# Affine structures on groups and semi-braces111This work is partially
supported by the Dipartimento di Matematica e Fisica “Ennio De Giorgi” -
Università del Salento. The author is partially supported by the ACROSS
project ARS01_00702 and is a member of GNSAGA (INdAM).

Paola STEFANELLI<EMAIL_ADDRESS>Dipartimento di Matematica e
Fisica “Ennio De Giorgi”
Università del Salento
Via Provinciale Lecce-Arnesano
73100 Lecce (Italy)

###### Abstract

We introduce affine structures on groups and show they form a category
equivalent to that of semi-braces. In particular, such a new description of
semi-braces includes that presented by Rump for braces. By specific affine
structures, we provide several instances of bi-skew braces, including some
that are not $\lambda$-homomorphic. Finally, we give a method for determining
affine structures on the Zappa product of two groups both endowed with affine
structures and prove that such a construction allows for obtaining semi-braces
that are not matched product of semi-braces.

###### keywords:

Quantum Yang-Baxter equation , set-theoretic solution , semi-brace , skew
brace , brace

###### MSC:

[2020] 16T25, 81R50, 20M18

## Introduction

The Yang-Baxter equation is one of the fundamental equations of statistical
mechanics that arose from two independent papers by Yang [43] and Baxter [7].
Motivated by its connection with many research areas, such as quantum groups,
knot theory, and Hopf algebras, many mathematicians started to study this
equation from different points of view. A complete description of all
solutions of the Yang-Baxter equation is still unknown, even if a large number
of papers is dedicated to this topic. In this context, Drinfel’d [22]
suggested the study of the special class of set-theoretic solutions. If $S$ is
a set, a map $r:S\times S\longrightarrow S\times S$ is said to be a _set-
theoretic solution of the Yang-Baxter equation_ , or, briefly, a _solution_ on
$S$, if the identity

$\displaystyle\left(r\times\operatorname{id}_{S}\right)\left(\operatorname{id}_{S}\times
r\right)\left(r\times\operatorname{id}_{S}\right)=\left(\operatorname{id}_{S}\times
r\right)\left(r\times\operatorname{id}_{S}\right)\left(\operatorname{id}_{S}\times
r\right).$

is satisfied. Writing $r(a,b)=(\lambda_{a}(b),\rho_{b}(a))$, with
$\lambda_{a}$ and $\rho_{b}$ maps from $S$ into itself, for all $a,b\in S$,
$r$ is said to be _left non-degenerate_ if $\lambda_{a}$ is bijective, for
every $a\in S$, _right non-degenerate_ if $\rho_{b}$ is bijective, for every
$b\in S$, and _non-degenerate_ if $r$ is both left and right non-degenerate.
Bijective solutions have been initially studied by Etingof, Schedler and
Soloviev [23], Gateva-Ivanova and Van den Bergh [24], and Lu, Yan, and Zhu
[33]. These pioneering works influenced and inspired the research from a more
algebraic perspective in the next years.

In 2007, Rump discovered that any Jacobson radical ring determines a solution.
This surprising fact motivated Rump to introduce _braces_ , algebraic
structures including radical rings that turned out to be significant in the
study of non-degenerate involutive solutions, namely, solutions $r$ such that
$r^{2}=\operatorname{id}_{S\times S}$. Later on, algebraic structures close to
braces, as skew braces and semi-braces, have been introduced for determining
solutions. Following [12] and [28], a set $B$ endowed with two binary
operations $+$ and $\circ$ is said to be a _(left) semi-brace_ if
$\left(B,+\right)$ is a semigroup, $\left(B,\circ\right)$ is a group, and the
identity

$\displaystyle a\circ\left(b+c\right)=a\circ b+a\circ\left(a^{-}+c\right)$
($\ast$)

is satisfied, for all $a,b,c\in B$, where $a^{-}$ is the inverse of $a$ with
respect to $\circ$. We highlight that in ($\ast$ ‣ Introduction) it is
implicit that the multiplication has higher precedence than the addition. If
$\left(B,+\right)$ is a left cancellative semigroup, we say that $B$ is a
_(left) cancellative semi-brace_. As shown in [12], if the additive structure
$\left(B,+\right)$ is a group, then ($\ast$ ‣ Introduction) is equivalent to
the relation

$\displaystyle a\circ\left(b+c\right)=a\circ b-a+a\circ c,$ ($\ast\ast$)

for all $a,b,c\in B$, namely, $B$ is a _skew brace_ , the structure introduced
in [26]. In addition, if the group $\left(B,+\right)$ is abelian, then $B$ is
a _brace_.
Any cancellative semi-brace $B$ gives rise to a left non-degenerate solution
$r_{B}$ that is the map $r_{B}:B\times B\to B\times B$ defined by
$r_{B}\left(a,b\right)=\left(a\circ\left(a^{-}+b\right),\,\left(a^{-}+b\right)^{-}\circ
b\right)$, for all $a,b\in B$. In particular, if $B$ is a skew brace, then
$r_{B}$ is a non-degenerate bijective solution that is involutive if and only
if $B$ is a brace. Under mild assumptions, semi-braces produce degenerate
solutions, among these the cubic ones, i.e., $r_{B}^{3}=r_{B}$.
Descriptions and classifications of these structures are object of
investigation in many recent papers, as [1, 2, 4, 17, 18, 21, 25, 27, 34],
just to name a few. In particular, in [37] Rump provided a description of
braces in terms of affine structures on groups. Using a different notation
from that adopted by Rump, given a group $G=\left(B,\circ\right)$ we say that
a map $\sigma:B\to\operatorname{Sym}_{B}$ is an _affine structure_ on $G$ if
$\sigma$ is an anti-homomorphism from $G$ to the symmetric group on the set
$B$ such that the following identity holds

$a\circ\sigma_{a}\left(b\right)=b\circ\sigma_{b}\left(a\right),$

for all $a,b\in B$. This notion allows to investigate isomorphism classes of
braces having a fixed group as multiplicative group. The study of braces from
the perspective of affine structures has led to new advances in the brace
theory itself. More specifically, in [37] Rump provided constructions of
finite braces by first proving that the determination of all affine structures
can be reduced to those of $p$,$q$-groups. Later on, in [38], Rump obtained
other instances of braces in terms of affine structures of decomposable
solvable groups and, in [39], proved what conjectured in [26, Conjecture
$6.9$] by classifying all the affine structures of a generalized quaternion
group of order greater than or equal to $32$.

This paper aims to introduce a new notion of affine structure on a group and
show how it is a useful tool for describing semi-braces and constructing
special instances of them. If $G=\left(B,\circ\right)$ is a group with
identity $0$, we say that a map $\sigma:B\to B^{B}$ is an _affine structure_
on $G$ if $\sigma$ is a semigroup anti-homomorphism from the group $G$ to the
set of all maps from $B$ into itself and the identity

$\sigma_{a}\left(b\circ\sigma_{b}\left(c\right)\right)=\sigma_{a}\left(b\right)\circ\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(c\right)$

is satisfied, for all $a,b,c\in B$. Moreover, if
$\sigma\left(B\right)\subseteq\operatorname{Sym}_{B}$, we say that $\sigma$ is
a _cancellative affine structure_. In such a case, if in addition
$\sigma_{a}\left(0\right)=0$, for every $a\in B$, we say that $\sigma$ is a
_groupal affine structure_. We prove that Rump’s affine structures are special
groupal affine structures that we rename _abelian affine structures_ for our
purposes.
We show the existence of a one-to-one correspondence between affine structures
of a given group $G$ and semi-braces having $G$ as multiplicative structure.
More specifically, cancellative affine structures are in one-to-one
correspondence with cancellative semi-braces and groupal affine structures
with skew braces. In this correspondence, isomorphism classes of semi-braces
correspond to equivalent classes of affine structures on $G$. In particular,
we prove that semi-braces and affine structures are equivalent as categories.

Part of this paper is dedicated to determine examples of affine structures and
to obtain new ones starting from fixed affine structures. Particular attention
has been posed to affine structures related to _bi-skew braces_ , namely, skew
braces that are also skew braces reversing the roles of the multiplication and
the sum. Bi-skew braces was introduced and studied by Child in [19] in the
finite case and after they were investigated also in [8], [10], and [29]. Note
that, recently, the authors in [5] focused on bi-skew braces $B$ that lie in
the class of $\lambda$-homomorphic skew braces, i.e., skew braces for which
$\lambda_{a+b}=\lambda_{a}\lambda_{b}$, where
$\lambda_{a}\left(b\right)=-a+a\circ b$, for all $a,b\in B$. Some of bi-skew
braces we give in this work do not belong to such a class, hence, as observed
in [6, Section 5], our examples are different from that studied in [19], [29,
30], and [10, 9], that are all of $\lambda$-homomorphic type.

Finally, starting from two affine structures $\sigma^{S}$ and $\sigma^{T}$ on
two groups $H$ and $K$, we provide sufficient conditions so that specific maps
$\alpha$ and $\beta$ make the map $\sigma^{S}\times\sigma^{T}$ into an affine
structure on a group $H\bowtie_{\alpha,\beta}K$ that is isomorphic to the
Zappa product of $H$ and $K$. In particular, such conditions are also
necessary in the cancellative case. The multiplicative structure of any semi-
brace obtained through such a method is the same obtained by means of the
matched product of semi-braces contained in [14, Theorem 6] (see also [28,
p.249]). However, specific examples show that, in general, semi-braces
obtained by these two constructions are not the same, up to isomorphism. In
addition, there exist maps $\alpha$ and $\beta$ that are suitable for
obtaining semi-braces through both the techniques above mentioned.

##  1 Affine structures on groups

This section aims to introduce affine structures on groups and clarify their
connection with the notion already given by Rump in [37]. Moreover, we provide
classes of examples of affine structures of different type.

Hereinafter, whenever we consider a group $G=\left(B,\circ\right)$, we will
denote its identity by $0$ and by $a^{-}$ the inverse of $a$, for every $a\in
B$. In addition, denoted by $B^{B}$ the set of maps from $B$ into itself, for
a map $\sigma:B\to B^{B}$, we write $\sigma_{a}$ to denote the image
$\sigma\left(a\right)$, for every $a\in B$.

Initially, let us recall the left-version of the notion of an affine structure
originally introduced by Rump in [37] (see also [38, p. 3]). We specify that
we rename such a notion as an _abelian affine structure_ only for our
purposes.

######  Definition 1.

Let $G=\left(B,\circ\right)$ be a group and
$\sigma:B\to\operatorname{Sym}_{B}$ an anti-homomorphism from $G$ to the
symmetric group on the set $B$. Then, $\sigma$ is said to be an _abelian
affine structure_ on $G$ if

$\displaystyle a\circ\sigma_{a}\left(b\right)=b\circ\sigma_{b}\left(a\right),$
(1)

for all $a,b\in B$.

Below, we introduce the notion of the affine structure and illustrate the link
with the previous definition.

######  Definition 2.

Let $G=\left(B,\circ\right)$ be a group and $\sigma:B\to B^{B}$ a semigroup
anti-homomorphism from the group $G$ to the set of all the maps from $B$ into
itself. Then, we say that $\sigma$ is an _affine structure_ on $G$ if the
identity

$\displaystyle\sigma_{a}\left(b\circ\sigma_{b}\left(c\right)\right)=\sigma_{a}\left(b\right)\circ\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(c\right)$
(2)

holds, for all $a,b,c\in B$. In particular, if
$\sigma\left(B\right)\subseteq\operatorname{Sym}_{B}$ (thus $\sigma$ is a
group anti-homomorphism), we say that $\sigma$ is a _cancellative affine
structure_. In this case, if in addition, $\sigma_{a}\left(0\right)=0$, for
every $a\in B$, we say that $\sigma$ is a _groupal affine structure_.

Observe that, in general, any affine structure satisfies
$\sigma_{0}\left(0\right)=0$. In fact, since $\sigma_{0}^{2}=\sigma_{0}$, by
(2), we have that
$\sigma_{0}\left(0\right)=\sigma_{0}\left(0\circ\sigma_{0}\left(0\right)\right)=\sigma_{0}\left(0\right)\circ\sigma_{\sigma_{0}\left(0\right)}\sigma_{0}\left(0\right)$,
hence
$\sigma_{\sigma_{0}\left(0\right)}\left(0\right)=\sigma_{\sigma_{0}\left(0\right)}\sigma_{0}\left(0\right)=0$.
It follows that
$\sigma_{0}\left(0\right)=\sigma_{\sigma_{0}\left(0\right)}\left(0\right)\circ\sigma_{\sigma_{\sigma_{0}\left(0\right)}\left(0\right)}\sigma_{\sigma_{0}\left(0\right)}\left(0\right)$
from which we obtain that
$\sigma_{0}\left(0\right)=\sigma_{\sigma_{0}\left(0\right)}\left(0\circ\sigma_{0}\left(0\right)\right)=\sigma_{\sigma_{0}\left(0\right)}\left(0\right)=0$.

Let us note that abelian affine structures are special cases of groupal affine
structures. Indeed, if $\sigma$ is an abelian affine structure on a group
$\left(B,\circ\right)$ and $a,b,c\in B$, we have that

$\displaystyle a\circ\sigma_{a}\left(b\circ\sigma_{b}\left(c\right)\right)$
$\displaystyle=b\circ\sigma_{b}\left(c\right)\circ\sigma_{b\circ\sigma_{b}\left(c\right)}\left(a\right)$
by (1)
$\displaystyle=b\circ\sigma_{b}\left(c\right)\circ\sigma_{\sigma_{b}\left(c\right)}\sigma_{b}\left(a\right)$
$\displaystyle=b\circ\sigma_{b}\left(a\right)\circ\sigma_{\sigma_{b}\left(a\right)}\sigma_{b}\left(c\right)$
by (1)
$\displaystyle=b\circ\sigma_{b}\left(a\right)\circ\sigma_{b\circ\sigma_{b}\left(a\right)}\left(c\right)$
$\displaystyle=a\circ\sigma_{a}\left(b\right)\circ\sigma_{a\circ\sigma_{a}\left(b\right)}\left(c\right)$
by (1)
$\displaystyle=a\circ\sigma_{a}\left(b\right)\circ\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(c\right),$

thus (2) holds. Moreover, $\sigma_{a}\left(0\right)=a^{-}\circ
a\circ\sigma_{a}\left(0\right)=a^{-}\circ
0\circ\sigma_{0}\left(a\right)=a^{-}\circ a=0$.

In addition, observe that there do not exist intermediate affine structures
between the general case and the cancellative one, since any affine structure
satisfying $\sigma_{0}=\operatorname{id}_{B}$ is cancellative.

######  Example 1.

Let $G=\left(B,\circ\right)$ be a group. If $|B|>1$, then the map
$\sigma:B\to\operatorname{Sym}_{B}$ given by
$\sigma_{a}\left(b\right)=a^{-}\circ b$, for all $a,b\in B$, is an anti-
homomorphism that satisfies (2) and, if $a\neq 0$, then
$\sigma_{a}\left(0\right)=a^{-}\neq 0$. Hence, $\sigma$ is an affine structure
that is cancellative but not groupal.

######  Example 2.

Let $G=\left(B,\circ\right)$ be a group and $f$ an idempotent endomorphism of
$G$. Then, the map $\sigma:B\to B^{B}$ given by $\sigma_{a}=f$, for every
$a\in B$, is an affine structure on $G$ that is not cancellative if
$f\neq\operatorname{id}_{B}$. Clearly, if $f=\operatorname{id}_{B}$, then
$\sigma$ is groupal and it is abelian if and only if the group $G$ is abelian.

######  Example 3.

Let $G=\left(B,\circ\right)$ be a group and $f$ an idempotent endomorphism of
$G$. Then, the map $\sigma:B\to\operatorname{Sym}_{B}$ defined by
$\sigma_{a}\left(b\right)=f\left(a\right)^{-}\circ b\circ f\left(a\right)$,
for all $a,b\in B$, is a groupal affine structure on $G$. Note that, if
$f=\operatorname{id}_{B}$, then $\sigma$ is abelian if and only $G$ is an
abelian group.

From now on, whenever we consider a cyclic group $G$, we assume that
$G=\langle\,g\,\rangle$, unless otherwise stated; in addition, if $G$ is
finite of order $m$, we will refer to $G$ by denoting it as $C_{m}$.

######  Examples 4.

Let $G=\left(B,\circ\right)$ be the infinite cyclic group.

  1. 1.

The map $\sigma:B\to\operatorname{Sym}_{B}$ defined by
$\sigma_{g^{k}}\left(g^{l}\right):=g^{\left(-1\right)^{k}l}$, for all
$k,l\in\mathbb{Z}$, is a groupal affine structure of $G$. Moreover, $\sigma$
is not abelian, since, for instance,
$g\circ\sigma_{g}\left(g^{2}\right)=g^{-1}$, instead
$g^{2}\circ\sigma_{g^{2}}\left(g\right)=g^{3}$.

  2. 2.

The map $\sigma:B\to\operatorname{Sym}_{B}$ defined by
$\sigma_{g^{k}}\left(g^{l}\right):=g^{k\left(-1+\left(-1\right)^{l}\right)+l}$,
for all $k,l\in\mathbb{Z}$ is a groupal affine structure on $G$ that is not
abelian. Indeed, one can check that the maps $\sigma_{g^{k}}$ are bijections
on $B$, for every $k\in\mathbb{Z}$, and that $\sigma$ is an anti-homomorphism
of groups. Note also that $\sigma_{g^{k}}\left(0\right)=0$. Furthermore, if
$k,l,m\in\mathbb{Z}$, then
$\left(-1\right)^{l\left(-1+\left(-1\right)^{m}\right)+m}=\left(-1\right)^{m}$,
hence we obtain that

$\displaystyle\sigma_{g^{k}}\left(g^{l}\right)\circ\,$
$\displaystyle\sigma_{\sigma_{g^{k}}\left(g^{l}\right)}\sigma_{g^{l}}\left(g^{m}\right)=g^{k\left(-1+\left(-1\right)^{l}\right)+l}\circ\sigma_{g^{k\left(-1+\left(-1\right)^{l}\right)+l}}\left(g^{l\left(-1+\left(-1\right)^{m}\right)+m}\right)$
$\displaystyle=g^{k\left(-1+\left(-1\right)^{l}\right)+l+\left(k\left(-1+\left(-1\right)^{l}\right)+l\right)\left(1+\left(-1\right)^{m}\right)+l\left(-1+\left(-1\right)^{m}\right)+m}$
$\displaystyle=g^{k\left(-1+\left(-1\right)^{l\left(-1\right)^{m}+m}\right)+l\left(-1\right)^{m}+m}$
$\displaystyle=\sigma_{g^{k}}\left(g^{l\left(-1\right)^{m}+m}\right)=\sigma_{g^{k}}\left(g^{l+l\left(-1+\left(-1\right)^{m}\right)+m}\right)$
$\displaystyle=\sigma_{g^{k}}\left(g^{l}\circ\sigma_{g^{l}}\left(g^{m}\right)\right),$

i.e., (2) holds. Finally, $\sigma$ is not abelian since
$g\circ\sigma_{g}\left(g^{2}\right)=g^{3}$ and
$g^{2}\circ\sigma_{g^{2}}\left(g\right)=g^{-1}$.

######  Example 5.

In the finite case, one can similarly define the maps $\sigma$ in $1.$ and
$2.$ of 4 on the cyclic group $G=C_{m}$ with $m$ even, and obtain that they
are affine structures that are not abelian if $m\nmid 4$.

The following proposition illustrates how to define affine structures on
isomorphic groups with each other.

######  Proposition 3.

Let $\sigma$ be an affine structure on a group $G=\left(B,\circ\right)$. If
$H=\left(C,\circ\right)$ is a group and $f:B\to C$ a group isomorphism from
$G$ to $H$, then the map $\varphi:C\to C^{C}$ given by

$\varphi_{u}:=f\sigma_{f^{-1}\left(u\right)}f^{-1},$

for every $u\in C$, is an affine structure on $H$. Moreover, if $\sigma$ is
cancellative, then $\varphi$ is cancellative; if in addition $\sigma$ is
groupal, then $\varphi$ is groupal.

###### Proof.

Clearly, $\varphi$ is a semigroup anti-homomorphism and if $u,v,w\in C$, we
get

$\displaystyle\varphi_{u}\left(v\circ\varphi_{v}\left(w\right)\right)$
$\displaystyle=f\sigma_{f^{-1}\left(u\right)}\left(f^{-1}\left(v\right)\circ\sigma_{f^{-1}\left(v\right)}f^{-1}\left(w\right)\right)$
$\displaystyle=f\left(\sigma_{f^{-1}\left(u\right)}f^{-1}\left(v\right)\circ\sigma_{\sigma_{f^{-1}\left(u\right)}f^{-1}\left(v\right)}\sigma_{f^{-1}\left(u\right)}f^{-1}\left(w\right)\right)$
by (2) $\displaystyle=\varphi_{u}\left(v\right)\circ
f\sigma_{f^{-1}\varphi_{u}\left(v\right)}\sigma_{f^{-1}\left(u\right)}f^{-1}\left(w\right)$
$\displaystyle=\varphi_{u}\left(v\right)\circ
f\sigma_{f^{-1}\varphi_{u}\left(v\right)}f^{-1}\varphi_{u}\left(w\right)$
$\displaystyle=\varphi_{u}\left(v\right)\circ\varphi_{\varphi_{u}\left(v\right)}\varphi_{u}\left(w\right),$

i.e., (2) holds. Finally, the remaining part of the proof is trivial. ∎

######  Definition 4.

Let $G=\left(B,\circ\right),H=\left(C,\circ\right)$ be groups and
$\sigma,\varphi$ affine structures on $G$ and $H$, respectively. If there
exists a group homomorphism $f$ from $G$ to $H$ such that

$\displaystyle\varphi_{f\left(a\right)}f=f\sigma_{a},$

for every $a\in B$, we say that $\sigma$ and $\varphi$ are _homomorphic via
$f$_ and $f$ is a _homomorphism between $\sigma$ and $\varphi$_. If in
addition $f$ is bijective, we say that $\sigma$ and $\varphi$ are _equivalent
via $f$_ and $f$ is an _isomorphism between $\sigma$ and $\varphi$_.

Given a group $G$, the following result provides sufficient conditions to
obtain a new affine structure on $G$ starting from two given ones on the same
group $G$. The subsequent 6 will show that, given an affine structure $\sigma$
on a group $G$, 5 turns out to be helpful to produce affine structures on $G$
that are not isomorphic to $\sigma$.

######  Theorem 5.

Let $G=\left(B,\circ\right)$ be a group and $\varphi,\omega:B\to B^{B}$ affine
structures on $G$. If the following conditions are satisfied

$\displaystyle\varphi_{a}\omega_{b}$ $\displaystyle=\omega_{b}\varphi_{a}$
($c1$) $\displaystyle\varphi_{b\circ\omega_{a}\left(b\right)^{-}}$
$\displaystyle=\omega_{\varphi_{a}\omega_{a}\left(b\right)\circ\omega_{a}\left(b\right)^{-}},$
($c2$)

for all $a,b\in B$, then the map $\sigma:B\to B^{B}$ given by
$\sigma_{a}:=\varphi_{a}\omega_{a}$, for every $a\in B$, is an affine
structure on $G$.

###### Proof.

Initially, note that $\sigma$ is a semigroup anti-homomorphism. Besides, if
$a,b,c\in B$,

$\displaystyle\sigma_{a}\left(b\circ\sigma_{b}\left(c\right)\right)$
$\displaystyle=\varphi_{a}\left(\omega_{a}\left(b\right)\circ\omega_{\omega_{a}\left(b\right)}\omega_{a}\varphi_{b}\left(c\right)\right)$
$\omega$ affine structure
$\displaystyle=\varphi_{a}\left(\omega_{a}\left(b\right)\circ\omega_{\omega_{a}\left(b\right)}\omega_{a}\varphi_{0}\varphi_{b}\left(c\right)\right)$
$\displaystyle=\varphi_{a}\left(\omega_{a}\left(b\right)\circ\varphi_{\omega_{a}\left(b\right)}\varphi_{\omega_{a}\left(b\right)^{-}}\omega_{\omega_{a}\left(b\right)}\omega_{a}\varphi_{b}\left(c\right)\right)$
by ($c1$)
$\displaystyle=\varphi_{a}\omega_{a}\left(b\right)\circ\varphi_{\varphi_{a}\omega_{a}\left(b\right)}\varphi_{a}\varphi_{\omega_{a}\left(b\right)^{-}}\omega_{\omega_{a}\left(b\right)}\omega_{a}\varphi_{b}\left(c\right)$
$\varphi$ affine structure
$\displaystyle=\varphi_{a}\omega_{a}\left(b\right)\circ\varphi_{\varphi_{a}\omega_{a}\left(b\right)}\varphi_{a}\varphi_{\omega_{a}\left(b\right)^{-}}\omega_{\omega_{a}\left(b\right)}\varphi_{b}\omega_{a}\left(c\right)$

and

$\displaystyle\sigma_{a}\left(b\right)\circ\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(c\right)$
$\displaystyle=\varphi_{a}\omega_{a}\left(b\right)\circ\varphi_{\varphi_{a}\omega_{a}\left(b\right)}\omega_{\varphi_{a}\omega_{a}\left(b\right)}\varphi_{a}\omega_{a}\left(c\right)$
$\displaystyle=\varphi_{a}\omega_{a}\left(b\right)\circ\varphi_{\varphi_{a}\omega_{a}\left(b\right)}\varphi_{a}\omega_{\varphi_{a}\omega_{a}\left(b\right)}\omega_{a}\left(c\right).$

Observing that

$\displaystyle\varphi_{\omega_{a}\left(b\right)^{-}}\omega_{\omega_{a}\left(b\right)}\varphi_{b}$
$\displaystyle=\omega_{\omega_{a}\left(b\right)}\varphi_{\omega_{a}\left(b\right)^{-}}\varphi_{b}=\omega_{\omega_{a}\left(b\right)}\varphi_{b\circ\omega_{a}\left(b\right)^{-}}$
$\displaystyle=\omega_{\omega_{a}\left(b\right)}\omega_{\varphi_{a}\omega_{a}\left(b\right)\circ\omega_{a}\left(b\right)^{-}}$
by ($c2$) $\displaystyle=\omega_{\varphi_{a}\omega_{a}\left(b\right)\circ
0}=\omega_{\varphi_{a}\omega_{a}\left(b\right)},$

we obtain that (2) is satisfied. Therefore, the claim follows. ∎

######  Corollary 6.

Let $G=\left(B,\circ\right)$ be a group and $\varphi,\omega:B\to B^{B}$
cancellative affine structures on $G$. If the following conditions are
satisfied

$\displaystyle\varphi_{a}\omega_{b}$ $\displaystyle=\omega_{b}\varphi_{a}$
($c1$) $\displaystyle\varphi_{\omega_{a^{-}}\left(b\right)\circ b^{-}}$
$\displaystyle=\omega_{\varphi_{a}\left(b\right)\circ b^{-}},$ ($c2^{\prime}$)

for all $a,b\in B$, then the map $\sigma:B\to B^{B}$ given by
$\sigma_{a}:=\varphi_{a}\omega_{a}$, for every $a\in B$, is a cancellative
affine structure on $G$. Moreover, if $\omega$ and $\varphi$ are groupal, then
$\sigma$ is groupal.

###### Proof.

It is a routine computation to verify that ($c2^{\prime}$) is equivalent to
($c2$). Besides, note that if $\omega$ and $\varphi$ are cancellative, then
$\sigma$ is a group anti-homomorphism. Finally, if $\omega$ and $\varphi$ are
groupal, it clearly holds that $\sigma_{a}\left(0\right)=0$, for every $a\in
B$. ∎

The following is an example of an abelian affine structure constructed as in 6
and starting from one that is groupal but not abelian.

######  Example 6.

Let $G=\left(B,\circ\right)$ be the cyclic group $C_{8}$ and consider
$\omega,\varphi:B\to\operatorname{Sym}_{B}$ both equal to the groupal affine
structure as in 4-$2.$, namely,
$\omega_{g^{k}}\left(g^{l}\right)=g^{k\left(-1+\left(-1\right)^{l}\right)+l}$,
for all $k,l\in\mathbb{Z}$. Then, the assumptions of 6 are satisfied. Indeed,
if $k,h,l\in\mathbb{Z}$, we clearly obtain that
$\omega_{g^{k}}\varphi_{g^{h}}=\varphi_{g^{h}}\omega_{g^{k}}$; in addition
setting
$u:=k+k\left(-1\right)^{l+1}+k\left(-1\right)^{h+1}+k\left(-1\right)^{h+l}$
and
$t:=-k+k\left(-1\right)^{l}+k\left(-1\right)^{h}+k\left(-1\right)^{h+l+1}$,

$\displaystyle\omega_{\varphi_{g^{k}}\left(g^{h}\right)}\left(g^{l}\right)$
$\displaystyle=\sigma_{g^{k\left(-1+\left(-1\right)^{h}\right)+h}}\left(g^{l}\right)=g^{\left(-k+k\left(-1\right)^{h}+h\right)\left(-1+\left(-1\right)^{l}\right)+l}=g^{u-h+h\left(-1\right)^{l}+l}$

and, since $\left(g^{k}\right)^{-}=g^{-k}$,

$\displaystyle\varphi_{\omega_{\left(g^{k}\right)^{-}}\left(g^{h}\right)}\left(g^{l}\right)$
$\displaystyle=\sigma_{g^{-k\left(-1+\left(-1\right)^{h}\right)+h}}\left(g^{l}\right)=g^{\left(k+k\left(-1\right)^{h+1}+h\right)\left(-1+\left(-1\right)^{l}\right)+l}=g^{t-h+h\left(-1\right)^{l}+l}.$

Note that if $h,l$ are odd, then $u=4k$ and $t=-4k$, thus ($c2$) holds.
Moreover, in all the the other cases $u=t=0$, hence ($c2$) is always
satisfied. Therefore, the map $\sigma:B\to\operatorname{Sym}_{B}$ given by
$\sigma_{g^{k}}:=\omega_{g^{k}}^{2}$ is an affine structure on the group $G$.
Finally, by distinguishing all the cases in which $k,l$ are even or odd, it is
a routine computation to check that (1) holds, namely, $\sigma$ is abelian.

##  2 Semi-braces and affine structures

In this section, we prove there exists a one-to-one correspondence between
affine structures on a given group $G$ and semi-braces having $G$ as
multiplicative structure. In particular, we show that semi-braces and affine
structures are equivalent (or isomorphic) as categories.

Initially, let us, briefly, recall some basics on the structure of the semi-
brace.

######  Definition 7 ([12], [28]).

A set $B$ endowed with two binary operations $+$ and $\circ$ is said to be a
_(left) semi-brace_ if $\left(B,+\right)$ is a semigroup,
$\left(B,\circ\right)$ is a group, and the identity

$\displaystyle a\circ\left(b+c\right)=a\circ b+a\circ\left(a^{-}+c\right)$
($\ast$)

holds, for all $a,b,c\in B$, where $a^{-}$ denotes the inverse of $a$ with
respect to $\circ$. Moreover, if $\left(B,+\right)$ is a (left) cancellative
semigroup, then $B$ is said to be a _left cancellative semi-brace_. The
structures $\left(B,+\right)$ and $\left(B,\circ\right)$ are named the
_additive_ and the _multiplicative structure_ of $B$, respectively.

Denoted by $0$ the identity of the group $\left(B,\circ\right)$, we have that
$0$ is an idempotent with respect to the sum. In the cancellative case, it
holds also that $0$ is a left identity of the semigroup $\left(B,+\right)$. If
the additive structure is a group, then $B$ is a _skew brace_ , the structure
introduced in [26]. In this case, the identity of the group $\left(B,+\right)$
is exactly $0$. Furthermore, if $\left(B,+\right)$ is an abelian group, then
$B$ is a _brace_.

Before illustrating the close link between affine structures and semi-braces,
let us recall two fundamental maps in the study of semi-braces and of the
solutions associated to them, namely, the maps

$\displaystyle\lambda_{a}:B\to B,\,b\mapsto
a\circ\left(a^{-}+b\right)\qquad\rho_{b}:B\to
B,\,a\mapsto\left(a^{-}+b\right)^{-}\circ b,$

for all $a,b\in B$. In the following proposition, we collect some properties
that are useful for our aims. For a fuller treatment, we refer the reader to
[26], [12], [28], and [11].

######  Proposition 8.

Let $B$ be a semi-brace. Then, the following statements hold:

  1. 1.

$\lambda_{a}$ is an endomorphism of $\left(B,+\right)$, for every $a\in B$;

  2. 2.

the map $\lambda:B\to\operatorname{End}(B,+),\,a\mapsto\lambda_{a}$ is a
homomorphism from the group $(B,\circ)$ to the monoid of the endomorphisms of
$\left(B,+\right)$;

  3. 3.

if $B$ is left cancellative, then $\lambda_{a}$ is bijective, for every $a\in
B$;

  4. 4.

if $B$ is left cancellative the map $\rho:B\to B^{B},\,b\mapsto\rho_{b}$ is a
semigroup anti-homomorphism from $(B,\circ)$ into the monoid of the maps from
$B$ into itself;

  5. 5.

if $B$ is a skew brace, then $\rho_{b}$ is bijective, for every $b\in B$.

######  Theorem 9.

Let $G=\left(B,\circ\right)$ be a group and $\sigma$ an affine structure on
$G$. Then, the binary operation given by

$\displaystyle a+b:=a\circ\sigma_{a}\left(b\right),$

for all $a,b\in B$, makes $B$ into a semi-brace that we call the _semi-brace
associated to $\sigma$_ and denote by $B_{\sigma}$. Moreover, if $\sigma$ is
cancellative then $B_{\sigma}$ is cancellative and we call it the
_cancellative semi-brace associated to $\sigma$_. If in addition $\sigma$ is
groupal, then $B_{\sigma}$ is a skew brace which we call the _skew brace
associated to $\sigma$_.

###### Proof.

At first, note that, by (2), we easily obtain that $\circ$ is associative,
hence $\left(B,\circ\right)$ is a semigroup. Moreover,

$\displaystyle a\circ b+a\circ\left(a^{-}+c\right)$ $\displaystyle=a\circ
b+\sigma_{a^{-}}\left(c\right)=a\circ b\circ\sigma_{a\circ
b}\sigma_{a^{-}}\left(c\right)=a\circ
b\circ\sigma_{b}\left(c\right)=a\circ\left(b+c\right),$

thus ($\ast$ ‣ Introduction) is satisfied. Now, if we assume that $\sigma$ is
cancellative and $a+b=a+c$, then
$a\circ\sigma_{a}\left(b\right)=a\circ\sigma_{a}\left(c\right)$ that implies
$b=c$, hence $\left(B,+\right)$ is a cancellative semigroup. Furthermore, if
we suppose that $\sigma$ is groupal, we obtain that
$a+0=a\circ\sigma_{a}\left(0\right)=a$ and, since
$\sigma_{0}=\operatorname{id}_{B}$, $0+a=a$. Finally,
$a+\sigma_{a^{-}}\left(a^{-}\right)=a\circ\sigma_{a}\sigma_{a^{-}}\left(a^{-}\right)=a\circ
a^{-}=0$ and

$\displaystyle\sigma_{a^{-}}\left(a^{-}\right)+a=\sigma_{a^{-}}\left(a^{-}\right)\circ\sigma_{\sigma_{a^{-}}\left(a^{-}\right)}\left(a\right)=\sigma_{a^{-}}\left(a^{-}\circ\sigma_{a^{-}}\sigma_{a}\left(a\right)\right)=\sigma_{a^{-}}\left(0\right)=0,$

namely, $a$ has opposite $-a=\sigma_{a^{-}}\left(a^{-}\right)$. Therefore the
statement is proved. ∎

######  Remark 1.

If $B_{\sigma}$ is the semi-brace associated to an affine structure $\sigma$,
by (2) we have that

$\displaystyle\sigma_{a}\left(b+c\right)=\sigma_{a}\left(b\circ\sigma_{b}\left(c\right)\right)=\sigma_{a}\left(b\right)\circ\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(c\right)=\sigma_{a}\left(b\right)+\sigma_{a}\left(c\right),$

for all $a,b,c\in B$, i.e., $\sigma_{a}\in\operatorname{End}\left(B,+\right)$,
for every $a\in B$.

######  Theorem 10.

Let $B$ be a semi-brace. Then, the map $\sigma:B\to
B^{B},a\mapsto\lambda_{a^{-}}$ is an affine structure of
$\left(B,\circ\right)$. Moreover, $a+b=a\circ\sigma_{a}\left(b\right)$, for
all $a,b\in B$, namely the semi-brace $B_{\sigma}$ coincides with $B$. In
addition, cancellative affine structures correspond to cancellative semi-
braces, and groupal affine structures correspond to skew braces.

###### Proof.

Clearly, by 8, we have that $\sigma$ is an anti-homomorphism from the
semigroup $\left(B,\circ\right)$ to the monoid $B^{B}$. If $a,b,c\in B$, we
have that

$\displaystyle\sigma_{a}\left(b\circ\sigma_{b}\left(c\right)\right)$
$\displaystyle=\lambda_{a^{-}}\left(b+c\right)=\lambda_{a^{-}}\left(b\right)+\lambda_{a^{-}}\left(c\right)=\lambda_{a^{-}}\left(b\right)\circ\sigma_{\lambda_{a^{-}}\left(b\right)}\lambda_{a^{-}}\left(c\right)$
$\displaystyle=\sigma_{a}\left(b\right)\circ\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(c\right),$

i.e., (2) is satisfied, and
$a\circ\sigma_{a}\left(b\right)=a\circ\lambda_{a^{-}}\left(b\right)=a+b$.
Furthermore, we trivially have that the sum defined as in 9 coincides with the
sum of the semi-brace $B$, hence $B_{\sigma}$ coincides with the semi-brace
$B$. Finally, the remaining part of the proof is trivial. ∎

From now on, given a semi-brace $B$, let us call the map $\sigma$ as in 10 the
_affine structure associated to $B$_ and denote it by $\sigma^{B}$.

######  Remark 2.

If $\sigma$ is an affine structure on a group $G=\left(B,\circ\right)$ and
$B_{\sigma}$ the semi-brace associated to $\sigma$, then we clearly have that
$\sigma^{B}=\sigma$.

As a consequence of 9 and 10, one can define a bijective correspondence
between the class of semi-braces having a fixed group $G$ as multiplicative
group and that of affine structures on $G$ itself.

######  Corollary 11.

Let $G=\left(B,\circ\right)$ be a group, $\mathcal{S}$ the class of semi-
braces having $G$ as multiplicative group, and $\mathcal{A}$ the class of
affine structures on $G$. Then, the map defined by

$\displaystyle\psi:\mathcal{S}\to\mathcal{A},\ B\mapsto\sigma^{B}$

is a bijection.

The following theorem shows a linkage between homomorphic classes of semi-
braces and homomorphic classes of affine structures.

######  Theorem 12.

If $B,C$ are semi-braces and $f$ is a homomorphism between $B$ and $C$, then
the affine structures $\sigma^{B}$ and $\sigma^{C}$ are homomorphic via $f$.
Conversely, if $\sigma,\varphi$ are affine structures on the groups
$G=\left(B,\circ\right)$ and $H=\left(C,\circ\right)$, respectively, and $f$
is a homomorphism between $\sigma$ and $\varphi$, then $f$ is a homomorphism
from the semi-brace $B_{\sigma}$ to $C_{\varphi}$.

###### Proof.

Initially, assume that $B,C$ are semi-braces and $f$ is an homomorphism
between $B$ and $C$. Then, if $a,b\in B$, we have that

$\displaystyle
f\sigma^{B}_{a}\left(b\right)=f\left(a^{-}\circ\left(a+b\right)\right)=f\left(a\right)^{-}\circ\left(f\left(a\right)+f\left(b\right)\right)=\sigma^{C}_{f\left(a\right)}f\left(b\right),$

i.e., $\sigma^{B}$ and $\sigma^{C}$ are homomorphic via $f$.
Conversely, let $\sigma,\varphi$ affine structures on $G$ and $H$,
respectively, and assume that they are homomorphic via a group homomorphism
$f$ from $G$ to $H$, thus $\varphi_{f\left(a\right)}f=f\sigma_{a}$, for every
$a\in B$. If $a,b\in B$, it follows that

$\displaystyle
f\left(a+b\right)=f\left(a\circ\sigma_{a}\left(b\right)\right)=f\left(a\right)\circ
f\sigma_{a}\left(b\right)=f\left(a\right)\circ\varphi_{f\left(a\right)}f\left(b\right)=f\left(a\right)+f\left(b\right),$

hence $f$ is a semi-brace homomorphism from $B$ to $C$. Therefore, the claim
follows. ∎

As a consequence of 12, we also have that the affine structures associated to
two isomorphic semi-braces are equivalent and vice versa.

Considering the class $\mathcal{O}b(\mathcal{SB})$ of semi-braces and the
class $\mathcal{M}or(\mathcal{SB})$ of all homomorphisms between semi-braces,
let us denote by $\mathcal{SB}$ the _category of semi-braces_. Moreover, if
$\mathcal{O}b(\mathcal{A})$ is the class of all affine structures and
$\mathcal{M}or(\mathcal{A})$ the class of all homomorphisms between affine
structures, let us denote by $\mathcal{A}$ the _category of affine
structures_.

######  Corollary 13.

The categories $\mathcal{SB}$ and $\mathcal{A}$ are equivalent.

###### Proof.

By 10 and 9, one can define the maps
$F_{1}:\mathcal{O}b(\mathcal{SB})\to\mathcal{O}b(\mathcal{A})$ and
$F_{2}:\mathcal{M}or(\mathcal{SB})\to\mathcal{M}or(\mathcal{A})$ such that
$F_{1}\left(B\right)=\sigma^{B}$, for every $B\in\mathcal{O}b(\mathcal{SB})$,
and $F_{2}\left(f\right)=f$, for every $f\in\mathcal{M}or(\mathcal{SB})$.
Thus, by 12, we have that $F:=\left(F_{1},F_{2}\right)$ is a functor from the
category $\mathcal{SB}$ to the category $\mathcal{A}$. Analogously, one can
define the maps $G_{1}:\mathcal{O}b(\mathcal{A})\to\mathcal{O}b(\mathcal{SB})$
and $G_{2}:\mathcal{M}or(\mathcal{A})\to\mathcal{M}or(\mathcal{SB})$ such that
$G_{1}\left(\sigma\right)=B_{\sigma}$, for every
$\sigma\in\mathcal{O}b(\mathcal{A})$, and $G_{2}\left(f\right)=f$, for every
$f\in\mathcal{M}or(\mathcal{A})$, so that $G:=\left(G_{1},G_{2}\right)$ is a
functor from the category $\mathcal{A}$ to the category $\mathcal{SB}$.
Furthermore, by 2 and 10, it follows that $FG=1_{\mathcal{A}}$ and
$GF=1_{\mathcal{SB}}$, where $1_{\mathcal{A}}$ and $1_{\mathcal{SB}}$ are the
identity functor on $\mathcal{A}$ and $\mathcal{SB}$, respectively. Therefore,
the claim follows. ∎

Let us conclude this section by illustrating that the existence of a
cancellative affine structure on a group $G$ is strictly linked to a Zappa-
Szép product of $G$ and $G$ as semigroups. For this purpose and to fix the
notation, let us recall the Zappa-Szép product of two semigroups (see [32] and
[44]). Let $\left(S,\circ\right),\left(T,\circ\right)$ be two semigroups, and
$\eta:T\to S^{S}$ and $\delta:S\to T^{T}$ maps. Set
${}^{u}a:=\eta\left(u\right)\left(a\right)$ and
$u^{a}:=\delta\left(a\right)\left(u\right)$, for all $a\in S$ and $u\in T$, if
the following conditions hold

${}^{u}(a\circ b)={}^{u}a\circ{}^{u^{a}}b$ ${}^{u\circ
v}a={}^{u}\left({}^{v}a\right)$ (Z1) $\displaystyle\left(u\circ
v\right)^{a}=u^{{}^{v}a}\circ v^{a}$ $\displaystyle u^{a\circ b}=(u^{a})^{b}$
(Z2)

for all $a,b\in S$ and $u,v\in T$, then $S\times T$ is a semigroup with
respect to the operation defined by

$\displaystyle\left(a,u\right)\circ\left(b,v\right)=\left(a\circ{}^{u}b,u^{b}\circ
v\right),$ (3)

for all $a,b\in S$ and $u,v\in T$, called the _Zappa-Szép product of $S$ and
$T$ via $\eta$ and $\delta$_. In addition, if $S$ and $T$ are groups and
$\eta(T)\subseteq\operatorname{Sym}(S)$ and
$\delta(S)\subseteq\operatorname{Sym}(T)$, then $S\times T$ endowed with the
operation (3) is a group.

Below we show that, given a group $G$, specific semigroup Zappa-Szép products
of $G$ and $G$ determine affine structures on $G$. Moreover, the converse is
true if we consider cancellative affine structures. We remark that this result
is intrinsic in [40, Lemma 6.9] and [20, Theorem 5.3] in the context of skew
braces, and in [38, Section 2] for braces.

######  Proposition 14.

Let $G=\left(B,\circ\right)$ be a group. If $G\bowtie G$ is the Zappa-Szép
product of $G$ and $G$ via $\eta:B\to B^{B}$ and $\delta:B\to B^{B}$ such that

$\displaystyle u\circ a={}^{u}a\circ{u}^{a},$ (4)

for all $a\in S$ and $u\in T$, then the map $\sigma:B\to B^{B}$ defined by
$\sigma_{u}\left(a\right)={}^{u^{-}}a$, for every $u\in B$, is an affine
structure on $G$.
Conversely, if $\sigma$ is a cancellative affine structure on $G$, considered
$\eta:B\to B^{B}$ and $\delta:B\to\operatorname{Sym}_{B}$ the maps defined by
${}^{u}a:=\sigma_{u^{-}}\left(a\right)$ and
${u}^{a}:=\left({}^{u}a\right)^{-}\circ u\circ a$, for all $a,u\in B$,
respectively, then (2) holds and $G\times G$ endowed with the operation (3) is
the Zappa-Szép product of $G$ and $G$ via $\eta$ and $\delta$.

###### Proof.

At first assume that $Z$ is the Zappa product of $G$ and $G$ via $\eta:B\to
B^{B}$ and $\delta:B\to B^{B}$ such that (4) is satisfied. Then, if $a,b,u\in
B$, we have that

$\displaystyle\sigma_{u}\left(a\circ\sigma_{a}\left(b\right)\right)$
$\displaystyle={}^{u^{-}}\left(a\circ{}^{a^{-}}b\right)={}^{u^{-}}a\circ{}^{\left(u^{-}\right)^{a}\circ
a^{-}}b=\sigma_{u}\left(a\right)\circ{}^{\sigma_{u}\left(a\right)^{-}\circ
u^{-}}\left(b\right)$
$\displaystyle=\sigma_{u}\left(a\right)\circ\sigma_{\sigma_{u}\left(a\right)}\sigma_{u}\left(b\right),$

hence $\sigma$ is an affine structure on $G$.
Conversely, suppose that $\sigma$ is a cancellative affine structure on $G$.
Then, (4) is obvious. Besides, if $a,b,u,v\in B$, clearly ${}^{u\circ
v}a={}^{u}\left({}^{v}a\right)$ and, since $\sigma_{0}=\operatorname{id}_{B}$,

$\displaystyle u^{a\circ b}$ $\displaystyle=\sigma_{u^{-}}\left(a\circ
b\right)^{-}\circ u\circ a\circ
b=\sigma_{\left(\sigma_{u^{-}}\left(a\right)^{-}\circ u\circ
a\right)^{-}}\left(b\right)^{-}\circ\sigma_{u^{-}}\left(a\right)^{-}\circ
u\circ a\circ b$
$\displaystyle=\left({}^{u^{a}}b\right)^{-}\circ\left({}^{u}a\right)^{-}\circ
u\circ a\circ b=\left({}^{u^{a}}b\right)^{-}\circ u^{a}\circ
b=\left(u^{a}\right)^{b}.$

In addition,

${}^{u}\left(a\circ
b\right)=\sigma_{u^{-}}\left(a\circ\sigma_{a}\sigma_{a^{-}}\left(b\right)\right)=\sigma_{u^{-}}\left(a\right)\circ\sigma_{\left(u^{a}\right)^{-}}\left(b\right)=\,^{u}a\circ\,^{u^{a}}b.$

and

$\displaystyle u^{{}^{v}a}\circ
v^{a}=\left(\sigma_{u^{-}}\sigma_{v^{-}}\left(a\right)\right)^{-}\circ u\circ
v\circ a=\sigma_{\left(u\circ v\right)^{-}}\left(a\right)^{-}\circ u\circ
v\circ a=\left(u\circ v\right)^{a},$

thus (Z1) and (Z2) are satisfied. Therefore, the claim follows. ∎

######  Remark 3.

Observe that the maps $\eta$ and $\delta$ involved in 14 coincide with the
known maps $\lambda$ and $\rho$ in 8, respectively. Indeed, by 10, we have
that ${}^{u}a=\lambda_{u}\left(a\right)$ and
$u^{a}=\lambda_{u}\left(a\right)^{-}\circ u\circ
a=\left(u^{-}+a\right)^{-}\circ a=\rho_{a}\left(u\right)$.

##  3 The special cases of skew braces

In this section, we analyze the skew braces obtained by the examples of
groupal affine structures provided in Section 1.

Initially, let us recall that, according to [19, Definition 2] (removing the
assumption of finiteness), a skew brace $B$ is said to be a _bi-skew brace_
if, other than condition ($\ast\ast$ ‣ Introduction), the identity

$\displaystyle a+b\circ c=\left(a+b\right)\circ a^{-}\circ\left(a+c\right)$
($\ast^{\prime}$)

holds, for all $a,b,c\in B$. Equivalently, we observe that a skew brace $B$ is
a bi-skew brace if the identity

$\displaystyle a+b\circ c=\left(a+b\right)\circ\left(a+\left(-a\right)\circ
c\right)$ ($\ast^{\prime\prime}$)

holds, for all $a,b,c\in B$, that is the relation consistent with ($\ast\ast$
‣ Introduction). Easy examples of bi-skew braces are those obtained starting
from a group $\left(B,\circ\right)$ and by setting $a+b:=a\circ b$ or
$a+b:=b\circ a$, for all $a,b\in B$, that, according to the terminology in
[31] and [15] are called _trivial skew brace_ and _almost trivial skew brace_
, respectively.

Below, we provide the following characterization that allows one for obtaining
all bi-skew braces on a given group. In particular, we have that any bi-skew
brace on a fixed group $G$ corresponds to specific semidirect products of $G$
with itself.

######  Proposition 15.

Let $G=\left(B,\circ\right)$ be a group and $\sigma$ a groupal affine
structure on $G$. Then, $B_{\sigma}$ is a bi-skew brace if and only if
$\sigma^{B}_{a}$ is an automomorphism of $\left(B,\circ\right)$, for every
$a\in B$.

###### Proof.

Observe that $B_{\sigma}$ is a bi-skew brace if and only if the condition
($\ast^{\prime}$ ‣ 3) is satisfied. Since $a+b\circ
c=a\circ\sigma^{B}_{a}\left(b\circ c\right)$ and

$\displaystyle\left(a+b\right)\circ
a^{-}\circ\left(a+c\right)=a\circ\sigma^{B}_{a}\left(b\right)\circ\sigma^{B}_{a}\left(c\right),$

for all $a,b,c\in B$, the statement follows by comparing the two equalities
obtained. ∎

In other words, as one can expect, a skew brace $B$ is a bi-skew brace if and
only if $\lambda_{a}\in\operatorname{Aut}\left(B,\circ\right)$, for every
$a\in B$. In addition, if $\psi_{B}$ is the groupal affine associated to the
skew brace $\left(B,\circ,+\right)$, then $\psi^{B}_{a}=\sigma^{B}_{a^{-}}$,
for every $a\in B$. In fact, if $a,b\in B$, then $a\circ
b=a+\psi^{B}_{a}\left(b\right)$ and so
$\psi^{B}_{a}\left(b\right)=\lambda_{a}\left(b\right)=\sigma^{B}_{a^{-}}\left(b\right)$.

The following is another characterization of skew braces that are bi-skew
braces. In particular, by 10, such a characterization is consistent with the
results contained in [8, Theorem 3.1] and in [6, Proposition 3.6].

######  Corollary 16.

Let $B$ be a skew brace. Then, $B$ is a bi-skew brace if and and only if
$\sigma_{a\circ\sigma_{a}\left(b\right)}=\sigma_{b\circ a}$, for all $a,b\in
B$, or, equivalently $\sigma^{B}$ is a homomorphism from $\left(B,+\right)$ to
$\operatorname{Aut}\left(B,+\right)$.

###### Proof.

Let $a,b,c\in B$. If $B$ is a bi-skew brace, by 15,

$\displaystyle\sigma^{B}_{a}\left(b\right)\circ\sigma^{B}_{\sigma^{B}_{a}\left(b\right)}\sigma^{B}_{a}\left(c\right)=\sigma^{B}_{a}\left(b\circ\sigma^{B}_{b}\left(c\right)\right)=\sigma^{B}_{a}\left(b\right)\circ\sigma^{B}_{a}\sigma^{B}_{b}\left(c\right)$

i.e., $\sigma^{B}_{a\circ\sigma_{a}\left(b\right)}=\sigma^{B}_{b\circ a}$.
Conversely, if we assume that
$\sigma^{B}_{a\circ\sigma_{a}\left(b\right)}=\sigma^{B}_{b\circ a}$, for all
$a,b\in B$, we obtain that

$\displaystyle\sigma^{B}_{a}\left(b\right)\circ\sigma^{B}_{a}\left(c\right)$
$\displaystyle=\sigma^{B}_{a}\left(b\right)\circ\sigma^{B}_{b\circ
a}\sigma^{B}_{b^{-}}\left(c\right)=\sigma^{B}_{a}\left(b\right)\circ\sigma^{B}_{a+b}\sigma^{B}_{b^{-}}\left(c\right)$
$\displaystyle=\sigma^{B}_{a}\left(b\right)\circ\sigma^{B}_{\sigma^{B}_{a}\left(b\right)}\sigma^{B}_{a}\left(\sigma^{B}_{b^{-}}\left(c\right)\right)=\sigma^{B}_{a}\left(b\circ\sigma^{B}_{b}\sigma^{B}_{b^{-}}\left(c\right)\right)=\sigma^{B}_{a}\left(b\circ
c\right),$

hence, by 15, $B$ is a bi-skew brace. Therefore the claim is proved. ∎

By the previous result, it follows also that any bi-skew brace satisfies
$\sigma^{B}_{a^{-}}=\sigma^{B}_{-a}$, for every $a\in B$, or, equivalently,
$\lambda_{a^{-}}=\lambda_{-a}$, for every $a\in B$.

In the next, we analyze skew braces obtained through the instances of groupal
affine structures provided in Section 1.

######  Example 7.

Given a group $G=\left(B,\circ\right)$, by 15 we obtain that the groupal
affine structures $\sigma:B\to\operatorname{Sym}_{B}$ in 3 defined by
$\sigma_{a}\left(b\right)=f\left(a\right)^{-}\circ b\circ f\left(a\right)$,
for all $a,b\in B$, with $f$ an idempotent endomorphism of $G$, give rise to
bi-skew braces $B_{\sigma}$ having $G$ as multiplicative group. Clearly, if
$f\left(a\right)=0$, for every $a\in B$, then $B_{\sigma}$ is the trivial skew
brace on $G$; if $f=\operatorname{id}_{B}$, then $B_{\sigma}$ is the almost
trivial skew brace on $G$.
Among these examples of bi-skew braces, one can find some that are not
$\lambda$-homomorphic, i.e., such that
$\lambda_{a+b}\neq\lambda_{a}\lambda_{b}$. In this way, by the observations in
[6, Section 5], one can deduce that these are skew braces different from those
provided in [19], [29, 30], and [10, 9]. To show this fact, at first note that
$B_{\sigma}$ is $\lambda$-homomorphic if and only if

$f\left(a\circ b\right)\circ c\circ f\left(a\circ b\right)^{-}=f\left(b\circ
a\right)\circ c\circ f\left(b\circ a\right)^{-},$

for all $a,b\in B$. Then, considered $C_{2}=\langle x\rangle$ the cyclic group
of order $2$ and
$\operatorname{Sym}_{3}=\langle\left(1\,2\,3\right),\left(1\,2\right)\rangle$
the symmetric group of degree $3$, let $G$ be the direct product
$G=C_{2}\times\operatorname{Sym}_{3}$ (in particular $G$ is isomorphic to the
dihedral group $D_{12}$) and $f:G\to G$ the idempotent endomorphism of $G$
given by $f\left(x,\left(1\,2\right)\right)=\left(x,\left(1\,2\right)\right)$,
$f\left(x,\operatorname{id}_{3}\right)=\left(1,\operatorname{id}_{3}\right)$,
and
$f\left(1,\left(1\,2\,3\right)\right)=\left(1,\left(1\,2\,3\right)\right)$. By
choosing $a=c=\left(x,\left(1\,2\right)\right)$ and
$b=\left(1,\left(1\,2\,3\right)\right)$, we obtain that $f\left(a\circ
b\right)\circ c\circ f\left(a\circ b\right)^{-}\neq f\left(b\circ
a\right)\circ c\circ f\left(b\circ a\right)^{-}$. Consequently, the bi-skew
brace $B_{\sigma}$ is not $\lambda$-homomorphic.

######  Examples 8.

  1. 1.

Let $G=\left(B,\circ\right)$ be the infinite cyclic group and
$\sigma:B\to\operatorname{Sym}_{B}$ the affine structure in 4-$1.$ defined by
$\sigma_{g^{k}}\left(g^{l}\right):=g^{\left(-1\right)^{k}l}$, for all
$k,l\in\mathbb{Z}$. Then, it is a routine computation to verify that
$\sigma_{g^{k}}$ is an automorphism of $G$, for every $k\in\mathbb{Z}$, thus,
by 15, $B_{\sigma}$ is a bi-skew brace. More specifically, the addition of
$B_{\sigma}$ is given by

$\displaystyle g^{k}\,+\,g^{l}=g^{k+\left(-1\right)^{k}l},$ (5)

for all $k,l\in\mathbb{Z}$, hence $\left(B,+\right)$ is isomorphic to the
infinite dihedral group $\mathbb{D}_{\infty}$. Note also that the skew brace
$\left(B,\circ,+\right)$ coincides with the structure of brace contained in
[36, Proposition 6].

  2. 2.

By similar observations in $1.$, we obtain that the first class of examples of
affine structure on the cyclic group $C_{m}$ with $m=2l$ in 5 are bi-skew
braces with additive group $\left(B,+\right)$ isomorphic to the dihedral group
$\mathbb{D}_{l}$. In particular, in the case of $m=6$, the skew brace
$\left(B,\circ,+\right)$ coincides with the brace of order $6$ obtained by
Rump in [35, Example 3].

  3. 3.

If $G=\left(B,\circ\right)$ is the infinite cyclic group and
$\sigma:B\to\operatorname{Sym}_{B}$ is the affine structure in 4-$2.$ defined
by
$\sigma_{g^{k}}\left(g^{l}\right):=g^{k\left(-1+\left(-1\right)^{l}\right)+l}$,
for all $k,l\in\mathbb{Z}$, then one can check that the maps $\sigma_{g^{k}}$
are not automorphisms of $G$, hence $B_{\sigma}$ is not a bi-skew brace.
Furthermore, the sum of such a skew brace is given by

$\displaystyle g^{k}\,+\,g^{l}=g^{l+\left(-1\right)^{l}k},$

for all $k,l\in\mathbb{Z}$. Besides, note that this structure of skew brace on
$\left(\mathbb{Z},+\right)$ is nothing but the opposite skew brace obtained in
$1)$, i.e., the skew brace obtained by considering the opposite sum of such a
skew brace (see [31, Proposition 3.1]).

  4. 4.

The affine structures on the cyclic group $C_{m}$ with $m=2l$ in the second
class of examples in 5 give rise to bi-skew braces if $m\mid 4$. In fact, it
is a routine computation to check that $\sigma_{g^{k}}$ is an automorphism of
$G$ if and only if $m\mid 4$.

######  Example 9.

The skew brace on the cyclic group $C_{8}$ obtained through the affine
structure in 6 is a non-trivial brace with additive group isomorphic to
$C_{8}$. In particular, this brace is obtained through the affine structure
$\omega$ which gives rise to a skew brace that has the multiplicative
structure isomorphic to $C_{8}$ and additive structure isomorphic to the
dihedral group $\mathbb{D}_{8}$.

######  Remark 4.

In [42, Problem 2.27], Vendramin posed the question of classifying isomorphism
classes of skew braces with multiplicative group isomorphic to
$\left(\mathbb{Z},+\right)$. In this regard, the skew brace in 8-$3)$
identifies a class of skew brace that is different to that in 8-$2)$ which is
already known. Note also that 8-$3)$ can be found in [41] where the problem
mentioned above is solved.

##  4 Affine structures on Zappa products of groups

This section aims to provide a construction of affine structures on groups
that are the Zappa product of two given ones both endowed with affine
structures. Furthermore, we show that such a construction allows to obtain
semi-braces that are different from those determined by the matched product of
semi-braces contained in [14, Theorem 6] (see also [28, p. 249]).

To this purposes, let us first present a construction of groups that is
equivalent to that of Zappa product of two groups (this one recalled at the
end of Section 2) that is already intrinsic in [3], [40], [13], [14], and
[16]. This alternative way to consider Zappa product of two groups allows one
to compare the construction provided in this work with the matched product of
semi-braces, skew braces, and braces contained in the papers mentioned above.

Let $H:=\left(S,\circ\right),K:=\left(T,\circ\right)$ be groups,
$\alpha:T\to\operatorname{Sym}_{S}$ a homomorphism from $H$ to the symmetric
group of $S$, and $\beta:S\to\operatorname{Sym}_{T}$ a homomorphism from $K$
to the symmetric group of $T$ such that

$\displaystyle\alpha_{u}{\left(\alpha^{-1}_{u}{\left(a\right)}\circ
b\right)}=a\circ\alpha_{\beta^{-1}_{a}{\left(u\right)}}{\left(b\right)}\qquad\quad\beta_{a}\left(\beta^{-1}_{a}\left(u\right)\circ
v\right)=u\circ\beta_{\alpha^{-1}_{u}{\left(a\right)}}\left(v\right)$ (6)

are satisfied, for all $a,b\in S$ and $u,v\in T$. Let us call the quadruple
$(S,T,\alpha,\beta)$ a _matched product system of groups_. Then, the binary
operation given by

$\displaystyle\left(a,u\right)\circ\left(b,v\right):=\left(a\circ\alpha_{\beta^{-1}_{a}{\left(u\right)}}{\left(b\right)},\
u\circ\beta_{\alpha^{-1}_{u}{\left(a\right)}}\left(v\right)\right),$ (7)

for all $\left(a,u\right),\left(b,v\right)\in S\times T$, makes $S\times T$
into a group with identity $\left(0,0\right)$ and such that any
$\left(a,u\right)\in S\times T$ has inverse given by
$\left(\alpha^{-1}_{\beta^{-1}_{a}\left(u\right)}\left(a^{-}\right),\,\beta^{-1}_{\alpha^{-1}_{u}\left(a\right)}\left(u^{-}\right)\right)=\left(\alpha^{-1}_{u}\left(a\right)^{-},\;\beta^{-1}_{a}\left(u\right)^{-}\right)$.
Finally, set $\sigma_{u}=\alpha_{u}$ and
$\delta_{a}\left(u\right)=\beta^{-1}_{\alpha_{u}\left(a\right)}\left(u\right)$,
it follows that $\left(S\times T,\cdot\right)$ is the Zappa product of $H$ and
$K$ via $\sigma$ and $\delta$ and the map $f:S\times T\to S\times T$ defined
by $\varphi\left(a,u\right)=\left(a,\beta^{-1}_{a}\left(u\right)\right)$, for
all $a\in S$ and $u\in T$, is an isomorphism from $G$ to the Zappa product of
$S$ and $T$. Furthermore, let us denote the group $G$ as
$H\bowtie_{\alpha,\beta}K$.

Now, we recall the construction of the matched product of semi-braces. Let
$S,T$ be semi-braces, $\alpha:T\to\operatorname{Aut}\left(S\right)$ a
homomorphism from $\left(T,\circ\right)$ to the automorphism group of
$\left(S,+\right)$, and $\beta:S\to\operatorname{Aut}\left(T\right)$ a
homomorphism from $\left(S,\circ\right)$ to the automorphism group of
$\left(T,+\right)$ such that identities in (6) are satisfied. Then,
$(S,T,\alpha,\beta)$ is called a _matched product system of semi-braces_. In
particular, the following conditions are satisfied

$\displaystyle\lambda_{a}\alpha_{\beta^{-1}_{a}\left(u\right)}=\alpha_{u}\lambda_{\alpha^{-1}_{u}\left(a\right)}\qquad\lambda_{u}\beta_{\alpha^{-1}_{u}\left(a\right)}=\beta_{a}\lambda_{\beta^{-1}_{a}\left(u\right)},$
(8)

for every $\left(a,u\right)\in S\times T$.

######  Theorem 17 (Theorem 6, [14]).

Let $\left(S,T,\alpha,\beta\right)$ be a matched product system of semi-
braces. Then, $S\times T$ with respect to the operations

$\displaystyle\left(a,u\right)\oplus\left(b,v\right)=\left(a+b,\,u+v\right)\qquad\left(a,u\right)\circ\left(b,v\right)=\left(a\circ\alpha_{\beta^{-1}_{a}\left(u\right)}\left(b\right),u\circ\beta_{\alpha^{-1}_{u}\left(a\right)}\left(v\right)\right),$

for all $\left(a,u\right),\left(b,v\right)\in S\times T$, is a left semi-brace
called the _matched product_ of $S$ and $T$ via $\alpha$ and $\beta$.

Clearly, the additive structure is the direct sum of $\left(S,+\right)$ and
$\left(T,+\right)$ and the multiplicative one is the Zappa product
$\left(S,\circ\right)\bowtie_{\alpha,\beta}\left(T,\circ\right)$.

From now on, to avoid overloading the notation, if $\sigma^{S}$ and
$\sigma^{T}$ are affine structures on two given groups
$H=\left(S,\circ\right)$ and $K=\left(T,\circ\right)$, respectively, we will
write $\sigma^{S}_{a}$ and $\sigma^{T}_{u}$ simply as $\sigma_{a}$ and
$\sigma_{u}$, respectively, for all $a\in S$ and $u\in T$. Moreover, we set
$\bar{a}:=\alpha^{-1}_{u}\left(a\right)$ and
$\bar{u}:=\beta^{-1}_{a}\left(u\right)$, for every pair $\left(a,u\right)\in
S\times T$. In addition, it is useful to recall that, if $B$ is a semi-brace,
then the map $\rho_{a}$ can be written in terms of affine structure as
$\rho_{b}\left(a\right)=\sigma_{a^{-}}\left(b\right)^{-}\circ a\circ b$, for
all $a,b\in B$.

The following theorem allows for constructing affine structures on groups
$H\bowtie_{\alpha,\beta}K$. Specifically, it provides sufficient conditions so
that the direct product $\sigma^{S}\times\sigma^{T}$ of two affine structures
$\sigma^{S}$ and $\sigma^{T}$ is an affine structure on a group of the type
$H\bowtie_{\alpha,\beta}K$.

######  Theorem 18.

Let $H=\left(S,\circ\right),K=\left(T,\circ\right)$ be groups endowed with
affine structures $\sigma^{S}$ and $\sigma^{T}$, respectively, and
$(S,T,\alpha,\beta)$ a matched product system of groups. If the following
conditions

$\displaystyle\sigma_{0}\alpha_{u}=\alpha_{u}\sigma_{0}$
$\displaystyle\qquad\sigma_{0}\beta_{a}=\beta_{a}\sigma_{0}$ (I)
$\displaystyle\sigma_{\alpha_{u}\left(a\right)}=\sigma_{a}$
$\displaystyle\qquad\sigma_{\beta_{a}\left(u\right)}=\sigma_{u}$ (II)
$\displaystyle\alpha_{\bar{v}}\sigma_{\rho_{b}\left(a^{-}\right)}=\sigma_{\rho_{b}\left(a^{-}\right)}\alpha_{\overline{V}}$
$\displaystyle\qquad\beta_{\bar{b}}\sigma_{\rho_{v}\left(u^{-}\right)}=\sigma_{\rho_{v}\left(u^{-}\right)}\beta_{\overline{B}}$
(III)

are satisfied, for all $a,b\in S$ and $u,v\in T$, where
$\overline{V}=\beta^{-1}_{\sigma_{a}\left(b\right)}\sigma_{u}\left(v\right)$
and
$\overline{B}=\alpha^{-1}_{\sigma_{u}\left(v\right)}\sigma_{a}\left(b\right)$,
then the map $\sigma:=\sigma^{S}\times\sigma^{T}$ is an affine structure on
the group $H\bowtie_{\alpha,\beta}K$.

###### Proof.

Let $\left(a,u\right),\left(b,v\right),\left(c,w\right)\in S\times T$. Then,
by (II),

$\displaystyle\sigma_{\left(a,u\right)\circ\left(b,v\right)}\left(c,w\right)$
$\displaystyle=\left(\sigma_{\alpha_{\bar{u}\left(b\right)}}\sigma_{a}\left(c\right),\,\sigma_{\beta_{\bar{a}\left(v\right)}}\sigma_{u}\left(w\right)\right)=\left(\sigma_{b}\sigma_{a}\left(c\right),\,\sigma_{v}\sigma_{u}\left(w\right)\right)$
$\displaystyle=\sigma_{\left(b,v\right)}\sigma_{\left(a,u\right)}\left(c,w\right),$

i.e., $\sigma$ is an anti-homomorphism. Moreover, set
$X:=\sigma_{\left(a,u\right)}\left(\left(b,v\right)\circ\sigma_{\left(b,v\right)}\left(c,w\right)\right)$
we have that

$\displaystyle
X=\left(\sigma_{a}\left(b\circ\alpha_{\bar{v}}\sigma_{b}\left(c\right)\right),\,\sigma_{u}\left(v\circ\beta_{\bar{b}}\sigma_{v}\left(w\right)\right)\right)$

where, the first component of $X$ can be written as

$\displaystyle\sigma_{a}\left(b\circ\alpha_{\bar{v}}\sigma_{0}\sigma_{b}\left(c\right)\right)$
$\displaystyle=\sigma_{a}\left(b\circ\sigma_{0}\alpha_{\bar{v}}\sigma_{b}\left(c\right)\right)$
by (I)
$\displaystyle=\sigma_{a}\left(b\circ\sigma_{b}\left(\sigma_{b^{-}}\alpha_{\bar{v}}\sigma_{b}\left(c\right)\right)\right)$
$\displaystyle=\sigma_{a}\left(b\right)\circ\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(\sigma_{b^{-}}\alpha_{\bar{v}}\sigma_{b}\left(c\right)\right)$
$\displaystyle=\sigma_{a}\left(b\right)\circ\sigma_{b^{-}\circ
a\circ\sigma_{a}\left(b\right)}\alpha_{\bar{v}}\sigma_{b}\left(c\right)$
$\displaystyle=\sigma_{a}\left(b\right)\circ\sigma_{\rho_{b}\left(a^{-}\right)^{-}}\alpha_{\bar{v}}\sigma_{b}\left(c\right).$

In addition, set
$Y:=\sigma_{\left(a,u\right)}\left(b,v\right)\circ\sigma_{\sigma_{\left(a,u\right)}\left(b,v\right)}\sigma_{\left(a,u\right)}\left(c,w\right)$,
we obtain that

$\displaystyle Y$
$\displaystyle=\left(\sigma_{a}\left(b\right),\sigma_{u}\left(v\right)\right)\circ\left(\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(c\right),\sigma_{\sigma_{u}\left(v\right)}\sigma_{u}\left(w\right)\right)$
$\displaystyle=\left(\sigma_{a}\left(b\right)\circ\alpha_{\overline{V}}\sigma_{\sigma_{a}\left(b\right)}\sigma_{a}\left(c\right),\,\sigma_{u}\left(v\right)\circ\beta_{\overline{B}}\sigma_{\sigma_{u}\left(v\right)}\sigma_{u}\left(w\right)\right)$
$\displaystyle=\left(\sigma_{a}\left(b\right)\circ\alpha_{\overline{V}}\sigma_{a\circ\sigma_{a}\left(b\right)}\left(c\right),\,\sigma_{u}\left(v\right)\circ\beta_{\overline{B}}\sigma_{u\circ\sigma_{u}\left(v\right)}\left(w\right)\right).$

Observing that

$\displaystyle\sigma_{\rho_{b}\left(a^{-}\right)^{-}}\alpha_{\bar{v}}\sigma_{b}$
$\displaystyle=\sigma_{\rho_{b}\left(a^{-}\right)^{-}}\alpha_{\bar{v}}\sigma_{0}\sigma_{b}$
$\displaystyle=\sigma_{0}\alpha_{\overline{V}}\sigma_{\rho_{b}\left(a^{-}\right)^{-}}\sigma_{b}$
by (III)
$\displaystyle=\alpha_{\overline{V}}\sigma_{b\circ\rho_{b}\left(a^{-}\right)^{-}\circ\,0}$
by (I)
$\displaystyle=\alpha_{\overline{V}}\sigma_{a\circ\sigma_{a}\left(b\right)},$

it follows that the first components of $X$ and $Y$, respectively, coincide.
Similarly, the second components are equal. Therefore, $\sigma$ is an affine
structure on $H\bowtie_{\alpha,\beta}K$. ∎

Note that, if in 18 the affine structures on the groups $H$ and $K$,
respectively, are cancellative, then the identities in (I) are trivially
satisfied and the affine structure on $H\bowtie K_{\alpha,\beta}$ is
cancellative. Moreover, if the affine structures on $H$ and $K$ are groupal,
then also the affine structure $H\bowtie K_{\alpha,\beta}$ is. More
specifically, we have the following characterization.

######  Corollary 19.

Let $H=\left(S,\circ\right),K=\left(T,\circ\right)$ be groups endowed with
cancellative affine structures and $(S,T,\alpha,\beta)$ a matched product
system. Then, the map $\sigma:S\times T\to S\times T$ defined by
$\sigma_{\left(a,u\right)}\left(b,v\right)=\left(\sigma_{a}\left(b\right),\,\sigma_{u}\left(v\right)\right)$,
for all $\left(a,u\right),\left(a,u\right)\in S\times T$, is a cancellative
affine structure on the group $H\bowtie_{\alpha,\beta}K$ if and only if the
identities in (II) and (III) are satisfied.

Now, note that if $B_{\sigma}$ is a semi-brace obtained through the affine
structure constructed by 18, then

$\displaystyle\left(a,u\right)+\left(b,v\right)=\left(a\circ\alpha_{\bar{u}}\sigma_{a}\left(b\right),\,u\circ\beta_{\bar{a}}\sigma_{u}\left(v\right)\right),$

for all $\left(a,u\right),\left(b,v\right)\in S\times T$. In general, it is
obvious that such a sum is not the direct sum of the additive structure of $H$
and $K$ as it is in 17.
We also have that the restrictions of $+$ and $\circ$ to the set
$S^{\ast}=\\{\left(a,0\right)\ |\ a\in S\\}$ are exactly component-wise.
Indeed, this is a consequence of the fact that $\sigma_{0}\left(0\right)=0$,
$\alpha_{u}\left(0\right)=0\circ\alpha_{\beta^{-1}_{0}\left(u\right)}\left(0\right)=\alpha_{u}\left(\alpha_{u}^{-1}\left(0\right)\circ
0\right)=0$, and, similarly, $\beta_{a}\left(0\right)=0$. Analogously, the
restriction of $+$ and $\circ$ to the set $T^{\ast}=\\{\left(0,u\right)\ |\
u\in T\\}$ are exactly component-wise.

######  Remarks 5.

Let $H:=\left(S,\circ\right)$ and $K:=\left(T,\circ\right)$ be groups,
$\sigma^{T}$ and $\sigma^{S}$ cancellative affine structures on $H$ and $K$,
respectively.

  1. 1.

If $\alpha:T\to\operatorname{Sym}_{S}$ and $\beta:S\to\operatorname{Sym}_{T}$
are group homomorphisms such that the identities (6) hold, then the conditions
in (8) hold if and only if $\alpha_{u}\in\operatorname{Aut}\left(S,+\right)$
and $\beta\in\operatorname{Aut}\left(T,+\right)$. Indeed, if the conditions in
(8) hold, since
$\alpha_{u}\left(a\right)^{-}=\alpha_{\beta^{-1}_{\alpha_{u}\left(a\right)}\left(u\right)}\left(a^{-}\right)$,
it follows that

$\displaystyle\alpha_{\beta^{-1}_{\alpha_{u}\left(a\right)}\left(u\right)}\lambda_{a^{-}}=\alpha_{\beta^{-1}_{\alpha_{u}\left(a\right)}\left(u\right)}\lambda_{\alpha^{-1}_{\beta^{-1}_{\alpha_{u}\left(a\right)}\left(u\right)}\alpha_{u}\left(a\right)^{-}}=\lambda_{\alpha_{u}\left(a\right)^{-}}\alpha_{\beta^{-1}_{\alpha_{u}\left(a\right)^{-}}\beta^{-1}_{\alpha_{u}\left(a\right)}\left(u\right)}=\lambda_{\alpha_{u}\left(a\right)^{-}}\alpha_{u}.$

Hence, we obtain that

$\displaystyle\alpha_{u}\left(a+b\right)$
$\displaystyle=\alpha_{u}\left(a\circ\lambda_{a^{-}}\left(b\right)\right)=\alpha_{u}\left(a\right)\circ\alpha_{\beta^{-1}_{\alpha_{u}\left(a\right)}}\lambda_{a^{-}}\left(b\right)=\alpha_{u}\left(a\right)\circ\lambda_{\alpha_{u}\left(a\right)^{-}}\alpha_{u}\left(b\right)$
$\displaystyle=\alpha_{u}\left(a\right)+\alpha_{u}\left(b\right),$

i.e., $\alpha_{u}\in\operatorname{Aut}\left(S,+\right)$. Similarly, one has
that $\beta\in\operatorname{Aut}\left(T,+\right)$. The converse part is
contained in [14, Remark 5].

  2. 2.

If $\alpha:T\to\operatorname{Sym}_{S}$ and $\beta:S\to\operatorname{Sym}_{T}$
are group homomorphisms, if $\alpha_{u}\in\operatorname{Aut}\left(S,+\right)$
and $\beta_{a}\in\operatorname{Aut}\left(T,+\right)$, and
$\lambda_{a}\alpha_{\bar{u}}=\alpha_{u}\lambda_{\bar{a}}$ and
$\lambda_{u}\beta_{\bar{a}}=\alpha_{a}\lambda_{\bar{u}}$, for all $a\in S$ and
$u\in T$, then the identities (6) are satisfied.

Now, let us note that if $\left(S,T,\alpha,\beta\right)$ is a matched product
system of semi-braces, then the affine structure associated to the group
$\left(S\times T,\circ\right)$, by 17, is the map $\bar{\sigma}:T\times S\to
T\times S$ given by

$\displaystyle\bar{\sigma}_{\left(a,u\right)}\left(b,v\right)=\left(\alpha^{-1}_{\bar{u}}\sigma_{a}\left(b\right),\,\beta^{-1}_{\bar{a}}\sigma_{u}\left(v\right)\right),$

for all $\left(a,u\right),\left(b,v\right)\in S\times T$. As shown by the
examples below, one can see that, in general, $\bar{\sigma}$ and
$\sigma^{S}\times\sigma^{T}$ are not equivalent.

######  Example 10.

Let $H=\left(S,\circ\right)$ be a group and $\sigma$ the affine structure on
$H$ given by $\sigma_{a}=f$, for every $a\in S$, with $f$ an idempotent
endomorphism of $H$. Consider $T=S$, $\sigma_{u}=f$, for every $u\in T$, and
$\alpha:T\to\operatorname{Sym}_{S}$ and $\beta:S\to\operatorname{Sym}_{T}$ the
maps given by $\alpha_{u}=\operatorname{id}_{S}$ and
$\beta_{a}\left(u\right)=f\left(a\right)\circ u\circ f\left(a\right)^{-}$, for
all $a\in S$ and $u\in T$, respectively. Then $\left(S,T,\alpha,\beta\right)$
is a matched product system of groups that satisfies the assumptions of 18.
Then, the affine structure
$\sigma_{\left(a,u\right)}=\left(f\left(b\right),f\left(v\right)\right)$ on
the group $G=H\bowtie_{\alpha,\beta}K$ gives rise to the semi-brace having $G$
as multiplicative structure, namely,

$\left(a,u\right)\circ\left(b,v\right)=\left(a\circ b,\,u\circ
f\left(a\right)\circ v\circ f\left(a\right)^{-}\right),$

and such that the sum is given by

$\displaystyle\left(a,u\right)+\left(b,v\right)=\left(a\circ b,\,u\circ
f\left(a\circ v\circ a^{-}\right)\right),$

for all $\left(a,u\right),\left(b,v\right)\in S\times T$. It is a routine
computation to check that $\alpha$ and $\beta$ satisfy also assumption of 17,
and that the sum $\oplus$ in 17 coincides with that in 18 if and only if the
group $\operatorname{Im}f$ is abelian.

The following example ensures that, in general, a semi-brace obtained by 18 is
not isomorphic to that obtained in 17.

######  Example 11.

Let $H=\left(S,\circ\right)$ be the cyclic group $C_{6}=\langle a\rangle$ and
$K=\left(T,\circ\right)$ the cyclic group $C_{2}=\langle u\rangle$. Let
$\sigma^{S}$ be the groupal affine structure on the group $H$ given by
$\sigma_{a^{k}}\left(a^{l}\right)=a^{\left(-1\right)^{k}l}$, for all
$k,l\in\mathbb{Z}$, and $\sigma^{T}$ the abelian affine structure on the group
$K$ given by $\sigma_{u^{t}}=\operatorname{id}_{T}$, for every
$t\in\mathbb{Z}$. Consider $\alpha:T\to\operatorname{Sym}_{S}$ the map defined
by $\alpha_{u^{t}}\left(a^{k}\right)=a^{\left(-1\right)^{t}k}$, for all
$t,k\in\mathbb{Z}$, and $\beta:S\to\operatorname{Sym}_{T}$ the map such that
$\beta_{a^{k}}=\operatorname{id}_{T}$, for every $k\in\mathbb{Z}$. Then, it is
a routine computation to verify that $\left(S,T,\alpha,\beta\right)$ is a
matched product system of groups that satisfies the hypotheses of 18. In
particular, the groupal affine structure obtained in this way, namely,
$\sigma_{\left(a^{k},u^{t}\right)}\left(a^{l},u^{s}\right)=\left(a^{\left(-1\right)^{k}l},\,u^{t}\right)$,
determines a structure of skew brace with

$\displaystyle\left(a^{k},u^{t}\right)+\left(a^{l},u^{s}\right)=\left(a^{k+\left(-1\right)^{t+k}l},\,u^{t+s}\right)\qquad\left(a^{k},u^{t}\right)\circ\left(a^{l},u^{s}\right)=\left(a^{k+\left(-1\right)^{t}l},\,u^{t+s}\right),$

for all $k,l,t,s\in\mathbb{Z}$. In particular, the multiplicative group of
$B_{\sigma}$ is isomorphic to $\mathbb{D}_{12}$. Moreover, observe that if
$k=0$, $l=t=1$, and $l=2$, then $a^{k+\left(-1\right)^{t+k}h}=a^{-2}=a^{4}$,
instead $a^{l+\left(-1\right)^{s+l}k}=a^{2}$, hence the additive group of
$B_{\sigma}$ is not abelian. Finally, by $1.$ in 5, the maps $\alpha$ and
$\beta$ satisfy the hypotheses of 17, hence we obtain a skew brace with
multiplicative group equal to that of $B_{\sigma}$ and abelian additive
structure that is the direct product of $C_{6}$ and $C_{2}$.

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# Zero-Shot Generalization using
Intrinsically Motivated Compositional Emergent Protocols

Rishi Hazra, Sonu Dixit11footnotemark: 1, Sayambhu Sen
{rishihazra, sonudixit<EMAIL_ADDRESS>
Indian Institute of Science, Bangalore  Equal Contribution

###### Abstract

Human language has been described as a system that makes use of finite means
to express an unlimited array of thoughts. Of particular interest is the
aspect of compositionality, whereby, the meaning of a compound language
expression can be deduced from the meaning of its constituent parts. If
artificial agents can develop compositional communication protocols akin to
human language, they can be made to seamlessly generalize to unseen
combinations. Studies have recognized the role of curiosity in enabling
linguistic development in children. In this paper, we seek to use this
intrinsic feedback in inducing a systematic and unambiguous protolanguage. We
demonstrate how compositionality can enable agents to not only interact with
unseen objects but also transfer skills from one task to another in a zero-
shot setting: Can an agent, trained to ‘pull’ and ‘push twice’, ‘pull twice’?.

##  1 Introduction

In the recent past, there has been a great deal of research in the field of
emergent language in artificial agents interacting in simulated environments
Kirby (2001); Havrylov and Titov (2017); Tieleman et al. (2019); Gupta et al.
(2020). However, the real question here is, to what extent do these evolved
protocols resemble natural language? Recent studies have revealed the
following about emergent languages: (i) they do not conform to Zipf’s Law of
Abbreviation Chaabouni et al. (2019); (ii) communication protocols either do
not follow compositionality patterns of natural language Kottur et al. (2017)
or are not always interpretable Lowe et al. (2019); (iii) emerged protocols
are sensitive to experimental conditions Lazaridou et al. (2018).

Although compositionality is not crucial to achieving generalization, more
compositional protocols have been shown to display higher zero-shot
performance Ren et al. (2020). While work on incorporating compositionality
into emergent languages is still in its early stages, certain works Mordatch
and Abbeel (2018); Chaabouni et al. (2020) have proposed to use limited
channel capacity as a means to achieve composition. However, we argue that
agents may fail to develop meaningful communication protocols in such a
restricted setting. Motivated by human behavior, we formulate intrinsic
rewards Gopnik et al. (2001); Baldassarre and Mirolli (2013) to provide
incentives to the agents for paying attention to communication despite having
a limited channel capacity. Forced to deal with it to earn more intrinsic
rewards, the agents must learn to use a more systematic and unambiguous
protolanguage.

As proof of concept, we push the boundaries of compositionality to a more
challenging multi-task settings, arguing that it can also support the
acquisition of a more complex repertoire of skills (performing a pull twice
task when it has been trained to pull, push and push twice), in addition to
generalizing over novel composition of object properties (pushing red square
when it has been trained to push a red circle and a blue square)
111demos:https://sites.google.com/view/compositional-comm.

##  2 Problem Setup

We analyze a typical signalling game Lewis (1969), comprising a stationary
Speaker-Bot (speaker) and a mobile Listener-Bot (listener), by modelling it in
form of a Markov Decision Process specified by the tuple
$(\mathcal{S},\mathcal{O},\mathcal{A},r,\mathcal{T},\gamma)$. At the beginning
of each round, the speaker receives a natural language instruction (push a red
circle) and communicates the same using discrete messages $m_{i=1}^{n_{m}}$,
sampled from a message space $\mathcal{M}$, to the listener over a
communication channel. Here, $d_{m}$ is the dimension of the message $m_{i}$,
and $n_{m}$ is the number of messages (these constitute the channel capacity,
$|\mathrm{C}|=\mathrm{c}_{d_{m}}^{n_{m}}$). At each step $t$, the listener
receives an observation $\mathbf{o}^{(t)}\in\mathcal{O}$, comprising the 2D
grid-view and the received messages $m_{i=1}^{n_{m}}$, and takes an action
$\mathbf{a}^{(t)}\in\mathcal{A}$. The goal of the listener is to choose
optimal actions according to a policy
$\bm{\pi}:(\mathcal{O},m_{i=1}^{n_{m}})\mapsto\Delta(\mathcal{A})$, to
maximize its long-term reward $\mathcal{R}=\sum_{t}\gamma^{t}r^{(t)}$. Here,
$\gamma$ is the discount factor and $\mathcal{T}$ is the transition function
$\mathcal{T}:\mathcal{S}\times\mathcal{A}\mapsto\mathcal{S}$. The environment
generates a 0-1 (sparse) reward, i.e., the listener gets a reward of $r=1$ if
it achieves the specified task, otherwise $r=0$. However, the listener has no
information about either the task or the target, and relies on the speaker for
the same. Given a language $\mathcal{L}(.):\mathcal{C}\mapsto\mathcal{M}$, we
use topographic similarity (topsim) Brighton and Kirby (2006) between
$\mathcal{C}$ (set of concepts) and $\mathcal{M}$ (set of messages) as a
measure of compositionality. Our work is contrasted with that of gSCAN Ruis et
al. (2020) which focuses on rule-based generalization using a supervised
learning framework.

##  3 Approach

###  3.1 Environment Description

In our experiments, we use a $4\times 4$ grid. Cells in the grid contain
objects characterized by certain attributes like shape, size, color and
weight. These objects can either be the target object or the distractor
objects. Distractors have either the same color or the same shape (or both) as
that of the target. We keep the number of distractors fixed ($=2$). The
listener and the objects may spawn at any random location on the grid. Given
an instruction, it is first processed using a parser to
$\langle\mathrm{VERB},\\{\mathrm{ADJ}_{i}\\}_{i=1}^{3},\mathrm{NOUN}\rangle$222$\mathrm{VERB}$:
task (‘walk’, ‘push’, ‘pull’); $\mathrm{ADJ}$: object attributes like color
(‘red’, ‘blue’, ‘yellow’, ‘green’), size (‘small’, ‘big’) and weight (‘light’,
‘heavy’); $\mathrm{NOUN}$: object shape (‘square’, ‘circle’, ‘cylinder’,
‘diamond’). The speaker transmits the same using a set of messages to the
listener which, then, processes the grid representation and the received
messages to achieve the given task. In our experiments, we use a
$\\{0,1\\}^{d_{grid}\times 4\times 4}$ vector array for the grid
representation, where each cell has a $d_{grid}$-dimensional encoding.

Figure 1: Model Description Figure 2: [Best viewed in color] Demonstration of
Intrinsic Speaker on the numeral split for task PULL TWICE. Here, the green
circle is heavy, hence the listener has to apply two units of force (TWICE) to
pull it.

###  3.2 Model Description

(1,2) The speaker receives the parsed input instruction parsed. (3) The
speaker uses an encoder to map the concept input to a hidden representation
$\in\mathbb{R}^{n_{m}\times d_{h}}$. From this representation, a set of one-
hot encoded messages $m_{i=1}^{n_{m}}\in\\{0,1\\}^{d_{m}}$ are sampled (during
training) using Categorical sampling, which are then transmitted over the
communication channel. The number of messages $n_{m}$ is set to
$|\langle\mathrm{VERB},\\{\mathrm{ADJ}_{i}\\}_{i=1}^{3},\mathrm{NOUN}\rangle|$.
During evaluation, sampling is replaced with an $\operatorname*{arg\,max}(.)$.
We use the Straight Through trick Jang et al. (2017) to retain
differentiability. (4) At each step, the grid input is mapped in the Grid
Encoder to an output $\mathcal{G}_{t}$ $\in\mathbb{R}^{d_{\mathcal{G}}\times
4\times 4}$. (5) Next, we compute the attention weights $\alpha_{i=1}^{16}$
for each grid cell by taking a normalized dot product between $z$ and
$\mathcal{G}_{t}^{i\times d_{\mathcal{G}}}$. A weighted combination is fed to
the sub-policy networks.

The listener learns to (i) identify the target object in the context of
distractors, (ii) interact with the target object by executing a task
specified by the speaker. We use a hierarchical-RL framework Sutton et al.
(1999) for training. There are two sub-policies corresponding to the PUSH and
the PULL tasks. (6, 7) In each round, the master policy selects either sub-
policies using the received messages333actions spaces: master policy: {A, B,
Null}; subpolicy A/B: {left, right, forward, backward, push/pull}. The sub-
policies have a shared input, which includes the grid encoder and the
attention network. The whole framework is trained end-to-end using REINFORCE
Williams (1992).

In order to induce a more efficient training, we keep a measure of the
Learning Progress (LP) of the listener for all tasks on a held-out set, where
LP for task $i$ is given as $\mathrm{LP}_{i}=|r_{i}-\mu_{i}|$. Here, $\mu_{i}$
denotes the running mean of rewards for task $i$. The tasks are sampled from a
Categorical distribution with probabilities
$p(i)=\frac{\mathrm{LP}_{i}}{\sum_{j}\mathrm{LP}_{j}}$ and, consequently,
episodes corresponding to the sampled tasks are generated. This way, the
listener can keep track of goals that are already learned, or can insist on
goals that are currently too hard.

###  3.3 Inducing Compositionality

We would ideally want the concept to message mapping to be injective (one-to-
one), i.e. $\forall
c,\tilde{c}\in\mathcal{C},\mathcal{M}(c)=\mathcal{M}(\tilde{c})\implies
c=\tilde{c}$. Furthermore, the messages in $\mathcal{M}$ must exhibit a
systematic structure (in holistic languages, one can satisfy the injective
property without being compositional). Studies on language evolution have
proposed limiting the channel capacity of communication as a constraint for
achieving compositionality Nowak and Krakauer (1999). Yet, in the course of
our experiments, on increasing $|\mathcal{C}|$, we observed rather predictably
that, with a limited channel capacity, it becomes increasingly difficult for
the speaker to converge upon a consistent and unambiguous mapping from
$\mathcal{C}$ to $\mathcal{M}$. Consequently, the listener would either ignore
the information from the speaker (speaker abandoning), or may exploit the
inadequate information (undercoverage444Inspired by machine translation works
Tu et al. (2016), we define coverage as a mapping from a particular concept
element to its appropriate message element. Full coverage refers to a distinct
mapping of the whole concept input to corresponding symbols in $\mathcal{M}$.)
to converge on a local optimum (learning a fixed sequence of actions). To that
end, we propose two types of intrinsic rewards to address these issues.

#### Undercoverage:

The limited channel capacity acts as an information bottleneck, impeding the
speaker’s ability to transmit unambiguously. Therefore, it becomes difficult
for the listener to infer the decoded messages at its end. To address this
issue, we formulate a notion of compositionality from recent works in
disentanglement Higgins et al. (2017). We propose to use the Mutual
Information (MI) between the concepts and the messages
$\mathrm{I}(\mathcal{C},\mathcal{M})$ as an intrinsic reward:

$\displaystyle\begin{split}\mathrm{I}(\mathcal{C},\mathcal{M})&=\mathrm{H}(\mathcal{C})-\mathrm{H}(\mathcal{C}|\mathcal{M})\\\
&=\mathrm{H}(\mathcal{C})+\mathds{E}_{c\sim\mathcal{C},m\sim\mathcal{M}(c)}\log
p(c|m)\end{split}$

Given that the training episodes are generated independent of the object
specifications, $\mathrm{H}(\mathcal{C})$ can be assumed to be constant. We
approximate the last term using Jensen’s inequality
($\mathds{E}_{c\sim\mathcal{C},m\sim\mathcal{M}(c)}\big{[}\log
p(c|m)\big{]}\geq\mathds{E}_{c\sim\mathcal{C},m\sim\mathcal{M}(c)}\big{[}\log
q_{\phi}(c|m)\big{]}$) to obtain a lower bound for
$\mathrm{I}(\mathcal{C},\mathcal{M})$. Here, $q_{\phi}(c|m)$ is a learned
discriminator module which takes the (concatenated) messages and tries to
predict the concept labels (i.e. elements of
$\langle\mathrm{VERB},\\{\mathrm{ADJ}_{i}\\}_{i=1}^{3},\mathrm{NOUN}\rangle$)
and $\mathds{E}_{c\sim\mathcal{C},m\sim\mathcal{M}(c)}\log q_{\phi}(c|m)$ is
its negative cross-entropy loss. The final intrinsic reward is:

$\mathrm{I}(\mathcal{C},\mathcal{M})\geq\mathrm{H}(\mathcal{C})+\mathds{E}_{c\sim\mathcal{C},m\sim\mathcal{M}(c)}\log
q_{\phi}(c|m)$ (1)

Intuitively, it suggests that it should be easy to infer the concepts from the
messages. Conversely, the confusion (high error) arising from the speaker’s
inability to express concepts will lead to lower rewards. Note, that the
reward will be highest when the conditions of full coverage and one-to-one
mapping are satisfied (the discriminator will then be able to predict all the
concept elements with high probability). We add the
$\mathrm{I}(\mathcal{C},\mathcal{M})$ reward at the last step of the episode,
given as: $r[-1]+\lambda_{1}\mathrm{I}(\mathcal{C},\mathcal{M})$, where
$\lambda_{1}$ is a tunable hyperparameter. The discriminator $q_{\phi}$ is
periodically trained using batches sampled from a memory buffer, where we
store the pair $\langle c_{i},m_{i}\rangle$. Note, that we block the
discriminator gradients to the speaker and use it merely as an auxiliary means
to provide intrinsic feedback.

#### Speaker Abandoning

Existing works Lowe et al. (2019) have shown that while training RL-agents
augmented with a communication channel, it is likely that the speaker fails to
influence the listener’s actions. To address this, we propose to add another
intrinsic reward to maximize the mutual information between the speaker’s
messages and the listener’s actions, given the grid information.

At each step, we simulate $k$ intermediate steps to sample pseudo messages
$\tilde{m}$ from the message distribution $\mathcal{M}$. Together with the
original message $m$, we compute two sets of probability values corresponding
to actions of the listener: (i) $\bm{\pi}(a_{t}|m,\mathcal{G}_{t})$ which is
the listener’s policy conditioned on both the messages and the output of the
grid encoder $\mathcal{G}_{t}$; (ii) $p(a_{t}|\mathcal{G}_{t})$ or the
probability distribution over the listener’s actions conditioned on just the
output of the grid encoder. We then calculate the mutual information for each
step as follows:

$\displaystyle\mathrm{I}(a_{t},m|\mathcal{G}_{t})$
$\displaystyle=\sum_{a_{t},m}p(a_{t},m|\mathcal{G}_{t})\log\frac{p(a_{t},m|\mathcal{G}_{t})}{p(a_{t}|\mathcal{G}_{t})p(m|\mathcal{G}_{t})}$
$\displaystyle=\sum_{a_{t},m}p(m|\mathcal{G}_{t})p(a_{t}|m,\mathcal{G}_{t})\log\frac{p(a_{t}|m,\mathcal{G}_{t})}{p(a_{t}|\mathcal{G}_{t})}$
$\displaystyle=\mathds{E}_{m\sim\mathcal{M}}[\mathrm{D}_{KL}(p(a_{t}|m,\mathcal{G}_{t})||p(a_{t}|\mathcal{G}_{t}))]$

Note that $p(m|\mathcal{G}_{t})=p(m)$ since messages and grid-view are
independently processed. Here $p(a_{t}|\mathcal{G}_{t})$ is obtained by
marginalizing over the joint probability distribution, given as,
$\sum_{\tilde{m}}p(a_{t},\tilde{m}|\mathcal{G}_{t})=\sum_{\tilde{m}}p(a_{t}|\tilde{m},\mathcal{G}_{t})p(m)$.
We use Monte Carlo approximation to replace the Expectation by sampling
messages from $\mathcal{M}$. The final reward equation for $k$ pseudo-steps is
given as:

$\mathrm{I}(a_{t},m|\mathcal{G}_{t})\\\
=\frac{1}{k}\sum_{m}\mathrm{D}_{KL}\big{[}\bm{\pi}(a_{t}|m,\mathcal{G}_{t})||\sum_{\tilde{m}}\bm{\pi}(a_{t}|\tilde{m},\mathcal{G}_{t})p(m)\big{]}$
(2)

Maximizing Equation 2 leads to a higher speaker influence on the listener’s
actions. The net reward at each step is given as:
$r_{t}+\lambda_{3}\mathrm{I}(a_{t},m|\mathcal{G}_{t})$, where $\lambda_{3}$ is
a tunable hyperparameter.

##  4 Experiments

#### Zero-Shot Generalization Splits:

(i) Visual split: All episodes not containing the ‘red square’ as a target
object, were used for training the model. During evaluation, we examine
whether the trained model can generalize to the following instructions: walk
to a red square; push/pull a red square. (ii) Numeral split: The training set
contains instructions with Push, Push Twice and Pull, whereas, test set
contains Pull Twice task. Here the modifier Twice is used to denote a heavier
object i.e., listener should execute two consecutive ‘pull’ actions to move
the object. The listener must infer from its training that a symbol
corresponding to heavy requires twice as many actions.

Figure 3: [Best viewed in color] Left: Comparison of Intrinsic Speaker with other baselines on a single policy module for WALK task. It can be observed that Intrinsic Speaker performs as well as the Perfect Speaker baseline; Right: Comparison of topsim metric of Intrinsic Speaker (with and without feedback) and Simple Speaker. All plots have been obtained by averaging the validation rewards obtained over 5 independent runs. [X-axis: 1 unit = 50 episodes] Task |  Model |  Zero-Shot Accuracy
---|---|---
walk to a red square (visual split) |  |  Simple Speaker |  $73.43\%$
---|---
Intrinsic Speaker |  $\mathbf{80.24\%}$
push a red square (visual split) |  |  Simple Speaker |  $67.17\%$
---|---
Intrinsic Speaker |  $\mathbf{72.45}\%$
pull a red square (visual split) |  |  Simple Speaker |  $66.80\%$
---|---
Intrinsic Speaker |  $\mathbf{73.29}\%$
pull a red square twice (numeral split) |  |  Simple Speaker |  $65.25\%$
---|---
Intrinsic Speaker |  $\mathbf{69.77}\%$
Table 1: Comparison of simple speaker and intrinsic speaker zero-shot
performance on different splits.

#### Baselines:

We compare our Intrinsic Speaker model with the following baselines. (i)
Oracle Listener: For each cell, we zero-pad the grid encoding with an extra
bit, and set it ($=1$) for the cell containing the target object. This way,
the listener has complete information about the target in context of the
distractors. We use this baseline as our upper limit of performance. (ii)
Perfect Speaker: The speaker uses an Identity matrix that channels the input
directly to the listener. Thus, it is perfectly compositional. (iii) Simple
Speaker: Here the speaker-listener is trained end-to-end without using the
intrinsic rewards.

##  5 Results

(i) The proposed Intrinsic Speaker outperforms the Simple Speaker in terms of
both, convergence rewards and topsim score (Figure 3). In fact, the Intrinsic
Speaker matches the performance of the Perfect Speaker, thus, showing that the
emergent communication is highly compositional ($\approx 0.9$). (ii) The zero-
shot generalization accuracy in Table 1 shows that the Intrinsic Speaker
consistently outperforms the Simple Speaker on both splits. (iii) In order to
test the effectiveness of intrinsic rewards in inducing compositionality, we
trained the Intrinsic Speaker with no external reward from the environment. As
shown in Fig 3 (right), the intrinsic rewards were alone capable of generating
a topsim score $\approx 0.6$. For further details, the readers are requested
to refer the paper Hazra et al. (2020).

## References

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  * Brighton and Kirby (2006) Henry Brighton and Simon Kirby. 2006.  Understanding linguistic evolution by visualizing the emergence of topographic mappings.  _Artif. Life_.
  * Chaabouni et al. (2020) Rahma Chaabouni, Eugene Kharitonov, Diane Bouchacourt, Emmanuel Dupoux, and Marco Baroni. 2020.  Compositionality and generalization in emergent languages.  _CoRR_ , abs/2004.09124.
  * Chaabouni et al. (2019) Rahma Chaabouni, Eugene Kharitonov, Emmanuel Dupoux, and Marco Baroni. 2019.  Anti-efficient encoding in emergent communication.  _CoRR_ , abs/1905.12561.
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