0,1]$ of $L$, there is no least
element in this block above $x$. See [19] for details.

######  Example 5.9.

Let $\mathcal{M}$ be a von Neumann subalgebra of the bounded operators
$\mathcal{B}(\mathcal{H})$ of a Hilbert space $\mathcal{H}$. The projections
$P(\mathcal{M})$ are a complete subalgebra of the complete oml
$P(\mathcal{H})$ of projection operators of $\mathcal{H}$. So this provides a
quantum monadic algebra. The underlying oml of this quantum monadic algebra is
$P(\mathcal{H})$ and for any $p\in P(\mathcal{H})$ we have $\exists p$ is the
least projection in $\mathcal{M}$ above $p$.

######  Example 5.10.

Let $L$ be a complete oml. Then its center, the elements that commute with all
others, is a complete subalgebra [7, 29]. Therefore, this provides a quantum
monadic algebra. When applied to the complete oml $P(\mathcal{M})$ of
projections of a von Neumann algebra $\mathcal{M}$, the least central element
$\exists p$ above a projection $p\in\mathcal{M}$ is commonly called the
central carrier of $p$.

######  Example 5.11.

Let $\mathcal{M}$ be a von Neumann subalgebra of the bounded operators
$\mathcal{B}(\mathcal{H})$ of a Hilbert space $\mathcal{H}$. For a projection
$p\in\mathcal{M}$, the corner $p\mathcal{M}p$ [4] is $*$-isomorphic to a von
Neumann subalgebra of $\mathcal{B}(p\mathcal{H})$. As they are von Neumann
algebras, $P(\mathcal{M})$ provides a quantum monadic structure on
$P(\mathcal{H})$ and $P(p\mathcal{M}p)$ such a structure on $P(p\mathcal{H})$.
In fact, this is an instance of Proposition 5.6 with the second an interval of
the first.

######  Example 5.12.

For $\mathcal{N}$ a von Neumann subalgebra of a von Neumann algebra
$\mathcal{M}$ the projections $P(\mathcal{N})$ are a complete subalgebra of
the complete oml $P(\mathcal{M})$ and therefore provides an example of a
quantum monadic algebra. This applies, in particular, when both $\mathcal{M}$
and $\mathcal{N}$ are factors, a situation called a subfactor. Here there is
an extensive literature [16, 24] including an associated Jones tower of
subfactors $\mathcal{N}\leq\mathcal{M}\leq\cdots$ that yields a corresponding
tower
$(L_{1},\exists_{\mathcal{N}})\leq(L_{2},\exists_{\mathcal{M}})\leq\cdots$ of
quantum monadic algebras.

The motivating example of a monadic algebra is the ba $B^{X}$ of all functions
from a set $X$ into a complete ba $B$ with $\exists f$ being the constant
function taking value $\bigvee\\{f(x):x\in X\\}$. A monadic algebra is
functional if it is a subalgebra of such. Halmos [17] showed that each monadic
algebra is isomorphic to a functional one. The situation for monadic ols and
quantum monadic algebras is open, but we make some comments.

######  Remark 5.13.

There is a notion of a monadic Heyting algebra [30] extending Halmos’ monadic
(Boolean) algebras. In [2] it was shown that every monadic Heyting algebra is
isomorphic to a functional one. The method of proof was different from Halmos’
result, using instead a construction involving strong amalgamations and
MacNeille completions. These tools are available for ols, so this is a natural
direction to approach the question.

Every oml is the underlying ol of a quantum monadic algebra, one simply takes
the subalgebra $\\{0,1\\}$. It is a long-open question whether every oml can
be embedded into a complete oml [19, 20], so it would be highly problematic to
show that every quantum monadic algebra can be embedded into a functional one.
One might consider the question for complete quantum monadic algebras whose
approximating subalgebra is Boolean. Here there is a slight opening provided
by the result that omls can be amalgamated over Boolean subalgebras [2], but
this situation is far from clear.

######  Definition 5.14.

A weak $I$-dimensional cylindric ol $(L,c_{i},d_{i,j})$ is an ol $L$ with
family of unary operators $c_{i}$ and constants $d_{i,j}$ where $i,j\in I$
such that for all $i,j,k\in I$

  * (C1)

$c_{i}$ is a quantifier,

  * (C2)

$c_{i}c_{j}x=c_{j}c_{i}x$,

  * (C3)

$d_{i,j}=d_{j,i}$ and $d_{i,i}=1$,

  * (C4)

if $j\neq i,k$ then $d_{i,k}=c_{j}(d_{i,j}\wedge d_{j,k})$,

This is an $I$-dimensional cylindric ol if it additionally satisfies

  * (C5)

if $i\neq j$ then $c_{i}(d_{i,j}\wedge x)\wedge c_{i}(d_{i,j}\wedge
x^{\perp})=0$.

Quantum weak cylindric algebras are those where the underlying ol is an oml.

The motivating example of a classical cylindric algebra is a cylindric set
algebra described in Example 2.5. In Section 6 we discuss a quantum analog of
a cylindric set algebra that informs our choice of axioms. We briefly discuss
the axioms below.

######  Remark 5.15.

The purpose of axiom (C5) in the classical setting, at least in part, is to
provide that the substitution operators $S_{j}^{\,i}x=c_{i}(d_{i,j}\wedge x)$
of Definition 2.8 are Boolean endomorphisms. Indeed, (C5) says that
$S_{j}^{\,i}x\wedge S_{j}^{\,i}x^{\perp}=0$. Since $c_{i}$ and the operation
$d_{i,j}\wedge(\,\cdot\,)$ preserve joins in the classical setting,
$S_{j}^{\,i}$ preserves joins, so (C5) provides that it also preserves
complements, and therefore is a Boolean endomorphism. As we will see, (C5)
does not hold in our quantum analog of cylindric set algebras. But even if it
did, the utility would be limited — in an oml the operation
$d_{i,j}\wedge(\,\cdot\,)$ does not preserve joins, and knowing that $x,y$ are
complements does not provide that they are orthocomplements. More problematic
is that in primary examples of omls, projections of a Hilbert space, there are
often limited supplies of endomorphisms. The story here is not settled, but we
remark that the operation
$S_{j}^{\,i}x=c_{i}(d_{i,j}\wedge(d_{i,j}^{\perp}\vee x))$ reduces to the
standard substitution operator in the classical case, and since quantifiers
and Sasaki products are residuated, this operation preserves joins.

The following section, Section 6, gives in detail a quantum version of a
cylindric set algebra. We conclude this section with a discussion of a
different sort of example. Here, the situation is not fully settled, but in
the simplest setting there is a link between quantum cylindric algebras and
what are called commuting squares of subfactors. For this discussion, we need
some additional background on von Neumann algebras.

######  Definition 5.16.

An element $x$ in a von Neumann algebra $\mathcal{M}$ is positive if
$x=a^{*}a$ for some $a\in\mathcal{M}$. There is a partial ordering $\leq_{sa}$
on the self-adjoint elements given by $x\leq y$ iff $y-x$ is positive. The
positive elements of $\mathcal{M}$ lying in the interval $[0,1]$ of
$\mathcal{M}$ are called effects.

Trivially every projection is positive, and since $1-p$ is the
orthocomplementary projection to $p$, every projection is an effect. The
partial ordering $\leq_{sa}$ when restricted to projections is the usual
ordering of projections. We will customarily write $\leq_{sa}$ simply as
$\leq$ since it will not cause confusion. The key to finer properties relating
to the ordering of effects and projections is what is known as the range
projection $P(x)$. The following is well-known, and is found for example in
[9, Lemma 3.1] and [43, p. 56].

######  Proposition 5.17.

If $x$ is an effect of $\mathcal{B}(\mathcal{H})$, then there is a least
projection $P(x)$ of $\mathcal{H}$ with $x\leq P(x)$. This $P(x)$ is given by
$P(x)={\displaystyle\lim_{n\to\infty}x^{1/n}}$ and belongs to the von Neumann
subalgebra of $\mathcal{B}(\mathcal{H})$ that is generated by $x$.

If $x$ is an effect of $\mathcal{B}(\mathcal{H})$, then $P(x)$ is the least
projection of $\mathcal{H}$ above $x$. If $\mathcal{M}$ is a von Neumann
subalgebra of $\mathcal{B}(\mathcal{H})$, as we noted in Example 5.9, there is
a least projection in $\mathcal{M}$ above the projection $P(x)$, hence a least
projection in $\mathcal{M}$ above $x$.

######  Definition 5.18.

If $x$ is an effect of $\mathcal{B}(\mathcal{H})$ and $\mathcal{M}$ is a von
Neumann subalgebra of $\mathcal{B}(\mathcal{H})$, let $\exists_{\mathcal{M}}x$
be the least projection in $\mathcal{M}$ above $x$.

Note that $\exists_{\mathcal{M}}x=\exists_{\mathcal{M}}P(x)$ and that if
$x\in\mathcal{M}$, then $\exists_{\mathcal{M}}x=P(x)$ since $P(x)$ belongs to
the von Neumann subalgebra generated by $x$ and hence belongs to
$\mathcal{M}$. If we restrict $\exists_{\mathcal{M}}$ to apply to projections
of $\mathcal{H}$, then we obtain the quantum monadic algebra of Example 5.9.
This extends further. If $\mathcal{M}$ is a von Neumann subalgebra of
$\mathcal{B}(\mathcal{H})$ and $\mathcal{N}\leq\mathcal{M}$ is a von Neumann
subalgebra, then consider the restriction of $\exists_{\mathcal{N}}$ to
$\mathcal{M}$. If this $\exists_{\mathcal{N}}$ is further restricted to the
projections of $\mathcal{M}$ we obtain the situation in Example 5.12. We will
not distinguish between these various restrictions, and leave it to context.

######  Definition 5.19.

If $\mathcal{N}$ is a von Neumann subalgebra of $\mathcal{M}$, a positive
linear map $E:\mathcal{M}\to\mathcal{N}$ is a conditional expectation if
$E(1)=1$ and $E(nmn^{\prime})=nE(m)n^{\prime}$ for all
$n,n^{\prime}\in\mathcal{N}$ and $m\in\mathcal{M}$.

For background on conditional expectations see [26, p. 147], [1, Sec. 9], [36,
p. 32.], and [24, 28]. Note that the defining conditions of a conditional
expectation imply that $En=n$ for each $n\in\mathcal{N}$, so a conditional
expectation is idempotent. The following is well-known and found for example
in [26, Thm. 7, Prop. 15].

######  Theorem 5.20.

If $\mathcal{M}$ is a von Neumann algebra with faithful tracial state $\tau$
and $\mathcal{N}$ is a von Neumann subalgebra of $\mathcal{M}$, then there is
a unique conditional expectation $E_{\mathcal{N}}:\mathcal{M}\to\mathcal{N}$
with $\tau E_{\mathcal{N}}=\tau$.

The following is result of Pimsner and Popa [34, Prop. 2.1] is of importance
(see also [31, p. 8]). To state this result, we mention the famous Jones index
[24]. While this provides insight to those familiar with this notion, a
knowledge of this index is not important for purposes here, it is enough to
know that in the our setting this index will be a real number with value at
least 1.

######  Theorem 5.21.

Let $\mathcal{M}$ be a type II1 factor, $\mathcal{N}\leq\mathcal{M}$ be a
subfactor of $\mathcal{M}$ of finite index $k=[\mathcal{M}:\mathcal{N}]$, and
$E_{\mathcal{N}}$ be the unique conditional expectation compatible with the
trace of $\mathcal{M}$. Then $\frac{1}{k}x\leq E_{\mathcal{N}}\,x$ for all
positive $x\in\mathcal{M}$.

This setting of a type II1 factor and a subfactor of finite index is the
important setting for much of Jones’ work related to planar algebras and knots
[24]. In this setting, we relate conditional expectations to quantifiers in
the following.

######  Proposition 5.22.

If $\mathcal{M}$ is a type II1 factor and $\mathcal{N}\leq\mathcal{M}$ is a
subfactor of finite index, then
$\exists_{\mathcal{N}}\,p=\exists_{\mathcal{N}}E_{\mathcal{N}}\,p=P(E_{\mathcal{N}}\,p)$
for each projection $p$ of $\mathcal{M}$.

###### Proof.

Note first that $E_{\mathcal{N}}\,p$ is an effect since $0\leq p\leq 1$ and
the conditional expectation $E_{\mathcal{N}}$ is positive, hence order
preserving, and $E_{\mathcal{N}}\,1=1$. By Theorem 5.21 there is a positive
$\lambda\leq 1$ with $\lambda p\leq E_{\mathcal{N}}\,p$. Since
$P(x)={\displaystyle\lim_{n\to\infty}x^{1/n}}$ for each effect $x$, it follows
that $p=P(\lambda p)\leq P(E_{\mathcal{N}}\,p)$. But
$E_{\mathcal{N}}\,p\in\mathcal{N}$ implies that the least projection
$P(E_{\mathcal{N}}\,p)$ above it belongs to $\mathcal{N}$, and therefore
$P(E_{\mathcal{N}}\,p)$ is the least projection in $\mathcal{N}$ above
$E_{\mathcal{N}}\,p$. This gives $p\leq
P(E_{\mathcal{N}}\,p)=\exists_{\mathcal{N}}E_{\mathcal{N}}\,p$. Then as
$\exists_{\mathcal{N}}E_{\mathcal{N}}\,p$ is a projection in $\mathcal{N}$
above $p$, we have
$\exists_{\mathcal{N}}\,p\leq\exists_{\mathcal{N}}E_{\mathcal{N}}\,p$. For the
other inequality, set $q=\exists_{\mathcal{N}}\,p$. Then $p\leq q$ gives
$\exists_{\mathcal{N}}E_{\mathcal{N}}\,p\leq\exists_{\mathcal{N}}E_{\mathcal{N}}\,q=\exists_{\mathcal{N}}\,q$
since $E_{\mathcal{N}}$ is the identity on $\mathcal{N}$. But
$\exists_{\mathcal{N}}\,q=q$ since $q$ is a projection in $\mathcal{N}$. Then
$q=\exists_{\mathcal{N}}\,p$ gives
$\exists_{\mathcal{N}}E_{\mathcal{N}}\,p\leq\exists_{\mathcal{N}}\,p$. ∎

######  Proposition 5.23.

If $\mathcal{L}$ is a type II1 factor and $\mathcal{M},\mathcal{N}$ are
subfactors of $\mathcal{L}$ of finite index such that the conditional
expectations $E_{\mathcal{M}}:\mathcal{L}\to\mathcal{M}$ and
$E_{\mathcal{N}}:\mathcal{L}\to\mathcal{N}$ commute, then the quantifiers
$\exists_{\mathcal{M}}:P(\mathcal{L})\to P(\mathcal{M})$ and
$\exists_{\mathcal{N}}:P(\mathcal{L})\to P(\mathcal{N})$ commute.

###### Proof.

We first note that $\exists_{\mathcal{M}}$ and $\exists_{\mathcal{N}}$
commute, iff (1) $q\in P(\mathcal{M})$ implies
$\exists_{\mathcal{N}}\,q\in\mathcal{M}\cap\mathcal{N}$ and (2) $r\in
P(\mathcal{N})$ implies
$\exists_{\mathcal{M}}\,r\in\mathcal{M}\cap\mathcal{N}$. Indeed, if the
quantifiers commute and $q\in P(\mathcal{M})$, then
$\exists_{\mathcal{N}}\,q=\exists_{\mathcal{N}}\exists_{\mathcal{M}}\,q=\exists_{\mathcal{M}}\exists_{\mathcal{N}}\,q$
hence belongs to both $\mathcal{M}$ and $\mathcal{N}$. This gives (1) and (2)
is similar. Conversely, suppose (1) and (2) hold. Then for $p\in
P(\mathcal{L})$, setting $q=\exists_{\mathcal{M}}\,p$ we have
$\exists_{\mathcal{N}}\,q$ is a projection in $\mathcal{M}\cap\mathcal{N}$
that lies above $p$ and hence lies above $\exists_{\mathcal{N}}\,p$ and
therefore also above $\exists_{\mathcal{M}}\exists_{\mathcal{N}}\,p$. So
$\exists_{\mathcal{M}}\exists_{\mathcal{N}}\,p\leq\exists_{\mathcal{N}}\exists_{\mathcal{M}}\,p$
and the other inequality follows from using condition (2) instead of (1).
Assume the conditional expectations $E_{\mathcal{M}}$ and $E_{\mathcal{N}}$
commute and $q\in P(\mathcal{M})$. By Proposition 5.22,
$\exists_{\mathcal{N}}\,q=P(E_{\mathcal{N}}\,q)$. Since $q\in\mathcal{M}$ we
have $q=E_{\mathcal{M}}\,q$, and then as conditional expectations commute
$E_{\mathcal{N}}\,q=E_{\mathcal{N}}E_{\mathcal{M}}\,q=E_{\mathcal{M}}E_{\mathcal{N}}\,q$,
giving that $E_{\mathcal{N}}\,q\in\mathcal{M}$. So
$P(E_{\mathcal{M}}\,q)\in\mathcal{M}$, showing that
$\exists_{\mathcal{N}}\exists_{\mathcal{M}}\,q\in\mathcal{M}$. This
establishes (1) and (2) is similar. So $\exists_{\mathcal{M}}$ and
$\exists_{\mathcal{N}}$ commute. ∎

######  Example 5.24.

Originating with a non-commutative version of independent $\sigma$-fields [35,
41], the notion of a commuting square of subfactors plays an important role in
their theory [16, p. 188]. Suppose $\mathcal{L}$ is a type II1 factor with a
finite normal tracial state $\tau$ and $\mathcal{M}$ and $\mathcal{N}$
subfactors of $\mathcal{L}$. If $\mathcal{M}\cap\mathcal{N}=\mathcal{K}$, then
$\mathcal{K},\mathcal{M},\mathcal{N},\mathcal{L}$ is a commuting square of
subfactors if the associated conditional expectations
$E_{\mathcal{M}}:\mathcal{L}\to\mathcal{M}$ and
$E_{\mathcal{N}}:\mathcal{L}\to\mathcal{N}$ commute.

$\mathcal{K}$$\mathcal{N}$$\mathcal{M}$$\mathcal{L}$$\subseteq$$\subseteq$

$\subseteq$

$\subseteq$

Such a commuting square gives a an oml $P(\mathcal{L})$ with two quantifiers,
and by Proposition 5.23 these quantifiers commute. Thus
$(P(\mathcal{L}),\exists_{\mathcal{M}},\exists_{\mathcal{N}})$ is a
2-dimensional diagonal-free quantum cylindric algebra.

##  6 Quantum cylindric set algebras and quantum predicate calculus

The motivating example of a cylindric algebra is that of a cylindric set
algebra. This is the powerset of a product $\prod_{I}X_{i}$ of sets with
cylindrification $c_{i}\,S$ being all choice functions $\alpha$ that agree
with some member of $S$ except possibly in the $i^{th}$-coordinate. If all
factors of this product are equal, so we have a power $X^{I}$, diagonals are
defined as $d_{i,j}=\\{\alpha:\alpha(i)=\alpha(j)\\}$. We provide a quantum
analog using tensor products and powers rather than Cartesian product. Our
treatment shares features with Weaver’s approach to quantum predicate logic
[42].

######  Definition 6.1.

For a Hilbert space $\mathcal{H}$ let $\mathcal{C}(\mathcal{H})$ be its oml of
closed subspaces.

While our eventual purpose is to work in a tensor product
$\mathcal{H}_{1}\otimes\cdots\otimes\mathcal{H}_{n}$, or even an infinite such
tensor product, our considerations involve the relation of one or two factors
to the whole. So it is more convenient to work with a binary or ternary tensor
product and use associativity. To this end, for $A,B$ closed subspaces of
$\mathcal{H}$ and $\mathcal{K}$, respectively, note that $A\otimes B$ is the
closed subspace of $\mathcal{H}\otimes\mathcal{K}$ generated by all $a\otimes
b$ where $a\in A$ and $b\in B$, and we use
$\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$ for the collection of all closed
subspaces of $\mathcal{H}\otimes\mathcal{K}$ of the form $\mathcal{H}\otimes
B$ where $B\in\mathcal{C}(\mathcal{K})$.

######  Proposition 6.2.

The map
$\alpha:\mathcal{C}(\mathcal{K})\to\mathcal{C}(\mathcal{H}\otimes\mathcal{K})$
where $\alpha(B)=\mathcal{H}\otimes B$ is a complete ol-embedding and its
image is $\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$.

###### Proof.

This map is well-defined, order preserving, and onto. Let
$B\in\mathcal{C}(\mathcal{K})$ and take onb’s $(e_{i})_{I}$ of $\mathcal{H}$,
$(f_{j})_{J}$ of $B$ and $(g_{k})_{K}$ of $B^{\perp}$. Then $(e_{i}\otimes
f_{j})_{I\times J}$ is an onb of $\mathcal{H}\otimes B$ and $(e_{i}\otimes
g_{k})_{I\times K}$ of $\mathcal{H}\otimes B^{\perp}$. Thus
$\mathcal{H}\otimes B$ and $\mathcal{H}\otimes B^{\perp}$ are
orthocomplements. Suppose $\bigvee_{I}B_{i}=B$. As $\alpha$ is order
preserving $\bigvee_{I}\mathcal{H}\otimes B_{i}\leq\mathcal{H}\otimes B$. For
$v\in B$ there is a member of the linear span of $\bigcup_{I}B_{i}$
arbitrarily close to $v$, so for any $u\in\mathcal{H}$ there are members of
the linear span of $\bigcup_{I}\mathcal{H}\otimes B_{i}$ arbitrarily close to
$u\otimes v$. It follows that $\mathcal{H}\otimes
B\leq\bigvee_{I}\mathcal{H}\otimes B_{i}$. This shows that $\alpha$ is a
complete ol-homomorphism onto $\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$. It
is easily seen that $\alpha(B)=0$ iff $B=0$. Since $\alpha$ is a homomorphism
between omls, it follows that $\alpha$ is an embedding. ∎

Since $\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$ is a complete subalgebra of
$\mathcal{C}(\mathcal{H}\otimes\mathcal{K})$ we may define the following.

######  Definition 6.3.

For $S\in\mathcal{C}(\mathcal{H}\otimes\mathcal{K})$ let

$\displaystyle\exists_{\mathcal{H}}S$ be the smallest in
$\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$ above $S$,
$\displaystyle\forall_{\mathcal{H}}S$ be the largest in
$\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$ below $S$.

We provide a more explicit description of these operations.

######  Proposition 6.4.

If $S\in\mathcal{C}(\mathcal{H}\otimes\mathcal{K})$, then
$\forall_{\mathcal{H}}S=\mathcal{H}\otimes S_{\mathcal{H}}$ where

$S_{\mathcal{H}}=\\{w\in\mathcal{K}:a\otimes w\in S\mbox{ for all
}a\in\mathcal{H}\\}.$

###### Proof.

It is easily seen that $S_{\mathcal{H}}\in\mathcal{C}(\mathcal{K})$. If $w\in
S_{\mathcal{H}}$, then $a\otimes w\in S$ for all $a\in\mathcal{H}$, so a
generating set of $\mathcal{H}\otimes S_{\mathcal{H}}$ is contained in $S$,
giving $\mathcal{H}\otimes S_{\mathcal{H}}\leq S$. Suppose
$T\in\mathcal{C}(\mathcal{K})$ and $\mathcal{H}\otimes T\leq S$. Then for
$w\in T$ we have $a\otimes w\in S$ for all $a\in\mathcal{H}$, so $T\leq
S_{\mathcal{H}}$. Thus $\mathcal{H}\otimes S_{\mathcal{H}}$ is largest in
$\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$ beneath $S$. ∎

Of course, $\exists_{\mathcal{H}}$ is given by
$\perp\forall_{\mathcal{H}}\perp$, but we can give an explicit description.
For this, note that for an onb $(a_{i})_{I}$ of $\mathcal{H}$, each element
$v\in\mathcal{H}\otimes\mathcal{K}$ can be uniquely expressed as
$\sum_{I}a_{i}\otimes v_{i}$ for some family $(v_{i})_{I}\in\mathcal{K}$. This
family depends on the choice of onb $(a_{i})_{I}$. We use $\langle T\rangle$
for the closed subspace generated by a set $T$.

######  Definition 6.5.

Let $(a_{i})_{I}$ be an onb of $\mathcal{H}$ and
$S\subseteq\mathcal{H}\otimes\mathcal{K}$. Set

$S^{\mathcal{H}}=\langle v_{i}:v\in S\mbox{ and }i\in I\rangle.$

######  Lemma 6.6.

For $(a_{i})_{I}$ an onb of $\mathcal{H}$ and
$S\subseteq\mathcal{H}\otimes\mathcal{K}$ we have $\langle
S\rangle^{\mathcal{H}}=S^{\mathcal{H}}$.

###### Proof.

One containment is obvious. For the other, assume $w\in\langle S\rangle$. Then
for $\epsilon>0$ there is $v=\lambda_{1}v^{1}+\cdots+\lambda_{n}v^{n}$ in the
linear span of $S$ with $\|w-v\|^{2}<\epsilon$. Since $(a_{i})_{I}$ is an onb
and $w-v=\sum_{I}a_{i}\otimes(w_{i}-v_{i})$

$\sum_{I}\|a_{i}\otimes(w_{i}-v_{i})\|^{2}=\|\sum_{I}a_{i}\otimes(w_{i}-v_{i})\|^{2}=\|w-v\|^{2}<\epsilon$

As
$\|a_{i}\otimes(w_{i}-v_{i})\|^{2}=\|a_{i}\|^{2}\|w_{i}-v_{i}\|^{2}=\|w_{i}-v_{i}\|^{2}$,
for any $i_{0}\in I$ we have

$\|w_{i_{0}}-(\lambda_{1}v_{i_{0}}^{1}+\cdots+\lambda_{n}v_{i_{0}}^{n})\|^{2}=\|w_{i_{0}}-v_{i_{0}}\|^{2}<\epsilon$

So $w_{i_{0}}\in S^{\mathcal{H}}$, and it follows that $\langle
S\rangle^{\mathcal{H}}\subseteq S^{\mathcal{H}}$. ∎

######  Proposition 6.7.

If $(a_{i})_{I}$ an onb of $\mathcal{H}$ and
$S\in\mathcal{C}(\mathcal{H}\otimes\mathcal{K})$, then
$\exists_{\mathcal{H}}S=\mathcal{H}\otimes S^{\mathcal{H}}$.

###### Proof.

By definition, $\mathcal{H}\otimes S^{\mathcal{H}}$ belongs to
$\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$. Let $v\in S$. Then $a_{i}\otimes
v_{i}\in\mathcal{H}\otimes S^{\mathcal{H}}$ for each $i\in I$, and as
$v=\sum_{I}a_{i}\otimes v_{i}$ we have $v\in\mathcal{H}\otimes
S^{\mathcal{H}}$. So $S\leq\mathcal{H}\otimes S^{\mathcal{H}}$. It remains to
show that this is the least member of
$\mathcal{H}\otimes\mathcal{C}(\mathcal{K})$ that contains $S$. Suppose
$T\in\mathcal{C}(\mathcal{K})$ and $S\leq\mathcal{H}\otimes T$. Since
$H\otimes T=\langle a_{i}\otimes b:i\in I,b\in T\rangle$, by Lemma 6.6
$(\mathcal{H}\otimes T)^{\mathcal{H}}=\langle(a_{i}\otimes b)_{j}:i,j\in
I,b\in T\rangle$. Thus $S^{\mathcal{H}}\leq(\mathcal{H}\otimes
T)^{\mathcal{H}}=T$. So $\mathcal{H}\otimes
S^{\mathcal{H}}\leq\mathcal{H}\otimes T$ as required. ∎

######  Corollary 6.8.

The definition of $S^{\mathcal{H}}$ is independent of the choice of onb of
$\mathcal{H}$.

###### Proof.

$\exists_{\mathcal{H}}S=\mathcal{H}\otimes S^{\mathcal{H}}$ and by Proposition
6.2 if $\mathcal{H}\otimes T=\mathcal{H}\otimes T^{\prime}$ then
$T=T^{\prime}$. ∎

Consider a tensor product of three Hilbert spaces
$\mathcal{H}\otimes\mathcal{K}\otimes\mathcal{M}$. Since tensor products are
commutative and associative, we can consider this to be a binary tensor
product in various ways, so our notation extends to allow
$\exists_{\mathcal{H}},\exists_{\mathcal{H}\otimes\mathcal{K}},S^{\mathcal{H}},S^{\mathcal{H}\otimes\mathcal{K}}$
and so forth.

######  Proposition 6.9.

For $S\in\mathcal{C}(\mathcal{H}\otimes\mathcal{K}\otimes\mathcal{M})$, we
have
$\exists_{\mathcal{H}}\exists_{\mathcal{K}}S=\exists_{\mathcal{H}\otimes\mathcal{K}}S=\exists_{\mathcal{K}}\exists_{\mathcal{H}}S$.

###### Proof.

Since
$\exists_{\mathcal{H}\otimes\mathcal{K}}S=\mathcal{H}\otimes\mathcal{K}\otimes
T$ for some $T\in\mathcal{C}(\mathcal{K})$, it belongs to both
$\mathcal{H}\otimes\mathcal{C}(\mathcal{K}\otimes\mathcal{M})$ and
$\mathcal{K}\otimes\mathcal{C}(\mathcal{H}\otimes\mathcal{M})$. So
$\exists_{\mathcal{K}}S$ lies beneath
$\exists_{\mathcal{H}\otimes\mathcal{K}}S$, hence
$\exists_{\mathcal{H}}\exists_{\mathcal{K}}S$ lies beneath
$\exists_{\mathcal{H}\otimes\mathcal{K}}S$.

Let $(a_{i})_{I}$ and $(b_{j})_{J}$ be onb’s of $\mathcal{H}$ and
$\mathcal{K}$. Then $(a_{i}\otimes b_{j})_{I\times J}$ is an onb of
$\mathcal{H}\otimes\mathcal{K}$. Each
$v\in\mathcal{H}\otimes\mathcal{K}\otimes\mathcal{M}$ has unique
representations

$v=\sum_{I\times J}a_{i}\otimes b_{j}\otimes v_{i,j}\quad\mbox{ and }\quad
v=\sum_{J}b_{j}\otimes v_{j}$

where $(v_{i,j})_{I\times J}$ is a family in $\mathcal{M}$ and $(v_{j})_{J}$
is a family in $\mathcal{H}\otimes\mathcal{M}$. Uniqueness gives
$v_{j}=\sum_{I}a_{i}\otimes v_{i,j}$. To show that
$\exists_{\mathcal{H}\otimes\mathcal{K}}S\leq\exists_{\mathcal{H}}\exists_{\mathcal{K}}S$,
it is sufficient to show that if $v\in S$ and $i_{0},i_{1}\in I$,
$j_{0},j_{1}\in J$, then $a_{i_{1}}\otimes b_{j_{1}}\otimes
v_{i_{0},j_{0}}\in\exists_{\mathcal{H}}\exists_{\mathcal{K}}S$ since such
elements generate the closed subspace
$\exists_{\mathcal{H}\otimes\mathcal{K}}S$. As $v\in S$ we have $v_{j_{0}}\in
S^{\mathcal{K}}$, and therefore $b_{j_{1}}\otimes
v_{j_{0}}\in\exists_{\mathcal{K}}S$. But

$b_{j_{1}}\otimes v_{j_{0}}=\sum_{I}a_{i}\otimes b_{j_{1}}\otimes
v_{i,j_{0}}.$

Since this element is in $\exists_{\mathcal{K}}S$, then $(b_{j_{1}}\otimes
v_{j_{0}})_{i_{0}}=b_{j_{1}}\otimes v_{i_{0},j_{0}}$ belongs to
$(\exists_{\mathcal{K}}S)^{\mathcal{H}}$. Therefore $a_{i_{1}}\otimes
b_{j_{1}}\otimes v_{i_{0},j_{0}}$ belongs to
$\exists_{\mathcal{H}}\exists_{\mathcal{K}}S$. This provides the equality of
$\exists_{\mathcal{H}}\exists_{\mathcal{K}}S$ and
$\exists_{\mathcal{H}\otimes\mathcal{K}}S$, the equality of
$\exists_{\mathcal{K}}\exists_{\mathcal{H}}S$ and
$\exists_{\mathcal{H}\otimes\mathcal{K}}S$ is by symmetry. ∎

Using the associativity of a finite tensor product
$\mathcal{H}_{1}\otimes\cdots\otimes\mathcal{H}_{n}$, or the fragment of
associativity for a complete infinite tensor product
$\bigotimes_{I}\mathcal{H}_{i}$ given in Proposition 3.6 we have the
following.

######  Theorem 6.10.

For a family $(\mathcal{H}_{i})_{I}$ of Hilbert spaces, the operations
$(\exists_{\mathcal{H}_{i}})_{I}$ are pairwise commuting quantifiers on the
oml $\mathcal{C}(\bigotimes_{I}\mathcal{H}_{i})$ of closed subspaces of the
complete tensor product. So $\mathcal{C}(\bigotimes_{I}\mathcal{H}_{i})$ is an
$I$-dimensional diagonal-free quantum cylindric algebra.

######  Example 6.11.

Let $\mathcal{L}$ be a first-order language consisting of a family
$\mathcal{R}$ of relation symbols, a set $(x_{i})_{I}$ of variables, and
connectives $\wedge,\vee,\perp,0,1,\exists_{x_{i}},\forall_{x_{i}}$. To each
basic formula $r(x_{i_{1}},\ldots,x_{i_{n}})$ we associate a closed subspace
of $\bigotimes_{I}\mathcal{H}_{i}$. Interpreting the connectives
$\wedge,\vee,\perp,0,1,\exists_{x_{i}},\forall_{x_{i}}$ as meet, join,
orthocomplementation, 0, 1, $\exists_{\mathcal{H}_{i}}$ and
$\forall_{\mathcal{H}_{i}}$ gives a subalgebra of the diagonal-free quantum
cylindric algebra described above.

We next consider diagonal elements. Here we require a tensor product
$\mathcal{H}_{1}\otimes\cdots\otimes\mathcal{H}_{n}$ with all factors equal to
the same Hilbert space $\mathcal{H}$. We often write this
$\mathcal{H}^{\otimes n}$. The treatment of diagonals follows the path of
Weaver [42] in using the symmetric tensor product
$\mathcal{H}\otimes_{s}\mathcal{H}$. For example,
$D_{1,2}=(\mathcal{H}_{1}\otimes_{s}\mathcal{H}_{2})\otimes\mathcal{H}_{3}\otimes\cdots\otimes\mathcal{H}_{n}$.
We introduce notation to facilitate computations with these diagonals.

######  Definition 6.12.

For a natural number $n$ and $F\subseteq\\{1,\ldots,n\\}$ let $\mbox{Perm}(n)$
be the group of permutations of $\\{1,\ldots,n\\}$ and $\mbox{Perm}_{n}(F)$ be
the members of $\mbox{Perm}(n)$ that fix all elements not in $F$.

Let $(a_{i})_{I}$ be an onb of $\mathcal{H}$, then the
$a_{\alpha(1)}\otimes\cdots\otimes a_{\alpha(n)}$, where $\alpha\in I^{n}$ is
an $n$-tuple of members of $I$, is an onb of $\mathcal{H}^{\otimes n}$. Thus
each $v\in\mathcal{H}^{\otimes n}$ has a unique representation

$v=\sum_{\alpha\in I^{n}}\lambda_{\alpha}\,a_{\alpha_{1}}\otimes\cdots\otimes
a_{\alpha_{n}}$

Note that if $\sigma\in$ Perm$(n)$ and $\alpha\in I^{n}$, the composite
$\alpha\sigma=(\alpha_{\sigma(1)},\ldots,\alpha_{\sigma(n)})$ is in $I^{n}$.

######  Definition 6.13.

For $(a_{i})_{I}$ an onb of $\mathcal{H}$ and $F\subseteq\\{1,\ldots,n\\}$ let

$D_{F}=\langle\,\sum_{I^{n}}\lambda_{\alpha}a_{\alpha(1)}\otimes\cdots\otimes
a_{\alpha(n)}:\lambda_{\alpha}=\lambda_{\alpha\sigma}\mbox{ for all
$\sigma\in$ {\em Perm}${}_{n}(F)$}\rangle$

$D_{F}$ is isomorphic in a natural way to the closed subspace of
$\mathcal{H}^{\otimes n}$ formed by replacing the tensor product of the
factors belonging to $F$ with their symmetric tensor product. So the
definition is independent of the choice of onb. We often use $D_{i,j}$ for
$D_{\\{i,j\\}}$ and so forth. Trivially $D_{i,j}=D_{j,i}$ and
$D_{i,i}=\mathcal{H}^{\otimes n}$. Also, since the group
$\mbox{Perm}_{n}(\\{i,j,k\\})$ is generated by the transposition of $i,j$ and
the transposition of $j,k$, it follows that $D_{i,j}\cap D_{j,k}=D_{i,j,k}$.

######  Proposition 6.14.

If $j\neq i,k$, then $D_{i,k}=\exists_{\mathcal{H}_{j}}(D_{i,j}\cap D_{j,k})$.

###### Proof.

For convenience assume $n=4$. We will show that
$D_{1,2}=\exists_{\mathcal{H}_{3}}(D_{1,3}\cap D_{3,2})$. That
$D_{1,1}=\exists_{\mathcal{H}_{2}}(D_{1,2}\cap D_{2,1})$ is similar and other
cases are symmetric. Note that $D_{1,2}=\langle a_{i}\otimes
a_{j}+a_{j}\otimes a_{i}:i,j\in
I\rangle\otimes\mathcal{H}_{3}\otimes\mathcal{H}_{4}$. So $D_{1,2}$ belongs to
$\mathcal{H}_{3}\otimes\mathcal{C}(\mathcal{H}_{1}\otimes\mathcal{H}_{2}\otimes\mathcal{H}_{4})$
and lies above $D_{1,2,3}$. Thus $\exists_{\mathcal{H}_{3}}(D_{1,3}\cap
D_{3,2})\leq D_{1,2}$. Let $p,q,r,s\in I$ and set

$v=(a_{p}\otimes a_{q}\otimes a_{q}+a_{q}\otimes a_{p}\otimes
a_{q}+a_{q}\otimes a_{q}\otimes a_{p})\otimes a_{s}.$

Then $v\in D_{1,2,3}$. To compute $(D_{1,2,3})^{\mathcal{H}_{3}}$ express
$v=\sum_{I}a_{i}\otimes v_{i}$ where $a_{i}$ appears in the third tensor
factor. If $p\neq q$, then $v_{q}=(a_{p}\otimes a_{q}+a_{q}\otimes
a_{p})\otimes a_{s}$ and if $p=q$ then $v_{q}=3a_{p}\otimes a_{p}\otimes
a_{p}\otimes a_{s}$. In either case, the generator $(a_{p}\otimes
a_{q}+a_{q}\otimes a_{p})\otimes a_{r}\otimes a_{s}$ of $D_{1,2}$ belongs to
$\mathcal{H}_{3}\otimes(D_{1,2,3})^{\mathcal{H}_{3}}=\exists_{\mathcal{H}_{3}}(D_{1,3}\cap
D_{3,2})$. ∎

######  Proposition 6.15.

$\exists_{\mathcal{H}_{1}}(D_{1,2}\cap
S)\cap\exists_{\mathcal{H}_{1}}(D_{1,2}\cap S^{\perp})=0$ need not hold.

###### Proof.

We give an example with $n=2$, but it can be modified for any $n\geq 2$. Let
$(a_{i})_{I}$ be an onb of $\mathcal{H}$ and $p,q,r$ be distinct elements of
$I$. Set $v=a_{p}\otimes a_{q}+a_{q}\otimes a_{p}$ and $w=a_{p}\otimes
a_{r}+a_{r}\otimes a_{p}$. Note that $v,w$ are orthogonal and both belong to
$D_{1,2}$. Set $S=\langle v\rangle$. Then $S^{\mathcal{H}_{1}}=\langle
a_{p},a_{q}\rangle$. So $\exists_{\mathcal{H}_{1}}(D_{1,2}\cap
S)=\exists_{\mathcal{H}_{1}}S=\mathcal{H}_{1}\otimes\langle
a_{p},a_{q}\rangle$. Since $w\in D_{1,2}\cap S^{\perp}$ we have
$a_{p}\in(D_{1,2}\cap S^{\perp})^{\mathcal{H}_{1}}$. It follows that
$\mathcal{H}_{1}\otimes\langle a_{p}\rangle$ is contained in both
$\exists_{\mathcal{H}_{1}}(D_{1,2}\cap S)$ and
$\exists_{\mathcal{H}_{1}}(D_{1,2}\cap S^{\perp})$. ∎

Remark 5.15 discussed substitution in relation to axiomatics of quantum
cylindric algebras. It would be of interest to explore the matter further with
an eye towards a version of quantum predicate calculus closed tied to its
origins in subspace lattices.

######  Remark 6.16.

We indicated that diagonal-free versions of quantum set algebras can be
obtained in the infinite-dimensional setting using the complete tensor product
$\bigotimes_{I}\mathcal{H}_{i}$. Our discussion of diagonals carries over to a
complete tensor power $\mathcal{H}^{\otimes I}$ without difficulty. If one
wanted, the incomplete tensor product could also be used with no essential
difficulties, but the complete tensor product seems more natural here.

######  Remark 6.17.

The path taken to realize diagonal-free quantum cylindric algebras can be
extended to the von Neumann algebra setting. Suppose
$\mathcal{M}_{1},\ldots\mathcal{M}_{n}$ are concrete von Neumann algebras.
Then we can define the quantifier $\exists_{\mathcal{M}_{1}}$ on the
projections $P(\mathcal{M}_{1}\otimes\cdots\otimes\mathcal{M}_{n})$ via the
complete subalgebra $1\otimes
P(\mathcal{M}_{2}\otimes\cdots\otimes\mathcal{M}_{n})$ and so forth. We won’t
develop this further here.

##  7 Kripke frames

In their work on canonical extensions, Jónsson and Tarski [25] developed a
general method to associate to any ba with additional operators $\mathcal{A}$
a relational structure $\mathcal{A}_{+}$, now commonly called a Kripke frame,
and to any relational structure $\mathcal{X}$, a complete atomic ba with
additional operators $\mathcal{X}^{+}$, such that
$\mathcal{A}\leq(\mathcal{A}_{+})^{+}$. They used this to show that each
cylindric algebra can be embedded into a complete atomic cylindric algebra.
The frame $\mathcal{A}_{+}$ is constructed from $\mathcal{A}$ via ultrafilters
of the ba $\mathcal{A}$ in a way that will not be directly applicable here.
Passage from $\mathcal{X}$ to $\mathcal{X}^{+}$ is given as follows.

######  Definition 7.1.

A relational structure $\mathcal{X}=(X,(R_{i})_{I})$ is a set $X$ with a
family $(R_{i})_{I}$ of $n_{i}+1$-ary relations on $X$. Its complex algebra
$\mathcal{X}^{+}$ is the powerset $P(X)$ with $n_{i}$-ary operations given by
relational image $f_{i}(A_{1},\ldots,A_{n_{i}})=R_{i}[A_{1},\ldots,A_{n_{i}}]$
where

$R_{i}[A_{1},\ldots,A_{n_{i}}]=\\{x:R_{i}(a_{1},\ldots,a_{n_{i}},x)\mbox{ for
some }a_{1}\in A_{1},\ldots,a_{n_{i}}\in A_{n_{i}}\\}.$

Conditions for $\mathcal{X}^{+}$ to be a monadic algebra are in [25] and for
$\mathcal{X}^{+}$ to be a cylindric algebra in [22, Thm. 2.7.40]. An
abbreviated account is given below.

######  Proposition 7.2.

For a binary relational structure $\mathcal{X}=(X,R)$, the complex algebra
$\mathcal{X}^{+}$ is a monadic algebra iff $R$ is an equivalence relation on
$X$. For a relational structure $\mathcal{X}=(X,(R_{i})_{I})$, the complex
algebra $\mathcal{X}^{+}$ is a diagonal-free cylindric algebra iff the $R_{i}$
are pairwise commuting equivalence relations.

We make first steps to develop the connection between monadic and cylindric
ol’s and frames. “Under the hood” our approach departs from that of Jónsson
and Tarski in that it is based on MacNeille completions rather than canonical
extensions.

######  Definition 7.3.

An orthoframe $\mathcal{X}=(X,\perp)$ is a set with an irreflexive, symmetric
binary relation called an orthogonality relation. For $A\subseteq X$ set

$A^{\perp}=\\{x\in X:a\perp x\mbox{ for all }a\in A\\}$

and let $\mathcal{L}(X,\perp)=\\{A:A=A^{\perp\perp}\\}$ be all biorthogonally
closed subsets of $X$.

It is well-known [29] that $\mathcal{L}(X,\perp)$ is a complete ol with meets
given by intersections, joins by the biorthogonal of the union, and with
$A^{\perp}$ the orthocomplement of $A$. If $L$ is an ol, then $(L^{*},\perp)$
is an orthoframe where $L^{*}=L\setminus\\{0\\}$ and $x\perp y$ iff $x\leq
y^{\perp}$. In this case $\mathcal{L}(L^{*},\perp)$ is the MacNeille
completion of $L$, and in particular is a complete ol containing $L$ as a sub-
ol. Unfortunately, it is not possible to give first-order properties of
$(X,\perp)$ equivalent to $\mathcal{L}(X,\perp)$ being an oml [15].

######  Definition 7.4.

A relational structure $(X,\perp,R)$ is a monadic orthoframe if $\perp$ and
$R$ are binary relations on $X$ that satisfy

  * (M1)

$\perp$ is an orthogonality relation,

  * (M2)

$R$ is reflexive and transitive,

  * (M3)

for each $x\in X$, the set $R[\\{x\\}]^{\perp}$ is closed under $R$.

Here (M3) means $R[R[\\{x\\}]^{\perp}]\subseteq R[\\{x\\}]^{\perp}$ for each
$x\in X$. Since (M2) gives reflexivity, set inclusion here can be replaced
with equality.

For any set $X$, the relation $x\perp y\Leftrightarrow x\neq y$ is an
orthogonality relation that we denote $\neq$. The ol $\mathcal{L}(X,\neq)$ is
the powerset of $X$. If we consider orthoframes restricted to having this
classical orthogonality relation, we obtain exactly the frames corresponding
to classical monadic algebras as the following result shows.

######  Proposition 7.5.

For any set $X$, the frame $(X,\neq,R)$ is a monadic orthoframe iff $R$ is an
equivalence relation on $X$.

###### Proof.

Suppose that $R$ is an equivalence relation on $X$. Clearly $\neq$ is an
orthogonality relation and $R$ is reflexive and transitive. For $x\in X$ we
have $R[\\{x\\}]$ is the equivalence class of $x$, and since $\perp$ is
$\neq$, we have $R[\\{x\\}]^{\perp}$ is the set-theoretic complement of this
equivalence class, which is indeed closed under $R$. So $(X,\neq,R)$ is a
monadic orthoframe. Conversely, suppose $(X,\neq,R)$ is a monadic orthoframe.
By definition $R$ is reflexive and transitive. Suppose $x\,R\,y$. If it is not
the case that $y\,R\,x$, then $y$ belongs to the set-theoretic complement of
$R[\\{x\\}]$ which in the current setting is given by $R[\\{x\\}]^{\perp}$.
Since $R[\\{x\\}]^{\perp}$ is closed under $R$ and $x\not\in
R[\\{x\\}]^{\perp}$ (since $R$ is reflexive and $\perp$ is irreflexive), it is
not the case that $y\,R\,x$. So $R$ is symmetric, hence an equivalence
relation. ∎

Note that Definition 7.4 is within the first-order language of the relational
structure since quantification in (M3) is over individuals in $X$ and not
subsets of $X$. However, the conditions are sufficient to give the version of
(M3) when quantified over all subsets and not just singletons. This is the
content of the first statement of the following result.

######  Lemma 7.6.

Let $(X,\perp,R)$ be a monadic orthoframe. Then for each $A\subseteq X$

  * (1)

$R[A]^{\perp}$ is closed under $R$,

  * (2)

$R[A]^{\perp\perp}$ is closed under $R$,

  * (3)

$R[A^{\perp\perp}]\subseteq R[A]^{\perp\perp}$.

Hence, as $R$ is reflexive, $R[R[A]^{\perp}]=R[A]^{\perp}$ and
$R[R[A]^{\perp\perp}]=R[A]^{\perp\perp}$.

###### Proof.

(1) Since $R[A]=\bigcup\\{R[\\{x\\}]:x\in A\\}$ then
$R[A]^{\perp}=\bigcap\\{R[\\{x\\}]^{\perp}:x\in A\\}$. So if $y\in
R[A]^{\perp}$ and $y\,R\,z$, then $y\in R[\\{x\\}]^{\perp}$ for all $x\in A$,
and since each $R[\\{x\\}]^{\perp}$ is closed under $R$, it follows that $z\in
R[A]^{\perp}$. This gives the first statement and as $R$ is reflexive, gives
$R[R[A]^{\perp}]=R[A]^{\perp}$. (2) It is convenient to write $A^{\perp}$ as
$OA$ and $R[A]$ as $RA$. Then the first statement gives $RORA=ORA$ for all
$A\subseteq X$. Then

$ROORA=RORORA=ORORA=OORA$

giving $R[R[A]^{\perp\perp}]=R[A]^{\perp\perp}$ and hence that
$R[A]^{\perp\perp}$ is closed under $R$. (3) Since $R$ is reflexive,
$A\subseteq R[A]$, and $A\subseteq A^{\perp\perp}$ since biorthogonal is a
closure operator. So $R[A^{\perp\perp}]\subseteq
R[R[A]^{\perp\perp}]=R[A]^{\perp\perp}$ where the final equality uses the
second statement. ∎

######  Definition 7.7.

For $\mathcal{X}=(X,\perp,R)$ an orthoframe with additional binary relation
$R$, let $\exists_{R}$ be the unary operation on $\mathcal{L}(X,\perp)$ where
$\exists_{R}A=R[A]^{\perp\perp}$. Let
$\mathcal{L}(\mathcal{X})=(\mathcal{L}(X,\perp),\exists_{R})$.

######  Proposition 7.8.

If $\mathcal{X}=(X,\perp,R)$ is a monadic orthoframe,
$\mathcal{L}(\mathcal{X})$ is a monadic ol.

###### Proof.

Equations (Q1) and (Q2) are trivial. For (Q3), note that $\exists_{R}$ is
order preserving, so $\exists_{R}A\vee\exists_{R}B\leq\exists_{R}(A\vee B)$.
For the other inequality, using Lemma 7.6.3 and that the biorthogonal is a
closure operator

$\exists_{R}(A\vee B)=R[(A\cup B)^{\perp\perp}]^{\perp\perp}\subseteq R[A\cup
B]^{\perp\perp\perp\perp}=R[A\cup B]^{\perp\perp}.$

Then since $R[A\cup B]=R[A]\cup R[B]$, using basic properties of the
biorthogonal

$R[A\cup B]^{\perp\perp}=(R[A]\cup
R[B])^{\perp\perp}\subseteq(R[A]^{\perp\perp}\cup
R[B]^{\perp\perp})^{\perp\perp}.$

This last expression is $\exists_{R}A\vee\exists_{R}B$, providing (Q3). Using
Lemma 7.6.2 and that biorthogonal is a closure operator gives

$\exists_{R}\exists_{R}A=R[R[A]^{\perp\perp}]^{\perp\perp}=R[A]^{\perp\perp\perp\perp}=R[A]^{\perp\perp}=\exists_{R}A,$

yielding (Q4). Finally, using Lemma 7.6.1 and the fact that
$A^{\perp\perp\perp}=A^{\perp}$

$\exists_{R}(\exists_{R}A)^{\perp}=R[R[A]^{\perp\perp\perp}]^{\perp\perp}=R[R[A]^{\perp}]^{\perp\perp}=R[A]^{\perp\perp\perp}=R[A]^{\perp}.$

This gives the final condition (Q5). ∎

######  Proposition 7.9.

Every monadic ol can be embedded into $\mathcal{L}(\mathcal{X})$ for some
monadic orthoframe $\mathcal{X}$ and every complete monadic ol is isomorphic
to some such $\mathcal{L}(\mathcal{X})$.

###### Proof.

Given a monadic ol $(L,\exists)$ set $\mathcal{X}=(L^{*},\perp,R)$ where
$L^{*}=L\setminus\\{0\\}$, $x\perp y$ iff $x\leq y^{\perp}$, and $x\,R\,y$ iff
$y\leq\exists x$. Since $\exists$ is increasing and idempotent, $R$ is
reflexive and transitive. Note that $R[\\{x\\}]=\\{y:y\leq\exists x\\}$, so
$R[\\{x\\}]^{\perp}=\\{y:y\leq(\exists x)^{\perp}\\}$. So if $y\in
R[\\{x\\}]^{\perp}$ and $y\,R\,z$, then $y\leq(\exists x)^{\perp}$ and
$z\leq\exists y$, so $z\leq\exists(\exists x)^{\perp}$. Then by (Q5)
$z\leq(\exists x)^{\perp}$, so $z\in R[\\{x\\}]^{\perp}$. Thus
$R[\\{x\\}]^{\perp}$ is closed under $R$.

It is well-known that $L$ embeds into $\mathcal{L}(X,\perp)$ via
$\alpha(a)={\downarrow a}=\\{x:x\leq a\\}$ and $\alpha$ is an ol-isomorphism
if $L$ is complete. It remains to show that $\alpha(\exists
a)=\exists_{R}(\alpha a)$. But $\exists_{R}(\alpha
a)=R[\,{\downarrow}a\,]^{\perp\perp}=({\downarrow}\exists
a)^{\perp\perp}={\downarrow}\exists a=\alpha(\exists a)$ since each principle
downset is biorthogonally closed. ∎

######  Definition 7.10.

For a set $I$, a relational structure
$\mathcal{X}=(X,\perp,(R_{i})_{I},(D_{i,j})_{I\times J})$ is an
$I$-dimensional weak cylindric orthoframe if for each $i,j,k\in I$

  * (W1)

$(X,\perp,R_{i})$ is a monadic orthoframe,

  * (W2)

$R_{i}$ commutes with $R_{j}$,

  * (W3)

$D_{i,j}=D_{j,i}=D_{i,j}^{\perp\perp}$ and $D_{i,i}=X$,

  * (W4)

if $j\neq i,k$ then $R_{j}[D_{i,j}\cap D_{j,k}]=D_{i,k}$.

The definition of $\mathcal{L}(\mathcal{X})$ for a weak cylindric orthoframe
$\mathcal{X}$ is the obvious one.

######  Proposition 7.11.

If $\mathcal{X}=(X,\perp,(R_{i})_{I},(D_{i,j})_{I\times J})$ is a weak
cylindric orthoframe, then $\mathcal{L}(\mathcal{X})$ is a weak cylindric ol
and each weak cylindric ol is a subalgebra of some $\mathcal{L}(\mathcal{X})$.

###### Proof.

To show $\mathcal{L}(\mathcal{X})$ is a weak cylindric ol we show that the
quantifiers $\exists_{R_{i}}$ commute. The other properties are fairly
obvious. Using Lemma 7.6.3 and that $R_{i},R_{j}$ commute,

$R_{i}[R_{j}[A]^{\perp\perp}]^{\perp\perp}\subseteq
R_{i}[R_{j}[A]]^{\perp\perp}=R_{j}[R_{i}[A]]^{\perp\perp}\subseteq
R_{j}[R_{i}[A]^{\perp\perp}]^{\perp\perp}.$

So $\exists_{R_{i}}\exists_{R_{j}}A\subseteq\exists_{R_{j}}\exists_{R_{i}}A$,
and by symmetry, we have equality. Conversely, given a weak cylindric ol
$(L,(\exists_{i})_{I},(d_{i,j})_{I\times J})$, let
$\mathcal{X}=(L^{*},\perp,(R_{i})_{I},(D_{i,j})_{I\times J})$ where $L^{*}$,
$\perp$, and each $R_{i}$ is as in the proof of Proposition 7.9 and
$D_{i,j}={\downarrow}\,d_{i,j}$. We show $R_{i},R_{j}$ commute, the other
properties are obvious. If $x\,R_{i}\,y$ and $y\,R_{j}z$, then
$y\leq\exists_{i}\,x$ and $z\leq\exists_{j}\,y$. Then
$z\leq\exists_{j}\exists_{i}\,x$. Since the quantifiers commute,
$z\leq\exists_{i}\exists_{j}\,x$, so for $w=\exists_{j}\,x$ we have
$x\,R_{j}\,w$ and $w\,R_{i}\,z$. Thus $R_{i}\circ R_{j}\subseteq R_{j}\circ
R_{i}$, and by symmetry they are equal. ∎

##  8 Conclusion

We introduced monadic and cylindric ol’s and developed their basic theory. The
connection between these structures and von Neumann algebras was illustrated
through a series of examples. A version of quantum cylindric set algebras
based in tensor powers of a Hilbert space was developed and related to
algebraic quantum predicate calculus. Issues with substitution in this context
were observed. A form of Kripke frames for these structures was developed.

Items related to the discussion deserve additional attention. A path was
suggested to determine if each monadic ol is isomorphic to a functional one.
The relationship between commuting squares of subfactors and quantum cylindric
algebras should be settled. Axiomatic issues were found in quantum cylindric
algebras related to substitution operations and quantum predicate calculus.
Kripke frames developed for monadic ols could be developed from the
perspective of canonical extensions rather than MacNeille completions as was
done here.

The broad aim is to use the structures introduced here as vehicles to view
modern developments in operator algebras from the perspective of order theory,
geometry, and logic.

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