Biquasigroups linear over a group
_Wieslaw A. Dudek and Robert A. R. Monzo_

Abstract. We determine the structure of biquasigroups $(Q,\circ,*)$ satisfying
variations of Polonijo’s Ward double quasigroup identity $(x\circ z)*(y\circ
z)=x*y$, including those that are linear over a group.  ††2010 Mathematics
Subject Classification: 20M15, 20N02 ††Keywords: Biquasigroups, linear
quasigroups, Ward quasigroups.

##  1\. Introduction

J.M. Cardoso and C.P. da Silva, inspired by Ward’s paper [11] on postulating
the inverse operations in groups, introduced in [1] the notion of Ward
quasigroups as quasigroups $(Q,\circ)$ containing an element $e$ such that
$x\circ x=e$ for all $x\in Q$, and satisfying the identity $(x\circ y)\circ
z=x\circ(z\circ(e\circ y))$. Polonijo [8] proved that these two conditions can
be replaced by the identity:

$\displaystyle(x\circ z)\circ(y\circ z)=x\circ y.$ (1)

In [1] it is proved that if $(Q,\circ)$ is a Ward quasigroup, then
$(Q,\cdot)$, where $x\cdot y=x\circ(e\circ y)$, is a group in which $e=x\circ
x$ and $x^{-1}=e\circ x$ for all $x\in Q$. Also, $x\circ e=x$, $e\circ(e\circ
x)=x$ and $e\circ(x\circ y)=y\circ x$. Conversely, if $(Q,\cdot)$ is a group,
then $Q$ with the operation $x\circ y=x\cdot y^{-1}$ is a Ward quasigroup (cf.
[11]). Other characterizations of Ward quasigroups can be found in [2] and
[10], some applications in [5]. Note that the Ward quasigroups corresponding
to commutative groups sometimes are called subtractive quasigroups (cf. [6]
and [12]). A Ward quasigroup $(Q,\circ)$ is subtractive if and only if it is
medial (that is, it satisfies the identity $(x\circ y)\circ(z\circ w)=(x\circ
z)\circ(y\circ w))$ if and only if it is left modular (that is, it satisfies
the identity $x\circ(y\circ z)=z\circ(y\circ x)$) (cf. Lemma 2.4, [3]).

A biquasigroup, i.e. an algebra of the form $(Q,\circ,*)$ where $(Q,\circ)$
and $(Q,*)$ are quasigroups, is called a Ward double quasigroup if it
satisfies the identity

$\displaystyle(x\circ z)*(y\circ z)=x*y.$ (2)

Obviously each Ward quasigroup $(Q,\circ)$ can be considered as a Ward double
quasigroup of the form $(Q,\circ,\circ)$. Ward double quasigroups have a
similar characterization as Ward quasigroups.

######  Theorem 1.1.

(cf. [7]) A biquasigroup $(Q,\circ,*)$ is a Ward double quasigroup if and only
if there is a group $(Q,+)$ and bijections $\alpha,\beta$ on $Q$ such that
$x\circ y=x-\beta y$ and $x*y=\alpha(x-y)$.

Note that Ward double quasigroups are distinct from the double Ward
quasigroups considered by Fiala (cf. [4]).

Let us consider the identity (2). Keeping the variables $x,y$ and $z$ the same
and varying only the quasigroup operations $\circ$ and $*$, there are sixteen
possible identities. Eight of these have reversible versions obtained by
replacing the operation $\circ$ with the operation $*$ and, simultaneously,
replacing the operation $*$ with the operation $\circ$.

For example, the identity $(x\circ z)*(y*z)=x\circ y$ has the reversible
version $(x*z)\circ(y\circ z)=x*y$. So, if we are to consider all possible
versions of Theorem 1.1, we need to explore the following identities:

$\displaystyle(x\circ z)\circ(y\circ z)=x\circ y,$ (3) $\displaystyle(x\circ
z)\circ(y\circ z)=x*y,$ (4) $\displaystyle(x\circ z)\circ(y*z)=x\circ y,$ (5)
$\displaystyle(x\circ z)*(y\circ z)=x\circ y,$ (6) $\displaystyle(x\circ
z)\circ(y*z)=x*y,$ (7) $\displaystyle(x\circ z)*(y*z)=x\circ y,$ (8)
$\displaystyle(x\circ z)*(y*z)=x*y.$ (9)

The biquasigroup $(Q,\circ,\circ)$ satisfies identity (3) if and only if
$(Q,\circ)$ is a Ward quasigroup. Our interest is in finding non-trivial
models of the other six identities, where ‘non-trivial’ means that the set $Q$
has more than one element. In particular, since Ward quasigroups are
unipotent, we will be interested in biquasigroups $(Q,\circ,*)$ where
$(Q,\circ)$ or $(Q,*)$ is unipotent, both are unipotent or when one or both
are Ward quasigroups.

Note that a biquasigroup $(Q,\circ,*)$, where $(Q,*)$ is a commutative group
and $x\circ y=x*y^{-1}$ satisfies identities (1) through (9) if and only if
$(Q,*)$ is a Boolean group.

##  2\. Main Results

We will now characterize the biquasigroups satisfying the identities (2) to
(9). First we will describe their general properties then we will characterize
biquasigroups linear over a group and satisfying identities (2) to (9).

1. Recall that a quasigroup $(Q,\cdot)$ is linear over a group (cf. [9]) if there exists a group $(Q,+)$, its automorphisms $\varphi,\psi$ and $a\in Q$ such that $x\cdot y=\varphi x+a+\psi y$ for all $x,y\in Q$. Consequently, a biquasigroup $(Q,\circ,*)$ will be called linear over a group if both its quasigroups $(Q,\circ)$ and $(Q,*)$ are linear over the same group, i.e. if there is a group $(Q,+)$, its automorphisms $\varphi,\psi,\alpha,\beta$ and elements $a,b\in Q$ such that

$x\circ y=\varphi x+a+\psi y\qquad{\rm and}\qquad x*y=\alpha x+b+\beta y.$

According to the Toyoda Theorem (cf. [9]), a quasigroup $(Q,\cdot)$ is medial
if and only if it is linear over a commutative group with commuting
automorphisms $\varphi,\psi$. In an analogous way we can shows that a
quasigroup $(Q,\cdot)$ is paramedial, that is it satisfies the identity
$(x\cdot y)\cdot(z\cdot u)=(u\cdot y)\cdot(z\cdot x)$ if and only if it is
linear over a commutative group with automorphisms $\varphi$, $\psi$ such that
$\varphi^{2}=\psi^{2}$. Based on these facts we say that a biquasigroup
$(Q,\circ,*)$ is medial (paramedial) if both its quasigroups $(Q,\circ)$ and
$(Q,*)$ are medial (paramedial) and linear over the same commutative group.

A biquasigroup $(Q,\circ,*)$ is unipotent if there is $q\in Q$ such that
$x\circ x=q=x*x$ for and all $x\in Q$. If both quasigroups $(Q,\circ)$ and
$(Q,*)$ are idempotent then we say that $(Q,\circ,*)$ is an idempotent
biquasigroup.

2. We will start with biquasigroups satisfying the identity (2).

A general characterization of such biquasigroups is given by Theorem 1.1. Now
we describe a biquasigroup linear over a group $(Q,+)$ and satisfying identity
(2).

From (2) for $x=y=z=0$ we obtain $\alpha a+b+\beta a=b$, This together with
(2) implies $\varphi=\varepsilon$ (the identity map). Thus

$\alpha x+\alpha a+\alpha\psi z+b+\beta y+\beta a+\beta\psi z=\alpha x+b+\beta
y.$

This for $z=0$ gives

$\alpha a+b+\beta y+\beta a=b+\beta y=\alpha a+b+\beta a+\beta y.$

So $\beta y+\beta a=\beta a+\beta y$, i.e. $a$ is in the center $Z(Q,+)$ of
the group $(Q,+)$. Thus using (2) and the above facts we obtain $\alpha\psi
z+b+\beta y+\beta\psi z=b+\beta y$. Hence $\alpha v+u+\beta v=u$ for all
$u,v\in Q$. Thus $\beta=-\alpha$ and consequently $\alpha v+u=u+\alpha v$ for
all $u,v\in Q$, which means that $(Q,+)$ is a commutative group.

In this way we have proved the “only if” part of the following Theorem. The
second part is trivial.

######  Theorem 2.1.

A biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ is a Ward double
quasigroup $($that is, it satisfies (2)$)$ if and only if $(Q,+)$ is a
commutative group, $x\circ y=x+\psi y+a$ and $x*y=\alpha x-\alpha y+b$.

Obviously such a biquasigroup is medial. The quasigroup $(Q,\circ)$ has a
right neutral element and the quasigroup $(Q,*)$ is unipotent. Moreover, a
biquasigroup $(Q,\circ,*)$ satisfying (2) is paramedial if and only if
$\psi^{2}=\varepsilon$.

3. Now consider biquasigroups satisfying the identity (3).

Since this identity contains only one operation, it is enough to examine the
quasigroup $(Q,\circ)$. Quasigroups satisfying (3) were characterized at the
beginning of this paper. If a quasigroup $(Q,\circ)$ linear over a group
$(Q,+)$ satisfies (3), then $\varphi=\varepsilon$ and $a+\psi a=0$. So (3) for
$y=0$, can be reduced to $\psi z+\psi^{2}z=0$. This means that $\psi z=-z$ and
$(Q,+)$ is a commutative group. Consequently $x\circ y=x-y+a$.

######  Theorem 2.2.

A quasigroup $(Q,\circ)$ linear over a group $(Q,+)$ satisfies (3) if and only
if $(Q,+)$ is a commutative group and $x\circ y=x-y+a$ for some fixed $a\in
Q$.

This quasigroup is medial, paramedial, unipotent and has a right neutral
element.

Note that $(Q,\circ)$ is a Ward quasigroup if and only if there is a group
$(Q,+)$ and an element $a\in Q$ such that $x\circ y=x-y+a$. The group $(Q,+)$
need not be commutative.

4. Will now consider a biquasigroup $(Q,\circ,*)$ satisfying the identity (4), i.e.

$\displaystyle(x\circ z)\circ(y\circ z)=x*y.$

######  Theorem 2.3.

If a biquasigroup $(Q,\circ,*)$ satisfies the identity $\eqref{e4}$, then both
quasigroups $(Q,\circ)$ and $(Q,*)$ are unipotent with $q\in Q$ such that
$x\circ x=q=x*x$ and $x*y=(x\circ y)\circ q=q\circ(y\circ x)$ for all $x,y\in
Q$.

###### Proof.

If $(Q,\circ)$ is idempotent, then $x=(x\circ x)\circ(x\circ x)=x*x$. So
$(Q,*)$ is idempotent too. If $(Q,*)$ is idempotent, then $x=x*x=(x\circ
z)\circ(x\circ z)$ for all $x,z\in Q$. In particular, for $z=x^{\prime}\in Q$
such that $x=x\circ x^{\prime}$ we obtain $x=x\circ x$. This shows that both
these quasigroups are idempotent or none of them are idempotent.

If both are idempotent, then $x\circ z=(x\circ z)\circ(x\circ z)=x*x=x=x\circ
x$ for all $x,z\in Q$, which implies $x=z$. Hence $Q$ has only one element. So
it is unipotent.

Now suppose both quasigroups $(Q,\circ)$ and $(Q,*)$ are not idempotent. Then
there exists $b\in Q$ such that $b*b=q\neq b$ and for any $x\in Q$ there exist
$x^{\prime},x^{\prime\prime}\in Q$ such that $b\circ x^{\prime}=x$ and $x\circ
x^{\prime\prime}=x$. Then $x\circ x=(b\circ x^{\prime})\circ(b\circ
x^{\prime})=b*b=q$ and $x*x=(x\circ x^{\prime\prime})\circ(x\circ
x^{\prime\prime})=x\circ x=q$. Hence, $(Q,\circ)$ and $(Q,*)$ are unipotent,
with $q=x\circ x=x*x$ for all $x\in Q$. Also, $x*y=(x\circ x)\circ(y\circ
x)=q\circ(y\circ x)$ and $x*y=(x\circ y)\circ(y\circ y)=(x\circ y)\circ q$. ∎

######  Corollary 2.4.

If a biquasigroup $(Q,\circ,*)$ satisfies $\eqref{e4}$ and $(Q,\circ)$ has a
right neutral element, then $(Q,\circ)=(Q,*)$ is a Ward quasigroup. If
$(Q,\circ)$ has a left neutral element, then $x*y=y\circ x$. If $(Q,\circ)$
has a neutral element, then $(Q,\circ)=(Q,*)$ is a commutative Ward
quasigroup.

Any medial unipotent quasigroup $(Q,\circ)$ can be ’extended’ to a medial
unipotent biquasigroup $(Q,\circ,*)$ satisfying the identity (4), as follows.

######  Proposition 2.5.

If $(Q,\circ)$ is a medial unipotent quasigroup, then $(Q,\circ,*)$, where
$x\circ x=q$ and $x*y=(x\circ y)\circ q$ for all $x,y\in Q$, is a biquasigroup
satisfying $\eqref{e4}$. Moreover, if $q$ is a left neutral element of
$(Q,\circ)$, then $x*y=y\circ x$.

###### Proof.

Indeed, $(Q,*)$ is a quasigroup and $x*y=(x\circ y)\circ q=(x\circ
y)\circ(z\circ z)=(x\circ z)\circ(y\circ z)$. Also, if $q$ is a left neutral
element of $(Q,\circ)$, then $x*y=(x\circ y)\circ q=(x\circ y)\circ(x\circ
x)=(x\circ x)\circ(y\circ x)=y\circ x$. ∎

Let $(Q,\circ,*)$ be a biquasigroup linear over a group $(Q,+)$. If it
satisfies (4), then $\varphi a+a+\psi a=b$ and $\alpha=\varphi^{2}$. So (4)
for $x=y=0$ and $\psi z=a$ gives $2\varphi a+a+2\psi a=b=\varphi a+a+\psi a$
which implies $a=b$. Consequently $a\circ a=a$. Thus, by Theorem 2.3, $a=q$
and $x*y=a\circ(y\circ x)$. Hence

$x*0=a\circ(0\circ x)=\alpha x+a=\varphi a+a+\psi a+\psi^{2}x=a+\psi^{2}x$

and

$a=x\circ x=\alpha x+a+\beta x=a+\psi^{2}x+\beta x.$

This gives $\psi^{2}x+\beta x=0$, i.e. $\beta=-\psi^{2}$. Hence
$x*y=\varphi^{2}x+a-\psi^{2}y$. Since $x\circ x=a=z*z$ we also have $\varphi
x+a=a-\psi x$ and $\varphi^{2}z+a=a+\psi^{2}z$. This for $x=\varphi z$ gives
$\varphi^{2}z+a=a-\psi\varphi z$. Hence $a+\psi^{2}z=a-\psi\varphi z$.
Consequently, $\psi=-\varphi$ and $x\circ y=\varphi x+a-\varphi y$. So
$a=x\circ x=\varphi x+a-\varphi x$. Thus $a\in Z(Q,+)$. Also
$\varphi^{2}=\psi^{2}$.

Therefore, $x\circ y=\varphi x+a-\varphi y$ and
$x*y=\varphi^{2}x+a-\varphi^{2}y$. Inserting these operations to (4) we obtain
$-\varphi^{2}z-\varphi^{2}y+\varphi^{2}z=-\varphi^{2}y$ for all $y,z\in Q$.
Hence $(Q,+)$ is a commutative group. Consequently $(Q,\circ,*)$ is medial and
unipotent. This proves the “only if” part of the Theorem 2.6 below. The proof
of the “if” part follows from a direct calculation and is omitted

######  Theorem 2.6.

A biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies the
identity (4) if and only if $(Q,+)$ is a commutative group, $x\circ y=\varphi
x+a-\varphi y$ and $x*y=\varphi^{2}x+a-\varphi^{2}y$.

It is clear that such a biquasigroup is medial and paramedial. If $a=0$ then
it is unipotent.

5. Will now consider a biquasigroup $(Q,\circ,*)$ satisfying the identity (5), i.e.

$(x\circ z)\circ(y*z)=x\circ y.$

######  Theorem 2.7.

If a biquasigroup $(Q,\circ,*)$ satisfies the identity $\eqref{e5}$, then both
quasigroups $(Q,\circ)$ and $(Q,*)$ have only one idempotent. This idempotent
is a right neutral element of these quasigroups. Moreover, $(Q,*)$ is
unipotent.

###### Proof.

For each $x\in Q$ there is uniquely determined $\overline{x}\in Q$ such that
$x\circ\overline{x}=x$. Then for $x,y\in Q$, by (5), we have

$x\circ y=(x\circ\overline{x})\circ(y*\overline{x})=x\circ(y*\overline{x}).$

So, $y=y*\overline{x}$ for each $y\in Q$. Also $y\circ
y=(y\circ\overline{x})\circ(y*\overline{x})=(y\circ\overline{x})\circ y$,
hence $y=y\circ\overline{x}$ for all $y\in Q$. Thus $e=\overline{x}$ is a
right neutral element of $(Q,\circ)$ and $(Q,*)$. There are no other
idempotents in $(Q,\circ)$ and $(Q,*)$. Indeed, if $a*a=a$, then for each
$x\in Q$

$x\circ a=(x\circ a)\circ(a*a)=(x\circ a)\circ a,$

so $x\circ a=x=x\circ e$. Hence $a=e$. Similarly for $a\circ a=a$ we have

$a\circ x=(a\circ a)\circ(x*a)=a\circ(x*a),$

which implies $x*a=x=x*e$, so also in this case $a=e$.

For each $x\in Q$ there exists $x^{\prime}\in Q$ such that $x*x=x\circ
x^{\prime}$. Thus, by (5),

$(x\circ x)\circ e=x\circ x=(x\circ x)\circ(x*x)=(x\circ x)\circ(x\circ
x^{\prime}),$

which implies $e=x\circ x^{\prime}=x*x$. So, $(Q,*)$ is unipotent. ∎

The following example shows that $(Q,\circ)$ may not be unipotent.

######  Example 2.8.

Let $(Q,\circ)$ be a group. Then $(Q,\circ,*)$, where $x*y=y^{-1}\circ x$ is
an example of a biquasigroup satisfying (5) in which only one of quasigroups
$(Q,\circ)$ and $(Q,*)$ has a left neutral element. Moreover, $(Q,*)$ is
unipotent but $(Q,\circ)$ is unipotent only in the case when it is a Boolean
group.

######  Corollary 2.9.

If in a biquasigroup $(Q,\circ,*)$ satisfying $\eqref{e5}$ one of quasigroups
$(Q,\circ)$ or $(Q,*)$ is idempotent, then $Q$ has only one element.

######  Proposition 2.10.

Let $(Q,\circ,*)$ be a biquasigroup satisfying (5). If $(Q,\circ)$ is a Ward
quasigroup, then $(Q,\circ)=(Q,*)$.

###### Proof.

Since $(Q,\circ)$ is a Ward quasigroup, there exists a group $(Q,\cdot)$ such
that $x\circ y=x\cdot y^{-1}$ and $e\circ(x\circ y)=y\circ x$, where $e$ is
the neutral element of the group $(Q,\cdot)$ (cf. [1]). Then $x\circ y=(x\circ
x)\circ(y*x)=e\circ(y*x)$ and so $x\circ y=e\circ(y\circ
x)=e\circ(e\circ(x*y))=x*y$. Hence $(Q,\circ)=(Q,*)$. ∎

######  Proposition 2.11.

Let $(Q,\circ,*)$ be a biquasigroup satisfying (5). If $(Q,\circ)$ is medial
and unipotent, then $(Q,\circ)=(Q,*)$.

###### Proof.

For every $x,y\in Q$ there exists $z\in Q$ such that $x*y=x\circ z$. Since
$(Q,\circ)$ is medial,

$(x\circ x)\circ e=x\circ x=(x\circ y)\circ(x*y)=(x\circ y)\circ(x\circ
z)=(x\circ x)\circ(y\circ z).$

Thus, $y\circ z=e=y\circ y$, where $e$ is the right neutral element of
$(Q,\circ)$. Therefore $y=z$ and consequently, $x*y=x\circ y$. ∎

######  Proposition 2.12.

A biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies (5) if and
only if a group $(Q,+)$ is a commutative group, $x\circ y=x+\psi y-\psi b$ and
$x*y=x-y+b$.

###### Proof.

If a biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies (5),
then $\varphi a+a+\psi b=a$ and $\varphi=\varepsilon$. Thus $a+\psi b=0$. This
together with (5) for $y=0$ gives $\psi z+\psi\beta z=0$. So, $\beta z=-z$ for
all $z\in Q$. Thus $(Q,+)$ is a commutative group. Consequently,
$\alpha=\varepsilon$. Therefore, $x\circ y=x+\psi y-\psi b$, $x*y=x-y+b$.

The proof of the converse follows from a direct calculation and is omitted. ∎

A biquasigroup $(Q,\circ,*)$ linear over a group and satisfying (5) is medial
and both its quasigroups $(Q,\circ)$ and $(Q,*)$ have the same right neutral
element. If $\psi^{2}=\varepsilon$ then this biquasigroup is also paramedial.

6. Will now consider a biquasigroup $(Q,\circ,*)$ satisfying the identity (6), i.e.

$\displaystyle(x\circ z)*(y\circ z)=x\circ y.$

######  Theorem 2.13.

A biquasigroup $(Q,\circ,*)$ satisfies the identity $(\ref{e6})$ if and only
if there is a group $(G,\cdot)$ and a bijection $\alpha$ on $Q$ such that
$x\circ y=(\alpha x)^{-1}\cdot(\alpha y)$ and $x*y=x\cdot y^{-1}$.

###### Proof.

$\Rightarrow$: Let $x,y,z\in Q$. Then for fixed $q\in Q$ there are
$x^{\prime},y^{\prime},z^{\prime}\in Q$ such that and $x=x^{\prime}\circ q$,
$y=y^{\prime}\circ q$ and $z=z^{\prime}\circ q$. Then, $x*z=(x^{\prime}\circ
q)*(z^{\prime}\circ q)=x^{\prime}\circ z^{\prime}$ and $y*z=(y^{\prime}\circ
q)*(z^{\prime}\circ q)=y^{\prime}\circ z^{\prime}$. So,

$(x*z)*(y*z)=(x^{\prime}\circ z^{\prime})*(y^{\prime}\circ
z^{\prime})=x^{\prime}\circ y^{\prime}=(x^{\prime}\circ q)*(y^{\prime}\circ
q)=x*y.$

Therefore, $(Q,*)$ is a Ward quasigroup and there exists a group $(Q,\cdot)$
such that $x*y=x\cdot y^{-1}$ and $x\cdot y=x*(e*y)$, where $e=w*w$ for any
$w\in Q$ and $x^{-1}=e*x$.

Let $x\in Q$. Then, $e=(x\circ z)*(x\circ z)=x\circ x$. So, $e*(x\circ
y)=(y\circ y)*(x\circ y)=y\circ x$, for any $y\in Q$. Let $\alpha x=e\circ x$.
Then, $(\alpha x)^{-1}=e*(e\circ x)=x\circ e$. Thus, $(\alpha
x)^{-1}\cdot(\alpha y)=(x\circ e)\cdot(e\circ y)=(x\circ e)*(e*(e\circ
y))=(x\circ e)*(y\circ e)=x\circ y$.

$\Leftarrow$: Let $x,y,z\in Q$. Then, $(x\circ z)*(y\circ z)=[(\alpha
x)^{-1}\cdot(\alpha z)]*[(\alpha y)^{-1}\\!\cdot(\alpha z)]=(\alpha
x)^{-1}\\!\cdot(\alpha z)\cdot(\alpha z)^{-1}\\!\cdot(\alpha y)=(\alpha
x)^{-1}\\!\cdot(\alpha y)=x\circ y$. ∎

######  Corollary 2.14.

If a biquasigroup $(Q,\circ,*)$ satisfies the identity $(\ref{e6})$, then it
is unipotent.

######  Corollary 2.15.

If in a biquasigroup $(Q,\circ,*)$ satisfying the identity $(\ref{e6})$ one of
quasigroups $(Q,\circ)$ and $(Q,*)$ is commutative, then also the second is
commutative. In this case both quasigroups are induced by the same Boolean
group.

######  Proposition 2.16.

A biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies the
identity $\eqref{e6}$ if and only if a group $(Q,+)$ is commutative, $x\circ
y=\varphi x-\varphi y+a$ and $x*y=x-y+a$.

###### Proof.

If a biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies the
identity $\eqref{e6}$ then $\alpha a+b+\beta a=a$ and $\alpha=\varepsilon$.
So, $b+\beta a=0$. Thus (6) for $x=y=0$ gives $\psi z+\beta\psi z=0$ which
means that $\beta v=-v$ for each $v\in Q$. Hence $(Q,+)$ is commutative and
$a=b$. Therefore $x*y=x-y+a$. Substituting this operation to (6) we obtain
$x\circ y=\varphi x-\varphi y+a$.

The converse statement is obvious. ∎

######  Corollary 2.17.

A linear biquasigroup satisfying the identity (6) is unipotent.

7. Will now consider a biquasigroup $(Q,\circ,*)$ satisfying the identity (7), i.e.

$(x\circ z)\circ(y*z)=x*y.$

A simple example of a biquasigroup $(Q,\circ,*)$ satisfying the identity (7)
is a commutative group $(Q,+)$ with the operations $x\circ y=y-x$ and
$x*y=x+y$. This biquasigroup is medial, both quasigroups $(Q,\circ)$ and
$(Q,*)$ have left neutral element but only the first is unipotent.

Suppose now that in a biquasigroup $(Q,\circ,*)$ satisfying the identity (7)
the first quasigroup is medial and the second is idempotent. Then, by Toyoda
Theorem (cf. [9]), there exists a commutative group $(Q,+)$ and its commuting
automorphisms $\varphi,\psi$ such that $x\circ y=\varphi x+\psi y+a$ for some
fixed $a\in Q$. Then $x*y=(x\circ y)\circ(y*y)=(x\circ y)\circ
y=\varphi^{2}x+\varphi\psi y+\psi y+\varphi a+a$. This, by (7), implies
$\varphi^{2}-\varphi=\varepsilon$, $\varphi+\varphi\psi+\psi=0$ and $\varphi
a=-a$. Thus $a=0$ and $x*y=\varphi^{2}x+\varphi\psi y+\psi y$. Since $(Q,*)$
is idempotent, $\varphi^{2}+\varphi\psi+\psi=\varepsilon$. Hence
$x*y=\varphi^{2}x+y-\varphi^{2}y=\varphi^{2}x-\varphi y$. Consequently $(Q,*)$
is medial. Therefore $(Q,\circ,*)$ is medial too.

In this way we have proved

######  Proposition 2.18.

If in a biquasigroup $(Q,\circ,*)$ satisfying the identity (7) the first
quasigroup is medial and the second is idempotent, then the second is medial
too and there exists a commutative group $(Q,+)$ and its commuting
automorphisms $\varphi,\psi$ such that $\varphi+\varphi\psi+\psi=0$,
$\varphi^{2}=\varphi+\varepsilon$, $x\circ y=\varphi x+\psi y$ and
$x*y=\varphi^{2}x-\varphi y$.

Conversely we have:

######  Proposition 2.19.

Let $(Q,+)$ be a commutative group and $\varphi,\psi$ be its commuting
automorphisms such that $\varphi+\varphi\psi+\psi=0$ and
$\varphi^{2}=\varphi+\varepsilon$. Then $(Q,\circ,*)$, where $x\circ y=\varphi
x+\psi y$ and $x*y=\varphi^{2}x-\varphi y$, is a medial biquasigroup
satisfying the identity $\eqref{e7}$.

###### Proof.

This is a straightforward calculation. ∎

As a consequence of the above results we obtain

######  Corollary 2.20.

An idempotent medial biquasigroup $(Q,\circ,*)$ satisfies the identity
$\eqref{e7}$ if and only if there exist a commutative group $(Q,+)$ and its
automorphism $\varphi$ such that $x\circ y=\varphi x+y-\varphi y$,
$x*y=\varphi^{2}x-\varphi y$ and $\varphi^{2}=\varphi+\varepsilon$.

In the case of quasigroups induced by the group $\mathbb{Z}_{n}$ we have
stronger result. For simplicity the value of the integer $t\geqslant 0$ modulo
$n$ will be denoted by $[t]_{n}$.

######  Corollary 2.21.

An idempotent medial biquasigroup induced by the group $\mathbb{Z}_{n}$
satisfies the identity $\eqref{e7}$ if and only if has the form
$(\mathbb{Z}_{n},\circ,*)$, where $x\circ y=[ax+(1-a)y]_{n}$,
$\,x*y=[a^{2}x+(1-a^{2})y]_{n}$ and $[a^{2}-a]_{n}=1$.

######  Corollary 2.22.

For every $a\geqslant 3$ there is an idempotent medial biquasigroup of order
$n=a^{2}-a-1$ satisfying $\eqref{e7}$. It has the form
$(\mathbb{Z}_{n},\circ,*)$, where $x\circ y=[ax+(1-a)y]_{n}$ and
$x*y=[(a+1)x-ay]_{n}$, or $x\circ y=[(1-a)x+ay]_{n}$ and
$x*y=[(2-a)x+(a-1)y]_{n}$.

######  Proposition 2.23.

A medial biquasigroup $(Q,\circ,*)$ satisfies the identity $\eqref{e7}$ if and
only if there exist a commutative group $(Q,+)$ and its commuting
automorphisms $\varphi,\psi$ such that $x\circ y=\varphi x+\psi y+c$,
$x*y=\varphi^{2}x-\varphi y+d$, $\varphi\psi+\varepsilon=0$ and $\varphi
c+\psi d+c=d$ for some fixed $c,d\in Q$.

######  Proposition 2.24.

A medial biquasigroup $(\mathbb{Z}_{n},\circ,*)$ satisfies the identity
$\eqref{e7}$ if and only if there exists $a,b,c,d\in\mathbb{Z}_{n}$ such that
$[ab+1]_{n}=0$, $[ac+bd+c]_{n}=d$, $x\circ y=[ax+by+c]_{n}$ and
$x*y=[a^{2}x-ay+d])_{n}$.

For linear biquasigroup we have the following result.

######  Theorem 2.25.

A biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies the
identity (7) if and only if $(Q,+)$ is a commutative group, $x\circ y=\varphi
x+\psi y+a$, $x*y=\varphi^{2}x+\psi\varphi^{2}y+b,$
$\varphi\psi+\psi^{2}\varphi^{2}=0$ and $\varphi a+a+\psi b=b$.

###### Proof.

If a biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies (7),
then $\varphi a+a+\psi b=b$ and $\varphi^{2}=\alpha$. Thus (7) can be reduced
to

$\varphi a+\varphi\psi z+a+\psi\alpha y+\psi b+\psi\beta z=b+\beta y,$

which for $z=0$ gives $\varphi a+a+\psi\alpha y+\psi b=b+\beta y=(\varphi
a+a+\psi b)+\beta y$. Hence $\psi\alpha y+\psi b=\psi b+\beta y$. So
$\psi\alpha y=\psi b+\beta y-\psi b$. Therefore the previous identity implies
$\varphi\psi z+a+\psi b+\beta y+\psi\beta z=a+\psi b+\beta y$. Since every
element $v\in Q$ can be presented in the form $v=a+\psi b+\beta y$, the last
identity means that $\varphi\psi z+v+\psi\beta z=v$ for all $v,z\in Q$. This
implies $\varphi\psi=-\psi\beta$. Hence $\varphi\psi z+v=v-\psi\beta
z=v+\varphi\psi z$. So, $(Q,+)$ is commutative. Applying these facts to (7) we
can see that $\beta=\psi\alpha$. Hence $x\circ y=\varphi x+\psi y+a$ and
$x*y=\varphi^{2}x+\psi\varphi^{2}y+b.$

The proof of the converse follows from a direct calculation and is omitted. ∎

8. Will now consider a biquasigroup $(Q,\circ,*)$ satisfying the identity (8), i.e.

$(x\circ z)*(y*z)=x\circ y.$

######  Proposition 2.26.

If a biquasigroup $(Q,\circ,*)$ satisfies (8), then $(Q,*)$ has no more than
one idempotent. If such idempotent exists then it is a right neutral element
of a quasigroup $(Q,*)$. Moreover, if $x\circ x=u$ for some $u\in Q$ and all
$x\in Q$, then $x\circ y=x*(y*x)$, $x*x=w$ and $w\circ u=w$ for all $x,y\in
Q$.

###### Proof.

Let $e*e=e$. Since $(Q,\circ)$ is a quasigroup, each $z\in Q$ can be expressed
in the form $z=x\circ e$. Thus $z=x\circ e=(x\circ e)*(e*e)=z*e$, so $e$ is a
right neutral element of $(Q,*)$. If $\bar{e}$ is the second idempotent of
$(Q,*)$, then $\bar{e}*e=\bar{e}=\bar{e}*\bar{e}$. Therefore $\bar{e}=e$, so
$(Q,*)$ has no more than one idempotent. If $x\circ x=u$ for all $x\in Q$,
then, by (8), $u*(x*x)=(x\circ x)*(x*x)=x\circ x=u$. Analogously $u*(y*y)=u$.
Thus, $x*x=y*y=w$ for some $w\in Q$, i.e. $(Q,*)$ is unipotent and $w$ is its
right neutral element. Then, $x\circ y=(x\circ x)*(y*x)=u*(y*x)$ and $w\circ
u=(w\circ w)*(u*w)=u*u=w$. ∎

Let $(Q,\circ,*)$ be linear over a group $(Q,+)$. If it satisfies (8), then
$\alpha a+b+\beta b=a$, $\alpha=\varepsilon$ and $\beta b=-b$. Thus (8) can be
reduced to

$\psi z+b+\beta y+\beta b+\beta^{2}z=\psi y.$ (10)

This for $y=0$ gives $\psi z+\beta^{2}z=0$. So, $\psi=-\beta^{2}$ and $\psi
b=-b=\beta b$.

Now putting $y=b$ in (10) we obtain $\psi z+b+\beta b+\beta b+\beta^{2}z=\psi
b$, i.e. $-\beta^{2}z+\beta b+\beta^{2}z=\psi b=\beta b$. So, $\beta
b+\beta^{2}z=\beta^{2}z+\beta b$. This means that $b$ is in the center of
$(Q,+)$. Thus putting $z=0$ in (10) and using the above facts, we obtain
$\beta=\psi=-\beta^{2}$. Hence $\beta=-\varepsilon$. So $(Q,+)$ is
commutative, $x\circ y=\varphi x-y+a$ and $x*y=x-y+b$.

In this way, we have proved the “only if” part of Theorem 2.27 below. The
proof of the converse part of Theorem 2.27 follows from a direct calculation
and is omitted.

######  Theorem 2.27.

A biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies (8) if and
only if $(Q,+)$ is commutative, $x\circ y=\varphi x-y+a$ and $x*y=x-y+b$.

######  Corollary 2.28.

A linear biquasigroup satisfying (8) is medial.

######  Corollary 2.29.

A medial biquasigroup induced by the group $\mathbb{Z}_{n}$ satisfies
$\eqref{e8}$ if and only if $x\circ y=[ax-y+c]_{n}$ and $x*y=[x-y+d]_{n}$ for
some $a,c,d\in\mathbb{Z}_{n}$ such that $(a,n)=1$.

9. Finally, let us consider a biquasigroup $(Q,\circ,*)$ satisfying the identity (9), i.e.

$(x\circ z)*(y*z)=x*y.$

######  Theorem 2.30.

In a biquasigroup $(Q,\circ,*)$ satisfying the identity (9) the quasigroups
$(Q,\circ)$ and $(Q,*)$ have no more than one idempotent. If such idempotent
exists then it is a common right neutral element of these quasigroups.

###### Proof.

Assume $(Q,\circ)$ has an idempotent $a$. Then $a*a=(a\circ a)*(a*a)=a*(a*a)$
and so $a*a=a$. Analogously, for $a*a=a$ we have $a*a=(a\circ a)*(a*a)=(a\circ
a)*a$, which implies $a\circ a=a$. So, $(Q,\circ)$ and $(Q,*)$ have the same
idempotent. Then for each $x\in Q$ $x*a=(x\circ a)*(a*a)=(x\circ a)*a$, which
implies $x=x\circ a$. Thus $a$ is a right neutral element of $(Q,\circ)$. On
the other hand, $x\circ a=x$ gives $x*x=(x\circ a)*(x*a)=x*(x*a)$, and
consequently $x=x*a$. Thus, $a$ is a right neutral element of $(Q,\circ)$ and
$(Q,*)$. ∎

######  Corollary 2.31.

If in a biquasigroup $(Q,\circ,*)$ satisfying $\eqref{e9}$ the quasigroup
$(Q,*)$ is unipotent, then $(Q,\circ)=(Q,*)$ and $(Q,\circ)$ is a Ward
quasigroup.

###### Proof.

Let $x*x=a$ for all $x\in Q$ and some $a\in Q$. Then $a*a=x*x=(x\circ
x)*(x*x)=(x\circ x)*a$. Therefore, $x\circ x=a$, i.e. $(Q,\circ)$ is
unipotent. Consequently, $a*(x*y)=(y\circ y)*(x*y)=y*x$, which implies $x\circ
y=(x\circ y)*a=(x\circ y)*(y*y)=x*y$. Hence $(Q,\circ)=(Q,*)$ and (9)
coincides with (1). This means that $(Q,\circ)$ is a Ward quasigroup. ∎

######  Theorem 2.32.

A biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfying the
identity (9) is medial and can be presented in the form $x\circ
y=x-\beta^{2}y-\beta b$ and $x*y=x+\beta y+b$, where $(Q,+)$ is a commutative
group, $\beta\in{\rm Aut}(Q,+)$ and $b\in Q$. This biquasigroup has a right
neutral element $e=-\varphi^{-1}b$.

Conversely, if $(Q,+)$ is a commutative group, $\beta\in{\rm Aut}(Q,+)$, $b\in
Q$, $x\circ y=x-\beta^{2}y-\beta b$ and $x*y=x+\beta y+b$, then the
biquasigroup $(Q,\circ,*)$ satisfies (9).

###### Proof.

If a biquasigroup $(Q,\circ,*)$ linear over a group $(Q,+)$ satisfies the
identity (9), then $\alpha a+b+\beta b=b$ and $\varphi=\varepsilon$. So, (9)
can be reduced to

$\alpha a+\alpha\psi z+b+\beta\alpha y+\beta b+\beta^{2}z=b+\beta y,$ (11)

which for $z=0$ gives $\alpha a+b+\beta\alpha y+\beta b=b+\beta y=\alpha
a+b+\beta b+\beta y$. Since $\alpha a=b=b-\beta b$, the last implies
$\beta\alpha y=\beta b+\beta y-\beta b$. This together with (11) (for $y=v$)
gives

$\alpha a+\alpha\psi z+b+\beta b+\beta v+\beta^{2}z=b+\beta v.$ (12)

Now adding $\beta v$ on the right side to (11) and putting $y=0$ we get

$\alpha a+\alpha\psi z+b+\beta b+\beta^{2}z+\beta v=b+\beta v.$

Comparing this identity with (12) we obtain $\beta
v+\beta^{2}z=\beta^{2}z+\beta v$ for all $v,z\in Q$. This shows that $(Q,+)$
is a commutative group. Consequently, $\beta\alpha y=\beta y$, so
$\alpha=\varepsilon$. This by $\alpha a+b+\beta b=b$ gives $a=-\beta b$. Again
putting $y=0$ in (11) and using the above facts we obtain $\psi=-\beta^{2}$.
Therefore, $x\circ y=x-\beta^{2}y-\beta b$ and $x*y=x+\beta y+b$.

The proof of the converse part of the Theorem follows from a direct
calculation and is omitted. ∎

######  Proposition 2.33.

A medial biquasigroup $(\mathbb{Z}_{n},\circ,*)$ satisfies the identity
$\eqref{e9}$ if and only if $x\circ y=[x-a^{2}y-ab]_{n}$, $x*y=[x+ay+b]_{n}$,
where $a,b\in\mathbb{Z}_{n}$ are fixed and $(a,n)=1$.

######  Example 2.34.

Let $n=a^{2}+1>4$. Then $(\mathbb{Z}_{n},\circ,*)$, where $x\circ y=[x+y]_{n}$
and $x*y=[x+ay]_{n}$ is an example of a biquasigroup satisfying (9).

10. Many authors study linear quasigroups of the second type, namely quasigroups $(Q,\cdot)$ where, in the definition of the operation, the constant element is not placed in the middle of the formula but at its end, i.e. $x\cdot y=\varphi x+\psi y+a$.

Biquasigroups of this type satisfying the identities $(2)-(9)$ coincide with
the quasigroups of the previous type. Namely, if a biquasigroup
$\widehat{Q}=(Q,\circ,*)$ with the operations $x\circ y=\varphi x+\psi y+a$
and $x*y=\alpha x+\beta y+b$, where $\alpha,\beta,\varphi,\psi$ are
automorphisms of a group $(Q,+)$, satisfies $(2)$ then $\alpha a+\beta a=0$
and $\varphi=\varepsilon$. Thus $\alpha\psi z+\alpha a+\beta y+\beta\psi
z+\beta a=\beta y$. This for $y=0$ and $\psi z=v$ gives $\alpha v=-\beta
a-\beta v-\alpha a$. Since $\alpha$ and $\beta$ are automorphisms of $(Q,+)$
the last expression for $v=u+w$ implies $\beta(u+w)=\beta w+\beta u$. Thus
$\beta u+\beta w=\beta u+\beta w$ for all $u,w\in Q$. Hence $(Q,+)$ is a
commutative group. Such biquasigroups are described in subsection 2.

If a biquasigroup $(Q,\circ,\circ)$ with $x\circ y=\varphi x+\psi y+a$
satisfies (3), then $\varphi a+\psi a=0$, $\varphi=\varepsilon$ and $\psi
z+a+\psi y+\psi^{2}z+\psi a=\psi y$, which for $y=a$ gives
$\psi=-\varepsilon$. This shows that $(Q,+)$ is a commutative group and
$x\circ y=x-y+a$. Also in the case when $(Q,\circ)$ with $x\circ y=\varphi
x+a+\psi y$ satisfies (1), the group $(Q,+)$ must be commutative and $x\circ
y=x-y+a$. This means that these two cases coincide.

If a biquasigroup $\widehat{Q}$ satisfies (4), then $\varphi a+\psi a+a=b$ and
$\alpha=\varphi^{2}$. Because by Theorem 2.3 we have $q=\varphi 0+\psi
0+a=\alpha 0+\beta 0+b$, must be $q=a=b$. Consequently, $a=\varphi x+\psi
x+a$. This implies $\varphi=-\psi$, which together with $(x\circ
0)\circ(x\circ 0)=a$ implies $\varphi^{2}x+\varphi a-\varphi^{2}x-\varphi
a=0$. Hence $\varphi x+a=a+\varphi x$ for all $x\in Q$. So, $a$ is in the
center of $(Q,+)$. Therefore this case is reduced to the case described in
subsection 4.

If a biquasigroup $\widehat{Q}$ satisfies (5), then by Theorem 2.7 the
quasigroup $(Q,*)$ has a right neutral element $e$. Thus $x=x*e=\alpha x+\beta
e+b$ for all $x\in Q$. In particular $0=0*e=\beta e+b$. Consequently,
$x=x*e=\alpha x$ and $x*y=x+\beta y+b$. Applying this formula to (5) we can
see that $\varphi=\varepsilon$ and $\varphi b=-a$. Therefore the identity (5)
can be written in the form

$\psi z+a+\psi y+\psi\beta z=\psi y+a.$

This for $z=0$ implies $a+\psi y=\psi y+a$. Hence $a$ is in the center of
$(Q,+)$. Also $b$ is in the center of $(Q,+)$ because $\varphi b=-a$. Thus
this case reduces to the case from subsection 5.

By Corollary 2.14 any quasigroup satisfying (6) is unipotent. Thus if
$\widehat{Q}$ satisfies (6), then $\alpha 0+\beta 0+b=b$ implies $b=x*x=\alpha
x+\beta x+b$, i.e. $\beta x=-\alpha x$ for all $x\in Q$. From (6) it follows
$\alpha=\varepsilon$. Thus $\beta x=-x$. Since $\beta$ is an automorphism of
$(Q,+)$, $(Q,+)$ is commutative. Hence this case reduces to subsection 6.

If a biquasigroup $\widehat{Q}$ satisfies (7), then $\varphi a+\psi b+a=b$
which together with (7) fort $x=y=0$ implies

$\varphi\psi z+\varphi a+\psi\beta z+\psi b+a=b=\varphi a+\psi b+a.$

Thus $\varphi\psi z=\varphi a-\psi\beta z-\varphi a$. Since $\varphi\psi$ and
$\psi\beta$ are automorphisms of $(Q,+)$ the last for $z=u+v$ gives

$\varphi\psi(u+v)=\varphi a-\psi\beta(u+v)-\varphi a$.

On the other side,

$\varphi\psi u+\varphi\psi v=\varphi a-\psi\beta u-\varphi a+\varphi
a-\psi\beta v-\varphi a=\varphi a-\psi\beta(v+u)-\varphi a$.

Comparing these two expression we obtain $\psi\beta(u+v)=\psi\beta(v+u)$.
Hence $(Q,+)$ is a commutative group and this case reduces to 7.

If a biquasigroup $\widehat{Q}$ satisfies (8), then $\alpha a+\beta b+b=a$ and
$\alpha=\varepsilon$. This together with (8) for $x=y=0$ implies $\psi
z+a+\beta^{2}z=a$. Hence $\beta^{2}z=-a-\psi z+a$. From this for $z=u+v$, in a
similar way as in the previous case, we obtain $\psi(u+v)=\psi(v+u)$.
Therefore $(Q,+)$ is a commutative group and this case reduces to 8.

The case when $\widehat{Q}$ satisfies (9) reduces to 9. Indeed, in this case
$\alpha a+\beta b=0$, which together (9) for $x=y=0$ shows that
$\beta^{2}z=-\alpha a-\alpha\psi z-\beta b$. From this we compute
$\alpha\psi(u+v)=\alpha\psi(v+u)$. Hence $(Q,+)$ is commutative.

## References

  * [1] J.M. Cardoso, C.P. da Silva, On Ward quasigroups, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat., 24 (1978), 231–233.
  * [2] S.K. Chatterjea, On Ward quasigroups, Pure Math. Manuscript, 6 (1987), 31–34.
  * [3] W.A. Dudek, R.A.R. Monzo, Double magma associated with Ward and double Ward quasigroups, Quasigroups and Related Systems, 27 (2019), 33–52.
  * [4] N.C. Fiala, Double Ward quasigroups, Quasigroups and Related Systems, 15 (2007), 261–262.
  * [5] K.W. Johnson, P. Vojtěchovský, Right division in groups, Dedekind-Frobenius group matrices, and Ward quasigroups, Abh. Math. Sem. Univ. Hamburg, 75 (2005), 121–136.
  * [6] J. Morgado, Definiç$\tilde{a}$o de quasigrupo subtractivo por único axioma, Gaz. Mat. (Lisboa) 92-93 (1963), 17–18.
  * [7] M. Polonijo, Ward double quasigroups, in Proc. Conf. "Algebra and Logic". Cetinje, 1986, 153–156, Univ. Novi Sad, 1987.
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  * [9] V. Shcherbacov, Elements of quasigroup theory and applications, Champan and Hall Book (2017).
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Received February 03, 2021

W.A. Dudek
Faculty of Pure and Applied Mathematics, Wroclaw University of Science and
Technology
50-370 Wroclaw, Poland
Email<EMAIL_ADDRESS>
R.A.R. Monzo
Flat 10, Albert Mansions, Crouch Hill, London N8 9RE, United Kingdom
E-mail<EMAIL_ADDRESS>


# Semantic Image Fusion

P.R. Hill
Visual Information Laboratory
University of Bristol, UK
<EMAIL_ADDRESS>&D.R. Bull
Visual Information Laboratory
University of Bristol, UK
<EMAIL_ADDRESS>

###### Abstract

Image fusion methods and metrics for their evaluation have conventionally used
pixel based or low level features. However, for many applications the aim of
image fusion is to effectively combine the semantic content of the input
images. This paper proposes a novel system for the semantic combination of
visual content using pre-trained CNN network architectures. Our proposed
semantic fusion is initiated through the fusion of the top layer feature map
outputs (for each input image) through gradient updating of the fused image
input (so called image optimisation). Simple ‘choose maximum’ and ‘local
majority’ filter based fusion rules are utilised for feature map fusion. This
provides a simple method to combine layer outputs and thus a unique framework
to fuse single channel and colour images within a decomposition pre-trained
for classification and therefore aligned with semantic fusion. Furthermore,
class activation mappings of each input image are used to combine semantic
information at a higher level. The developed methods are able to give
equivalent low level fusion performance to state of the art methods while
providing a unique architecture to combine semantic information from multiple
images.

_Keywords_ Image Fusion $\cdot$ CNN

##  1 Introduction

Image fusion is the combination of multiple images into a single image that
aims to combine the most important visual information from all sources [1].
Image fusion has been motivated by the need to improve visual representations,
visualisation, scene understanding and situational awareness in multi-sensor
and multi-camera applications such as remote sensing [2], medicine [3] and
surveillance [4].

Image fusion has been driven by device and sensor limitations. For example,
not all important visual information can be captured by one type of sensor
(e.g. IR, visible etc.) or within one single shooting setting (i.e. focus,
angle etc.). Furthermore, complementary imaging modalities co-exist within
domains such as remote sensing and medicine that contain very different and
important visual information. The effective combination of all the visual
information from all image sources is therefore the aim of image fusion. Such
a combined image is often effective for subsequent tasks such as scene
understanding and target recognition.

Image fusion has been a highly researched area over the last half century.
During this time image fusion is performed at or decision-level, feature-level
and pixel-level [5]. Simple signal processing based pixel-level image fusion
has given excellent results in preceding decades and continues to be used due
to its high-efficiency and lack of a need for training data [5]. Pixel-level
fusion can be further classified into decomposition based methods or Sparse
Representation (SR) based methods [5]. In decomposition based techniques, the
input images are decomposed into transform domains using methods such as
complex wavelets (DT-CWT) [1], the Discrete Wavelet Transform (DWT) [6] and
the contourlet transform [7]. Within the transform domain, coefficients are
combined using suitably defined fusion rules (such as weighted-averaging [8]
and "choose maximum" [9]).

More recently, state-of-the-art image fusion techniques have focused on the
use of network based methods [10, 11, 12, 13, 14, 15, 16, 17]. Due to the
requirements of needing training data these methods are often domain focused
with state-of-the-art results reported for IR/Visible fusion [12, 13], remote
sensing (multispectral) [14] and multi-focus areas [15, 16]. Recent work has
also been focused on universal image fusion methods that can effectively
combine multiple sources within all of these domains [17].

There have only been a very small number of previously developed fusion
methods that utilise semantic information for image fusion [18, 19]. However,
these methods do not use the joint classification and class activation maps to
semantically fuse the input sources as proposed in our work. These previous
works also only use semantic information in a very limited way.

###  1.1 Contributions

The contributions and characteristics of our work are summarised as follows.

  * •

An unsupervised fusion technique is proposed that can combine the outputs of
pre-trained low level feature maps using choose maximum and majority filter
based fusion rules (using image optimisation).

  * •

An unsupervised semantic fusion method that uses class activation maps.

  * •

An unsupervised fusion method that combines both direct semantic fusion using
class activation maps together with the combination of low-level network layer
outputs (i.e. combining the above two approaches).

##  2 Feature Map Fusion using Image Optimisation

Feature map based loss functions within pre-trained networks for image
optimisation have been used extensively in Neural Style Transfer (NST) [20]
and "Deep Dream" methods [21]. This field was initiated by the seminal work in
NST by Gatys et al. [22]. Gatys’ NST system applied previously developed
texture synthesis methods [23, 24] to the combination of two images (a
"content" image and a "style" image) through an image optimisation method
utilising a pre-trained Convolutional Neural Network (CNN): VGG19 [25].
Although these methods have provided amazing visual results they have yet to
be utilised for general image processing applications such as image fusion and
denoising. NST using image optimised can be summarised through the generation
of the output style transferred image ($I^{*}$) using the minimisation of a
loss function:

$\displaystyle I^{*}$
$\displaystyle=\operatorname*{arg\,min}_{I}\mathcal{L}_{total}(I|I_{c},I_{s})$
(1)
$\displaystyle=\operatorname*{arg\,min}_{I}\alpha\mathcal{L}_{c}(I|I_{c})+\beta\mathcal{L}_{s}(I|I_{s})$
(2)

where $I^{*}$ is the output image, $I_{s}$ is the "style" image and $I_{c}$ is
the "content" image and $\alpha$ and $\beta$ are the loss weighting
parameters. The content and style losses ($\mathcal{L}_{c},\mathcal{L}_{s}$)
compare the content and style representations between the style and content
images ($I_{s}$, $I_{c}$).

As an initial step, each of these two images are decomposed into the layer
outputs of a VGG19 CNN network [25]. The two losses are calculated as
functional comparisons between the layer outputs of the two images. Image
optimisation is then achieved through gradient updates through the network
using the gradient on the image w.r.t. the total loss $\mathcal{L}_{total}$.
Since the original paper, many updates and optimisations to NST have been
reported. Li and Wand [26] have proposed a Markov Random Field (MRF) based
loss function that generates more plausible visual outputs. Computational
optimised NST methods include Johnson et al. [27] and Ulyanov et al. [28].
These methods are similar to the Gatys’ method in principle but are
implemented using a single forward pass of a pre-trained network. Multiple-
Style-Per-Model NST methods have included Dumoulin et al. [29], Li et al. [30]
and Zhang and Dana [31]. Finally GANs [32], CycleGANs [33] and image
transformers [34] have been recently used for NST. Although there have been
many advances in this field, the Gatys method is still considered to be the
gold standard by most researchers in terms of the quality of its results [20].
Therefore, although it is not computationally optimised or based on more
complex transforms (GANs or Image Transformers), we have based our work on
this type of image gradient update as it gives excellent results and is
conceptually easy to understand and manipulate. Computational optimisation of
our developed methods can be implemented as future work.

###  2.1 Fusion Rules and Loss Functions for Image Optimisation based Image
Fusion

The image optimisation methods described above have almost exclusively been
used for Neural Style Transfer. However, we propose such feature map image
optimisation methods for image fusion. For image fusion, a variety of fusion
rules and loss functions have been tested over an exhaustive set of layer
subsets of various pre-trained CNN networks. It was found that taking losses
across single layers gave the most effective results. Furthermore, it is
recognised that as the layers get further from the input images the semantic
information increases. This gives the possibility that fusion at such layers
can combine more and more abstract semantic information. However, such high
level layers have a reduced resolution and therefore increased spatial support
of each feature in the feature map. It was found that fusing such layers
generated unwanted artefacts such as banding. Layers with the same resolution
as the input images were found to give the most effective results (e.g. the
first two layers of the VGG19 CNN).

Although very sophisticated fusion rules and loss functions have been
considered, it was found that a simple choose maximum fusion rule combined
with a $l_{2}$ loss function in most cases gave the best results. The choice
and utility of such a choose maximum fusion rule is motivated by its use
within the wavelet transform domain [1, 6] i.e. large magnitude coefficients
(or feature map outputs) correlates with perceptually important content.

Algorithm 1 illustrates the gradient update method used within the feature map
based image fusion method. This algorithm is based on the key concept of
updating the input image $I^{*}$, input to a network $N$ through a gradient
update with respect to a loss function such as (6) (as implemented within the
style transfer methods and the deep dream method). The loss function within
the style transfer method is a weighted combination of the "content" and
"style" losses defined through the comparison of the content and style image
inputs. The loss function within the DeepDream method [21] is just based on
the absolute magnitude of a chosen network output, layer or layer output. Our
fusion loss function is given in (4). This is just the $L_{2}$ norm distance
between the fused layer outputs (calculated just once outside the optimisation
loops) and the iterated layer output feature map $F$. The actual image fusion
optimisation process is defined in (3) being very similar to the NST equation
(1) (but with the input images being the images to be fused $I_{0}$ and
$I_{1}$ rather than the style and content images: $I_{s}$ and $I_{c}$).

####  2.1.1 Loss Functions

The fused image $I^{*}$ is generated using image minimisation of the fusion
loss function $\mathcal{L}_{f}$:

$\displaystyle I^{*}$
$\displaystyle=\operatorname*{arg\,min}_{I}\mathcal{L}_{f}(I|I_{0},I_{1})$ (3)
$\displaystyle\mathcal{L}_{f}(I|I_{0},I_{1})$
$\displaystyle=\sum_{l\in\\{l_{f}\\}}||\left(\Psi\left((N^{l}(I_{0}),N^{l}(I_{1})\right)-N^{l}(I)\right)||^{2}_{2}$
(4)

where $N^{l}(I)$ is the network feature map output at layer $l$ when the
network is input image $I$. $l_{f}$ is the set of layers to calculate the loss
over and $\Psi$ is the fusion rule (defined below). Although an exhaustive
combination of layers have been tested we have only utilised the top layer for
our VGG19 CNN i.e. $l_{f}=\\{^{\prime}conv1\\_1^{\prime}\\}$.

####  2.1.2 Fusion Rules

Our initial fusion rule $\Psi_{0}$ for combining the layer outputs is the
choose maximum rule:

$\displaystyle\Psi_{0}(F_{0},F_{1})=max\\{F_{x}:x=0...1\\}$ (5)

where $F_{0}$ and $F_{1}$ are the network outputs for the chosen layer (or set
of layers) for input images 0 and 1 respectively. The maximum operator
operates on a feature by feature basis in the selected output feature map. The
output is therefore a tensor the same size as the two input tensors where the
tensor elements are selected according to the maximum absolute magnitude of
the two corresponding tensor elements.

The choice between each of the spatial positions within the network layer
outputs is effectively a binary decision mask. This mask can be noisy and
contain spatial inconsistencies. We therefore use a localised majority filter
as an alternative fusion rule $\Psi_{1}$.

$\displaystyle\Psi_{1}(F_{0},F_{1})=med_{c}\\{|F_{0,\bm{i}+\bm{r},c}|>|F_{1,\bm{i}+\bm{r},c}|:\bm{r}\in
W\\}$ (6)

$F_{0,\bm{i},c}$ is the feature map value at vector spatial position $\bm{i}$
and channel $c$ when when image $I_{0}$ is input into the network, $\bm{r}$ is
the spatial index vector of a sliding window ($3\times 3$ in our case) and
$med_{c}$ is the median value of the $3\times 3$ boolean map on a channel by
channel basis. This median filter applied on a boolean map therefore gives a
spatial majority filter on a channel by channel basis for the considered
feature map.

input : Input Images: $I_{0}$, $I_{1}$

output : Fused Image: $I^{*}$

Initialise $I^{*}$: $I^{*}=mean(I_{0},I_{1})$

Fuse layer outputs: $G^{l}=\Psi\left((N^{l}(I_{0}),N^{l}(I_{1})\right)$

for  _$e$ Epochs_ do

Update ADAM learning rate: $\lambda$;

for  _$k$ Iterations_ do

Calculate layer feature loss:
$\mathcal{L}_{f}(I^{*}|I_{0},I_{1})=\sum_{l\in\\{l_{f}\\}}||\left(G^{l}-N^{l}(I^{*})\right)||^{2}_{2}$

Update input image:
$I^{*}=I^{*}+\lambda\nabla_{I^{*}}\left(\mathcal{L}_{f}(I^{*}|I_{0},I_{1})\right)$

end for

end for

Algorithm 1 Layer-Based Fusion using Gradient Update: We k = 100, e = 100, in
our experiments.

Image 1

Image 2

FM0

FM1

FM2

FM3

U2Fusion

Figure 1: Fused image results. From top to bottom the fusion pairs are
labelled “Clock”, “Head”, “Remote”, “Bottle”, “Gingerbread”,
“Goldfinch/Indigo_bunting”, “Beagle/Ferret”, “Great_Pyrenees/Bee-Eater”.
U2Fusion results are not available for the last three rows as it is not
possible to be used for colour images.

##  3 Semantic Fusion Using Class Activation Mappings (CAMs)

The use of the more abstract layers within a CNN theoretically provides the
ability to semantically fuse images using the image optimisation method
described above. However, due to these feature maps having very limited
spatial resolution such fusion results in significant edge based artefacts. We
have therefore chosen to utilise Class Activation Mappings (CAMs) that give a
spatial mapping of how relevant each pixel has been in generating a
classification of a given class.

###  3.1 Class Activation Mappings

CAMs were first introduced by Zhou et al. [35] and generate an importance
spatial mapping according to a class $c$ through the sum of output weights of
an input image for the last layer of a CNN. This method was generalised to the
use of backpropagated class gradients Grad-CAMs [36]. These methods have been
further extended using methods such as Eigen-CAM [37], Grad-CAM using Vision-
Transformers [38] etc. However, for our work the mappings from Grad-CAM gave
the best results (due to their spatial consistency).

Given the definitions of a Grad-CAM mapping by Selvaraju et al. [36] a spatial
mapping (the same resolution as the input image) can be represented as
$M_{c}^{l}(x,y)$ where $c$ is the class under consideration, $l$ is the
analysed layer (usually the spatial layer closest to the actual classification
layer). We found that using the lowest resolution feature map, the mappings
give the least spatial localisation. Using higher resolution feature map
layers gives better spatial localisation but less accurately reflects the
abstract class activation’s as the lower (more abstract) layers. We therefore
combine the lowest three spatial layers as follows:

$\displaystyle P_{c}(x,y)$ $\displaystyle=Norm\left(\prod_{l\in
lset}M_{c}^{l}(x,y)\right)$ (7)

where $Norm$ is the normalisation function:

$\displaystyle Norm(z)$ $\displaystyle=\frac{z-min(z)}{max(z)-min(z)}$ (8)

As the $Norm$ function normalises the combination of the CAM mappings to the
range [0,1] it can be considered as the probability $P_{c}$ of each pixel
contributing to the classification of the top object recognised in each image
(class $c_{0}$ for image $I_{0}$ and class $c_{1}$ for image $I_{1}$). For our
utilised VGG19 CNN, $lset$ is the set of the last three coarsest resolution
layers i.e.
$lset=\\{^{\prime}conv3\\_4^{\prime},^{\prime}conv4\\_4^{\prime},^{\prime}conv5\\_4^{\prime}\\}$.

These probabilities are termed $P_{c_{0},I_{0}}(x,y)$ and
$P_{c_{0},I_{1}}(x,y)$ for input images $I_{0}$ and $I_{1}$ at spatial
positions $(x,y)$. From these probabilities we need to generate a mixing ratio
for the image fusion (also in the form of a probability termed
$P_{M_{0}}(x,y)$: the probability that the output image should contain input
image $I_{0}$). $P_{M_{0}}(x,y)$ is generated as an exclusive combination of
$P_{c_{0},I_{0}}(x,y)$ and $P_{c_{1},I_{1}}(x,y)$ i.e.

  * •

When $P_{c_{0},I_{0}}(x,y)$ and $P_{c_{1},I_{1}}(x,y)$ are approximately equal
(for all values between 0 and 1), $P_{M_{0}}(x,y)$ should give an equal mix of
each image in the output (i.e. $P_{M_{0}}(x,y)\approx 0.5$).

  * •

When $P_{c_{0},I_{0}}(x,y)$ is small (i.e. near 0) and $P_{c_{1},I_{1}}(x,y)$
is large (i.e. near 1) then $P_{M_{0}}(x,y)$ should approximate 0).

  * •

When $P_{c_{1},I_{1}}(x,y)$ is small (i.e. near 0) and $P_{c_{0},I_{0}}(x,y)$
is large (i.e. near 1) then $P_{M_{0}}(x,y)$ should approximate 1).

This is achieved by defining $P_{M_{0}}(x,y)$ as (illustrated in figure 2):

$\displaystyle P_{M_{0}}(x,y)=$ $\displaystyle
0.5(1+P_{c_{0},I_{0}}(x,y)-P_{c_{1},I_{1}}(x,y))$ (9) $\displaystyle
P_{M_{1}}(x,y)=$ $\displaystyle 1-P_{M_{0}}(x,y)$ (10)

Figure 2: Generating function (9) to obtain mixing probability
$P_{M_{0}}(x,y)$ from $P_{c_{0},I_{0}}(x,y)$ and $P_{c_{1},I_{1}}(x,y)$

Fusion is therefore achieved using $P_{M_{0}}(x,y)$ and $P_{M_{1}}(x,y)$ as
the mixing ratios of the two input images. The fused image $I^{*}$ is
therefore defined as (dropping the $x,y$ indices).

$\displaystyle I^{*}=$ $\displaystyle P_{M_{0}}I_{0}+P_{M_{1}}I_{1}$ (11)

##  4 Results

Table 1 shows objective results for all the methods (defined below) for the
single channel input images. These have been compared to state of the art
methods U2Fusion [17] and GMC [39]. The fusion metrics include: Wang ($Q_{0}$
[40]), Xydeas ($P_{e}$ [41]) and Piella ($Q$ [42]). These metrics can only be
effectively used for single channel images; colour images are therefore not
included in this table. This table shows that for the majority of the image
pairs and metrics, the FM0 method (image optimisation based image fusion)
gives the best performance. The GMC method gives the best results for all
metrics for the head image pair. This is unsurprising as GMC was specifically
designed to be used in this domain [39].

###  4.1 Method Definitions

FM0,

Fusion Method 0: This method utilises image optimisation using (3) and (4) and
fusion rule (choose maximum) $\Psi_{0}$ defined in (5).

FM1,

Fusion Method 1: This method utilises image optimisation using (3) and (4) and
fusion rule (majority filter) $\Psi_{1}$ defined in (6).

FM2,

Fusion Method 2: This combines method FM0 and FM3 (i.e. feature maps $F_{0}$
and $F_{1}$ are multiplied by CAM probabilities $P_{c_{0},I_{0}}$ and
$P_{c_{1},I_{1}}$ respectively).

FM3,

Fusion Method 3: Fusion using CAMs utilising (7),(8),(9),(10) and (11).

###  4.2 Classification for Semantic Image Fusion

In order to generate the Class Activation Mappings (CAMs) the chosen network
(VGG19) was used to get the top classification class $c$ for each of the input
images i.e. class $c_{0}$ is the top classification class for input image
$I_{0}$ and $c_{1}$ is the top classification class for input image $I_{1}$.
These classifications don’t always make sense for all the considered
applications and domains (e.g. the "head" and "remote" images in Figure 1).
However, table 2 shows the top classification classes for the image pairs
shown in figure 3. Figure 3 also shows the CAM mappings for the top classes
for each of the image pairs. This figure shows how the different semantic
objects are highlighted by the class activation mappings and how this is
utilised to combine and semantically fuse the input images. This figure shows
how the utilisation of CAM mappings can distinguish between spatial regions
that are in focus in multi-focus fusion pairs (see the clock CAMs in the top
row).

Dataset | Method | $Q_{0}$ | $P_{e}$ | $Q$
---|---|---|---|---
Clock | FM0 | 0.8294 | 0.6308 | 0.9580
FM1 | 0.8293 | 0.6262 | 0.9572
FM2 | 0.8296 | 0.6199 | 0.9544
FM3 | 0.8293 | 0.5881 | 0.9495
| U2Fusion | 0.8267 | 0.5409 | 0.8970
| GMC | 0.8286 | 0.5208 | 0.9325
Head | FM0 | 0.8035 | 0.3425 | 0.3045
FM1 | 0.8034 | 0.3411 | 0.3049
FM2 | 0.8045 | 0.5539 | 0.6614
FM3 | 0.8066 | 0.4753 | 0.7117
| U2Fusion | 0.8072 | 0.2732 | 0.7056
| GMC | 0.8216 | 0.6660 | 0.8002
Remote | FM0 | 0.8100 | 0.6490 | 0.8640
FM1 | 0.8103 | 0.6394 | 0.8604
FM2 | 0.8106 | 0.6364 | 0.8597
FM3 | 0.8121 | 0.5827 | 0.8544
| U2Fusion | 0.8097 | 0.5458 | 0.8487
| GMC | 0.8156 | 0.6210 | 0.7708
Bottle | FM0 | 0.8196 | 0.7112 | 0.9491
FM1 | 0.8191 | 0.6993 | 0.9470
FM2 | 0.8199 | 0.6960 | 0.9476
FM3 | 0.8164 | 0.4688 | 0.8819
| U2Fusion | 0.8168 | 0.6111 | 0.9286
| GMC | 0.8166 | 0.4830 | 0.8725
Gingerbread | FM0 | 0.8217 | 0.6681 | 0.9522
FM1 | 0.8215 | 0.6633 | 0.9517
FM2 | 0.8230 | 0.6497 | 0.9489
FM3 | 0.8208 | 0.5218 | 0.9190
| U2Fusion | 0.8204 | 0.5947 | 0.9430
| GMC | 0.8205 | 0.5518 | 0.9033
Table 1: The objective results of different methods (these comparisons are only possible for single channel image pairs). Metrics: Wang ($Q_{0}$ [40]), Xydeas ($P_{e}$ [41]), Piella ($Q$ [42]). FM0-3 defined in section 4.1. Comparison techniques: U2Fusion [17] and GMC [39]. Fusion Pair |  $c_{0}$, P($c_{0}$) |  $c_{1}$, P($c_{1}$) |  $c_{FM3}$, P($c_{FM3}$)
---|---|---|---
Clocks (Top row figure 3) | Analog_Clock, | Analog_Clock, | Analog_Clock,
0.524 | 0.722 | 0.663
Beagle/Ferret (Second row figure 3) | Beagle, | Black-Footed_Ferret, | Beagle,
0.970 | 0.449 | 0.927
Great_Pyrenees/Bee-Eater (Third row figure 3) | Great_Pyrenees, | Bee-Eater, | Great_Pyrenees,
0.858 | 0.991 | 0.656
Goldfinch/Indigo_bunting (Fourth row figure 3) | Goldfinch, | Indigo_Bunting, | Goldfinch,
0.9999 | 0.999 | 0.780
Table 2: VGG19 top classification results (class and probability) for input
and FM3 fused images (from figure 3).

$I_{0}$

$I_{1}$

$P_{c_{0},I_{0}}$

$P_{c_{1},I_{1}}$

$P_{M_{0}}\times I_{0}$

$P_{M_{1}}\times I_{1}$

($(I_{0}+I_{1})/2$)

FM3

Figure 3: Fused image results. From top to bottom the fusion pairs are
labelled "Clock", "Beagle/Ferret", "Great_Pyrenees/Bee-Eater" and
"Goldfinch/Indigo_bunting". This image shows the input image, CAM mappings,
the CAM based mixtures from each image, the averaged image output and the
fused image output (FM3).

##  5 Conclusion

This paper has defined four novel fusion methods that utilise: image
optimisation; choose maximum and majority filter fusion rules; and class
activation mappings for semantic fusion. The image optimisation method was
able to achieve state of the art image fusion results measured using
conventional image fusion metrics in multiple domains. This method is also
directly extendable to colour images (not historically a major focus of image
fusion).

Furthermore the CAM based methods can, for the first time, directly exploit
semantic information from the top classified class in each of the fused images
to generate true semantic level image fusion. It is conjectured that combining
regions that are important to the classification of a set of images will most
effectively combine the semantic information from the input images. Images
combined in this semantically meaningful way are shown to retain the important
semantic information from both images.

This method would easily extend to multiple images and the fusion of the top-5
classes in each image.

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