# Reflections to set-theoretic solutions of the Yang-Baxter equation
11footnotemark: 1

Andrea ALBANO  Marzia MAZZOTTA  Paola STEFANELLI  Dipartimento di Matematica e
Fisica “Ennio De Giorgi”
Università del Salento
Via Provinciale Lecce-Arnesano
73100 Lecce (Italy)

###### Abstract

The main aim of this paper is to determine _reflections_ to bijective and non-
degenerate solutions of the Yang-Baxter equation, by exploring their
connections with their derived solutions. This is motivated by a recent
description of left non-degenerate solutions in terms of a family of
automorphisms of their associated left rack. In some cases, we show that the
study of reflections for bijective and non-degenerate solutions can be reduced
to those of derived type. Moreover, we extend some results obtained in the
literature for reflections of involutive non-degenerate solutions to more
arbitrary solutions. Besides, we provide ways for defining reflections for
solutions obtained by employing some classical construction techniques of
solutions. Finally, we gather some numerical data on reflections for bijective
non-degenerate solutions associated with skew braces of small order.

###### keywords:

Yang-Baxter equation , Reflection equation , Set-theoretic solution ,
Structure shelf

###### MSC:

[2020] 16T25 , 81R50 , 20N02

## Introduction

The _Yang-Baxter equation_ was first introduced by Yang [30] to study a gas of
spin with a $\delta$ interaction and, independently, by Baxter [1] to
investigate a two-dimensional solvable statistical model. Much of the
attention has been directed to the so-called _quantum_ Yang-Baxter equation,
which is fundamental in mathematical physics since it occupies a central role
in the theory of quantum integrable systems [20]. The search for solutions to
this equation has attracted numerous authors, and it is still an open problem.
Due to the complexity of the solutions, Drinfel’d [13] suggested studying a
simplified class. Into the specific, if $X$ is a set and $r:X\times X\to
X\times X$ is a map, the pair $(X,r)$ is said to be a set-theoretic solution
of the Yang-Baxter equation, briefly a _solution_ , if the identity

$\displaystyle\left(r\times\operatorname{id}_{X}\right)\left(\operatorname{id}_{X}\times
r\right)\left(r\times\operatorname{id}_{X}\right)=\left(\operatorname{id}_{X}\times
r\right)\left(r\times\operatorname{id}_{X}\right)\left(\operatorname{id}_{X}\times
r\right)$ (YBE)

holds. Writing
$r\left(x,y\right)=\left(\lambda_{x}\left(y\right),\rho_{y}\left(x\right)\right)$,
where $\lambda_{x}$ and $\rho_{y}$ are maps from $X$ into itself, for all
$x,y\in X$, then $r$ is a solution if and only if

$\displaystyle\lambda_{x}\lambda_{y}(z)=\lambda_{\lambda_{x}\left(y\right)}\lambda_{\rho_{y}\left(x\right)}\left(z\right)$
(Y1)
$\displaystyle\lambda_{\rho_{\lambda_{y}\left(z\right)}\left(x\right)}\rho_{z}\left(y\right)=\rho_{\lambda_{\rho_{y}\left(x\right)}\left(z\right)}\lambda_{x}\left(y\right)$
(Y2)
$\displaystyle\rho_{z}\rho_{y}(x)=\rho_{\rho_{z}\left(y\right)}\rho_{\lambda_{y}\left(z\right)}\left(x\right)$
(Y3)

for all $x,y,z\in X$. With this notation, a solution is bijective if $r$ is a
bijective map; involutive if $r^{2}=\operatorname{id}_{X\times X}$;
_idempotent_ if $r^{2}=r$; left non-degenerate if
$\lambda_{x}\in\operatorname{Sym}_{X}$, for all $x\in X$; right non-degenerate
if $\rho_{x}\in\operatorname{Sym}_{X}$, for all $x\in X$, and _non-degenerate_
if it is both left and right non-degenerate. In the beginning, many authors
mainly focused on the class of bijective non-degenerate solutions. In the
years that followed, the attention has been shifted to more arbitrary classes
of solutions and a wide range of papers has been published in the literature.

Similarly to the YBE, the _reflection equation_ is an important tool in the
theory of quantum groups and integrable systems. This equation was first
studied by Cherednik [6] to encode the reflection on the boundary of particles
in quantum field theory and by Sklyanin [27] to prove the integrability of
quantum models with boundaries. The set-theoretic version of this equation was
formulated by Caudrelier and Zhang in [5], who also provided the first
examples of solutions. It turns out that the set-theoretic reflection equation
jointly with the YBE ensures the factorization property of the interactions of
$N$-soliton solutions on the half-line. For an explanatory illustration, see
Figs. 1-2 in [26].
If $(X,r)$ is an arbitrary solution, a map $\kappa:X\to X$ is a set-theoretic
solution of the reflection equation, or briefly a _reflection_ , for $(X,r)$
if it holds

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

One can easily see that (RE) is equivalent to requiring that, for all $x,y\in
X$,

$\displaystyle\lambda_{\lambda_{x}(y)}\kappa\rho_{y}\left(x\right)=\lambda_{\lambda_{x}(\kappa(y))}\kappa\rho_{\kappa(y)}\left(x\right),$
(T)
$\displaystyle\rho_{\kappa\rho_{\kappa(y)}(x)}\lambda_{x}{\kappa(y)}=\kappa\rho_{\kappa\rho_{y}(x)}\lambda_{x}\left(y\right).$
(Q)

The interplay between solutions to the YBE and reflections was deepened by
Caudrelier, Crampé, and Zhang in [4]. In particular, they provided a method
for constructing all the reflection maps associated with the quadrirational
Yang–Baxter maps. In the following years, Smoktunowicz, Vendramin, and Weston
[28] proposed a first more systematic approach to the set-theoretic case that
makes use of ring-theoretic methods, and more generally methods coming from
the theory of braces, to produce families of reflections. The reflections they
provide belong to the class of what we call _$\lambda$ -centralizing maps_,
namely, maps $\kappa$ such that $\lambda_{x}\kappa=\kappa\lambda_{x}$, for all
$x\in X$. Furthermore, De Commer [8] introduces a notion of braided action of
a group with braiding and shows that it provides reflections. Besides, Doikou
and Smoktunowicz [12] investigate connections between set-theoretic
Yang–Baxter and reflection equations and quantum integrable systems. More
recently, Lebed and Vendramin [26] have constructed a family of solutions
$r^{(\kappa)}$ indexed by a reflection $\kappa$ for a right non-degenerate
solution $(X,r)$, including the solution $r$ itself and its derived solution.
They also show that if $(X,r)$ is an involutive left non-degenerate solution,
a map $\kappa$ is a reflection if and only if (T) holds; similarly, if $(X,r)$
is right non-degenerate, $\kappa$ is a reflection if and only if (Q) holds.
Thus, $\lambda$-centralizing maps are reflections for involutive left-non
degenerate solutions, while maps $\kappa$ satisfying
$\rho_{\kappa(x)}=\rho_{x}$, for all $x\in X$, that we call _$\rho$
-invariant_, are reflections for the involutive right non-degenerate ones. We
mention other interesting works on the reflection equation as [22, 7, 10, 21,
23, 9].

The main aim of this paper is to study reflections for non-degenerate
solutions employing the left and the right shelf structures associated with
them. We briefly recall that a _left shelf_ is a set $X$ endowed with a left
self-distributive binary operation $\triangleright$, namely, the identity
$x\triangleright\left(y\triangleright z\right)=\left(x\triangleright
y\right)\triangleright\left(x\triangleright z\right)$ is satisfied, for all
$x,y,z\in X$. Equivalently, for all $x\in X$, the left multiplication
$L_{x}:X\to X$ defined by $L_{x}\left(y\right)=x\triangleright y$, for all
$y\in X$, is a shelf homomorphism. In particular, if $\left(X,r\right)$ is a
left non-degenerate solution, set
$x\triangleright_{r}y:=\lambda_{x}\rho_{\lambda_{y}^{-1}(x)}(y)$, for all
$x,y\in X$, one obtain that $\left(X,\triangleright_{r}\right)$ is a left
shelf that gives rise to a solution $\left(X,r_{\triangleright}\right)$, with
$r_{\triangleright}\left(x,y\right)=\left(y,y\triangleright_{r}x\right)$, for
all $x,y\in X$, called the _left derived solution of $r$_. Analogously, one
can define the structure $\left(X,\triangleleft\right)$ of _right shelf_ ,
introduce the right multiplications $R_{x}:X\to X$ given by
$R_{x}(y)=y\triangleleft x$, for all $y\in X$, and the link with right non-
degenerate solutions. For our purposes, it will also be useful to introduce
classical tools of this theory, such as the left multiplication group of a
left shelf $(X,\triangleright)$ given by
$\operatorname{LMlt}(X,\,\triangleright)=\langle L_{x}\,\mid\ x\in X\rangle$.
For more details on this topic, see [16, 17, 24, 25].

Our study is motivated by a recent description of left non-degenerate
solutions in terms of left shelves structures provided in [11, Theorem 2.15].
Indeed, any left non-degenerate solution can be written in terms of a family
of automorphisms of the left shelf associated with it, i.e., the _Drinfel’d
twist_ , and of the shelf operation itself. This naturally suggests splitting
the study of reflections into their behaviour with respect to the maps
$\lambda_{x}\in\operatorname{Aut}\left(X,\triangleright_{r}\right)$ and left
multiplications $L_{x}$. Hence, reflections further motivate the investigation
of the endomorphism monoid of shelves.

We first focus on reflections for solutions that are only left or right non-
degenerate since, in our analysis, they turn out to be different from each
other. Following some existing approaches mentioned before, we pay particular
attention to $\lambda$-centralizing and $\rho$-invariant reflections and
characterize them in the left and right non-degenerate cases in 2.1 and in
3.1, respectively. Besides, we reformulate the results achieved in terms of
the respective multiplication groups of their associated shelf structure (see
2.10 and 3.4). Then, we deal with the non-degenerate bijective case, by
combining the observations made separately for the left and right cases, and
developing a theory for such solutions. In this context, in Appendix A, we
gather some numerical data on reflections for bijective non-degenerate
solutions associated with skew braces, dividing them by classes of belonging.

The last section is devoted to constructing reflections starting from known
ones. More specifically, considering construction techniques for arbitrary
solutions, as the _matched product_ [2] and the _strong semilattice of
solutions_ [3], we define reflections for the new solution obtained, starting
from reflections on each solution involved in the construction itself.
Moreover, we focus on the _twisted union_ , a method for obtaining involutive
non-degenerate solutions provided in [15]. In particular, we first extend this
techinque to arbitrary solutions and then we show how to construct a
reflection for new solutions determined.

##  1 Preliminaries

In this section, for the ease of the reader, we provide the necessary
background for the rest of the paper.

###  1.1 Basics on the reflection equation

######  Definition 1.1.

[4, Definition 1.1] Let $(X,r)$ be a solution. A map $\kappa:X\to X$ is a set-
theoretic solution of the reflection equation, or briefly a _reflection_ , for
$(X,r)$ if it holds

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

Moreover, we say that the reflection $\kappa$ is _involutive_ if
$\kappa^{2}=\operatorname{id}_{X}$.

We denote with $\mathcal{K}(X,r)$ the set of all reflections for a solution
$(X,r)$. Moreover, $\mathcal{K}_{\text{inj}}(X,r)$ and
$\mathcal{K}_{\text{bij}}(X,r)$ will denote the set of all _injective_ and
_bijective_ reflections for $(X,r)$, respectively. If the solution is clear
from the context, we will briefly write $\mathcal{K}(X)$. Note that
$\mathcal{K}(X)$ is non-empty since the identity map $\operatorname{id}_{X}$
is always a reflection for all solutions defined on the set $X$.

######  Remarks 1.2.

If $(X,r)$ is a solution, let us observe some easy but essential properties:

  1. 1.

If $(Y,s)$ is equivalent to $(X,r)$ via a bijection $\varphi:X\to Y$ (see
[15]), namely, $(\varphi\times\varphi)r=s(\varphi\times\varphi)$, and
$\kappa\in\mathcal{K}(X,r)$, then
$\kappa_{\varphi}:=\varphi\kappa\varphi^{-1}\in\mathcal{K}(Y,s)$.

  2. 2.

If $(X,r)$ is bijective and $\kappa\in\mathcal{K}_{\text{bij}}(X,r)$, then
$\kappa^{-1}\in\mathcal{K}_{\text{bij}}(X,r^{-1})$.

In the following, we provide some examples of reflections of solutions.

######  Examples 1.3.

Assume that $X$ is a set.

  1. 1.

Any map $\kappa:X\to X$ is a reflection for the solution
$\left(X,\operatorname{id}_{X}\right)$.

  2. 2.

Let $c\in X$ and let $\left(X,r\right)$ be the idempotent solution given by
$r\left(x,y\right)=\left(c,c\right)$, for all $x,y\in X$. Then, it is easy to
check that any map $\kappa:X\to X$ such that $\kappa\left(c\right)=c$ is a
reflection for $\left(X,r\right)$.

  3. 3.

Let $\left(X,r\right)$ be the idempotent left non-degenerate solution with
$r\left(x,y\right)=\left(y,y\right)$, for all $x,y\in X$. Then, a map
$\kappa:X\to X$ is a reflection for $\left(X,r\right)$ if and only if
$\kappa^{2}=\kappa$.

  4. 4.

The unique reflection for the idempotent right non-degenerate solution
$\left(X,s\right)$ given by $s\left(x,y\right)=\left(x,x\right)$, for all
$x,y\in X$, is the identity map $\operatorname{id}_{X}$.

Note that even though the solutions $(X,r)$ and $(X,s)$ in 1.3-$3.\&4.$ are
such that $r=\tau s\tau$, with $\tau$ the flip map, their reflections do not
share any relevant property.

######  Example 1.4.

[28, Example 1.4] Let $X$ be a set, $f,g\in\operatorname{Sym}_{X}$ such that
$fg=gf$, and consider the bijective non-degenerate solution
$r(x,y)=(f(y),g(x))$ belonging to the class of Lyubashenko’s solutions [13]. A
map $\kappa:X\to X$ is a reflection for $(X,r)$ if and only if $\kappa$
commutes with $fg$. In particular, if $(X,r)$ involutive, i.e., $g=f^{-1}$ ,
any map $\kappa:X\to X$ is a reflection for $(X,r)$.

######  Example 1.5.

[28, Example 1.3] Let $X=\\{1,2,3\\}$ and
$r(x,y)=\left(f_{x}\left(y\right),f_{y}\left(x\right)\right)$ the involutive
non-degenerate solution given by $f_{1}=f_{2}=\operatorname{id}_{X}$ and
$f_{3}=(1\ 2)$. Then, there exist five reflection maps. To write them, we
denote the image of $\kappa$ as the string $\kappa(1)\kappa(2)\kappa(3)$. They
are given by

$\displaystyle\kappa_{1}=123=\operatorname{id}_{X},\quad\kappa_{2}=113,\quad\kappa_{3}=213,\quad\kappa_{4}=223,\quad\kappa_{5}=333.$

Note that $\kappa_{j}$ is involutive if and only if $j\in\\{1,3\\}$.

In [28, Lemma 1.7], it is introduced the following notation to study the
reflections of a solution $(X,r)$.

######  Lemma 1.6.

Let $(X,r)$ be a solution and let $\kappa:X\to X$ be a map. Set

$\displaystyle
t(x,y):=\lambda_{\lambda_{x}(y)}\kappa\rho_{y}\left(x\right)\quad\text{and}\quad
q(x,y):=\rho_{\kappa\rho_{y}(x)}\lambda_{x}\left(y\right),$

for all $x,y\in X$. Then, $\kappa\in\mathcal{K}(X,r)$ if and only if the
following hold

$\displaystyle t(x,\kappa(y))=t(x,y)$ (T) $\displaystyle
q(x,\kappa(y))=\kappa(q(x,y))$ (Q)

for all $x,y\in X$.

In [26, Theorem 3.2], it is shown that for involutive non-degenerate
solutions, it is sufficient to check only one of the relations (T) and (Q).

######  Proposition 1.7.

  1. 1.

Let $(X,r)$ be an involutive left non-degenerate solution and $\kappa:X\to X$
a map. Then, $\kappa\in\mathcal{K}(X,r)$ if and only if (T) holds.

  2. 2.

Let $(X,r)$ be an involutive right non-degenerate solution and $\kappa:X\to X$
a map. Then, $\kappa\in\mathcal{K}(X,r)$ if and only if (Q) holds.

The authors in [28, Theorem 1.9] give sufficient conditions for a map $\kappa$
to be a reflection for an involutive left non-degenerate solution. To recall
such a result, let us introduce the following notion.

######  Definition 1.8.

Let $(X,r)$ be a solution. Then, a map $\kappa:X\to X$ is said to be
_$\lambda$ -centralizing_ if

$\displaystyle\forall\,x\in X\quad\lambda_{x}\kappa=\kappa\lambda_{x}.$ (L)

Similarly one can define the notion of _$\rho$ -centralizing_ map.

Notice that this notion was first given in [28] in terms of
$\mathcal{G}(X,r)$-equivariance, where $\mathcal{G}(X,r)$ is the involutive
Yang-Baxter group generated by the maps $\lambda_{x}$.

######  Proposition 1.9.

Let $(X,r)$ be an involutive left non-degenerate solution. Then, any
$\lambda$-centralizing map is a reflection for $(X,r)$.

We underline that, in general, not all the reflections for an involutive left
non-degenerate solution are $\lambda$-centralizing (see, for instance,
$\kappa_{2}$ in 1.5). Regarding the right non-degenerate case, we recall the
following result contained in [26, Theorem 3.2].

######  Proposition 1.10.

Let $(X,r)$ be an involutive right non-degenerate solution and let us assume
that $\kappa:X\to X$ is a map such that

$\displaystyle\forall\,x\in X\quad\rho_{x}=\rho_{\kappa(x)},$ (R)

then $\kappa\in\mathcal{K}(X,r)$.

Also in this case, this is only a sufficient condition, since the reflection
$\kappa_{5}$ in 1.5 does not satisfy (R). To study such a kind of maps, we
introduce the following terminology:

######  Definition 1.11.

Let $(X,r)$ be a solution. Then, we call _$\rho$ -invariant_ any map
$\kappa:X\to X$ satisfying the property (R). Similarly, one defines the notion
of $\lambda$_-invariance_.

######  Remark 1.12.

The properties (L) and (R) are invariant under equivalence. Indeed, if $(X,r)$
and $(Y,s)$ are equivalent solutions via a bijection $\varphi:X\to Y$ and
$\kappa\in\mathcal{K}(X,r)$, then the following hold:

  1. 1.

$\kappa$ is $\lambda$-centralizing if and only if $\kappa_{\varphi}$ is
$\lambda$-centralizing;

  2. 2.

$\kappa$ is $\rho$-invariant if and only if $\kappa_{\varphi}$ is
$\rho$-invariant.

More generally, the following can be proven for a $\lambda$-invariant or
$\rho$-invariant reflection.

######  Lemma 1.13.

Let $(X,r)$ be a solution and $\kappa:X\to X$ a map.

  1. 1.

If $\left(X,r\right)$ is left non-degenerate and $\kappa$ is $\rho$-invariant
for $(X,r)$, then:

$\displaystyle\kappa\ \text{is $\lambda$-invariant for
$(X,r)$}\iff\forall\,x,y\in
X\quad\lambda_{\lambda_{x}\kappa\left(y\right)}=\lambda_{\lambda_{x}\left(y\right)}.$

  2. 2.

If $\left(X,r\right)$ is right non-degenerate and $\kappa$ is
$\lambda$-invariant, for every $x\in X$, then:

$\displaystyle\kappa\ \text{is $\rho$-invariant for
$(X,r)$}\iff\forall\,x,y\in
X\quad\rho_{\rho_{x}\kappa\left(y\right)}=\rho_{\rho_{x}\left(y\right)}.$

The following are sufficient conditions for obtaining reflections of an
involutive left (resp. right) non-degenerate solution $(X,r)$ belonging to the
class of $\rho$-invariant maps.

######  Proposition 1.14.

Let $(X,r)$ be an involutive solution and let $\kappa:X\to X$ be a map.

  1. 1.

If $(X,r)$ is left non-degenerate and it holds that
$\lambda_{\lambda_{x}\kappa\left(y\right)}=\lambda_{\lambda_{x}\left(y\right)}$,
for all $x,y\in X$, then $\kappa\in\mathcal{K}\left(X,r\right)$ and is
$\rho$-invariant.

  2. 2.

If $(X,r)$ is right non-degenerate and it holds that
$\rho_{\rho_{x}\kappa\left(y\right)}=\rho_{\rho_{x}\left(y\right)}$, for all
$x,y\in X$, then $\kappa\in\mathcal{K}\left(X,r\right)$ and is
$\rho$-invariant.

###### Proof.

The first statement follows by 1.7-$1.$. Moreover, since $(X,r)$ is
involutive, by the assumptions, $\kappa$ is $\rho$-invariant. For the second
one, it is enough to observe that
$\lambda_{x}=\lambda_{\kappa\left(x\right)}$, for every $x\in X$. Thus, by
1.7-$2.$ and Lemma 1.13, $\kappa\in\mathcal{K}\left(X,r\right)$ and $\kappa$
is $\rho$-invariant. ∎

###  1.2 Some basics on shelf structures

Let us start this section by recalling some basics on shelves and racks. For
further details, we refer to [16, 17, 24, 25].

A _left shelf_ $(X,\triangleright)$ is a set $X$ equipped with a left self-
distributive binary operation $\triangleright$ such that

$\forall\,x,y,z\in X\quad x\triangleright(y\triangleright z)=(x\triangleright
y)\triangleright(x\triangleright z)\,.$ (1)

A _shelf homomorphism_ between two shelves is defined as any map preserving
the shelf operations on each underlying set. Hereafter, we will denote with
$\operatorname{End}(X,\triangleright)$ and
$\operatorname{Aut}(X,\triangleright)$ the endomorphism monoid and the
automorphism group of a given shelf $(X,\triangleright)$, respectively. If we
denote with $L_{x}:X\to X,y\mapsto x\triangleright y$ the operator of left
multiplication by $x$, for all $x\in X$, then equation (1) is equivalent to
require that $L_{x}$ is a shelf homomorphism.

A left shelf $(X,\triangleright)$ is a _left rack_ if all the maps $L_{x}$ are
bijective. Moreover, a rack $(X,\triangleright)$ is a _left quandle_ if, in
addition, $L_{x}(x)=x$, for all $x\in X$.

In a similar way, it is possible to define a _right shelf_
$(X,\triangleleft)$, and analogously a _right rack_ as a right shelf
$(X,\triangleleft)$ having all the right multiplications $R_{x}$ bijective.

Below, we give some useful examples of left shelves and left racks, some of
which are well-known in literature. Clearly, one can derive their right
version.

######  Examples 1.15.

  1. 1.

If $X$ is a set and $a\in X$, set $x\triangleright y:=a$, for all $x,y\in X$,
then $(X,\triangleright)$ is a left shelf.

  2. 2.

Let $X$ be a set, $f:X\to X$ an idempotent map, and define $x\triangleright
y:=f(x)$, for all $x,y\in X$, then $(X,\triangleright)$ is a left shelf.

  3. 3.

Let us consider $(\mathbb{Z}_{6},+)$ and define $x\triangleright y:=2x+2y$
$\left(\text{mod}\,\,6\right)$, for all $x,y\in\mathbb{Z}_{6}$. Then,
$\left(\mathbb{Z}_{6},\triangleright\right)$ is a left shelf, cf. [11, Example
2.2].

######  Examples 1.16.

  1. 1.

(Permutation Rack) Let $X$ be a set, $f\in\operatorname{Sym}_{X}$, and set
$x\triangleright y:=f(y)$, for all $x,y\in X$. If $f=\operatorname{id}_{X}$,
then $(X,\triangleright)$ is the _trivial rack_.

  2. 2.

(Cyclic Rack of order n) Let $n\in\mathbb{N}$ and $X=\mathbb{Z}_{n}$. Define
$x\triangleright y:=y+1$ $\left(\text{mod}\,\,n\right)$, for all $x,y\in X$.
Then, $(X,\triangleright)$ is a left rack which is a quandle if and only if
$n=1$.

  3. 3.

(Conjugation quandle) Let $G$ be a group and define $x\triangleright
y:=x^{-1}yx$, for all $x,y\in G$. Then, the pair $(G,\triangleright)$ is a
left quandle. More generally, any possible union of conjugacy classes of $G$
inherits the structure of a left quandle.

  4. 4.

(Dihedral quandle) Let $n\in\mathbb{N}_{0}$ and $X=\mathbb{Z}_{n}$. Define a
binary operation on $X$ by setting $i\triangleright j:=2i-j$
$\left(\text{mod}\,\,n\right)$, for all $i,j\in X$. Then, $(X,\triangleright)$
is a left quandle. In particular, the case $n=0$ provides the infinite
dihedral quandle.

The following results explain the relationship between shelf structures and
solutions that will be useful for our purposes.

######  Proposition 1.17.

Let $(X,\triangleright)$ be a left shelf. Then, the map
$r_{\triangleright}:X\times X\to X\times X$ given by
$r_{\triangleright}(x,y)=\left(y,y\triangleright x\right)$, for all $x,y\in
X$, is a left non-degenerate solution on $X$, called the _solution associated
to the left shelf $(X,\triangleright)$_.
Conversely, if $(X,r)$ is a left non-degenerate solution, set

$x\triangleright_{r}y:=\lambda_{x}\rho_{\lambda_{y}^{-1}(x)}(y),$

for all $x,y\in X$, then the structure $(X,\triangleright_{r})$ is a left
shelf, called the _left shelf associated to the solution $(X,r)$_. The
solution associated with $\left(X,\triangleright_{r}\right)$ is called the
_left-derived solution_ of $(X,r)$.

The correspondence between right non-degenerate solutions and right shelves is
analogous and easily obtained by exchanging the roles of the maps $\lambda$
and $\rho$. Specifically, if $(X,r)$ is right non-degenerate, then its
associated right shelf $(X,\triangleleft_{r})$ is defined by setting

$\forall\ x,y\in X\qquad
x\triangleleft_{r}y:=\rho_{y}\lambda_{\rho^{-1}_{x}(y)}(x).$

######  Proposition 1.18.

Let $(X,\triangleright)$ be a shelf. Then, the solution
$\left(X,r_{\triangleright}\right)$ is non-degenerate if and only if
$(X,\triangleright)$ is a rack.

According to [11, Theorem 2.15], it is possible to describe any left (or
right) non-degenerate solution in terms of its associated shelf. We recall
such a result in the direction useful for our purposes.

######  Theorem 1.19.

  1. 1.

Let $(X,r)$ be a left non-degenerate solution and let $(X,\triangleright_{r})$
its associated shelf. Then,
$\lambda_{x}\in\operatorname{Aut}\left(X,\triangleright_{r}\right)$ and
$\rho_{y}\left(x\right)=\lambda^{-1}_{\lambda_{x}\left(y\right)}\left(\lambda_{x}\left(y\right)\triangleright_{r}x\right)$,
for all $x,y\in X$.

  2. 2.

Let $(X,r)$ be a right non-degenerate solution and let $(X,\triangleleft_{r})$
its associated shelf. Then,
$\rho_{x}\in\operatorname{Aut}\left(X,\triangleleft_{r}\right)$ and
$\lambda_{x}\left(y\right)=\rho^{-1}_{\rho_{y}\left(x\right)}\left(y\triangleleft_{r}\rho_{y}(x)\right)$,
for all $x,y\in X$.

Let us observe that, in particular, any left non-degenerate involutive
solution on a set $X$ is the solution associated with the trivial rack.

It turns out to be useful to introduce the following group associated with a
left rack structure. Similarly, one can define it for a right rack.

######  Definition 1.20.

Let $(X,\triangleright)$ be a left rack. The _left multiplication group_ of
$(X,\triangleright)$ is the normal subgroup of $\operatorname{Aut}(X)$ defined
by

$\displaystyle\operatorname{LMlt}(X,\triangleright):=\langle L_{x}\mid x\in
X\rangle\,.$

This group is an important invariant of a rack and computing it, in general,
is a fundamental task in the theory (for example, see [17, 14]).

##  2 Reflections for left non-degenerate solutions

In this section, we focus on reflections for left non-degenerate solutions
and, consequently, on left shelf structures. Specifically, we describe
$\lambda$-centralizing reflections for this class of solutions, i.e.,
reflections commuting with all the maps $\lambda_{x}$.

######  Theorem 2.1.

Let $(X,r)$ be a left non-degenerate solution, $(X,\triangleright_{r})$ its
associated shelf, and $\kappa:X\to X$ a $\lambda$-centralizing map. Then,
$\kappa\in\mathcal{K}(X,r)$ if and only if, for all $x,y\in X$, the following
hold:

  1. 1.

$\kappa L_{x}=\kappa L_{\kappa\left(x\right)}$,

  2. 2.

$\kappa L_{L_{x}(y)}(x)=L_{\kappa L_{x}(y)}\kappa(x)$ .

###### Proof.

Initially, using 1.19, observe that, by an easy computation, the equation in
1. is equivalent to (T). Now, notice that

$\displaystyle\kappa q\left(x,y\right)=\lambda^{-1}_{t\left(x,y\right)}\kappa
L_{t\left(x,y\right)}\lambda_{x}\left(y\right)\quad\text{and}\quad
q\left(x,\kappa\left(y\right)\right)=\lambda^{-1}_{t\left(x,\kappa\left(y\right)\right)}L_{t\left(x,\kappa\left(y\right)\right)}\lambda_{x}\kappa\left(y\right),$

for all $x,y\in X$. Thus, if $\kappa$ is a reflection for $(X,r)$, we get

$\displaystyle\forall\,x,y\in X\quad\kappa q(x,y)=q(x,\kappa(y))$
$\displaystyle\Leftrightarrow\forall\,x,y\in X\quad\kappa
L_{t\left(x,y\right)}\lambda_{x}\left(y\right)=L_{t\left(x,y\right)}\lambda_{x}\kappa\left(y\right)$
$\displaystyle\Leftrightarrow\forall\,x,y\in X\quad\kappa L_{\kappa
L_{\lambda_{x}(y)}(x)}\lambda_{x}\left(y\right)=L_{\kappa
L_{\lambda_{x}(y)}(x)}\lambda_{x}\kappa\left(y\right)$
$\displaystyle\underset{\eqref{eq:1}}{\Leftrightarrow}\forall\,x,y\in
X\quad\kappa L_{L_{\lambda_{x}(y)}(x)}\lambda_{x}\left(y\right)=L_{\kappa
L_{\lambda_{x}(y)}(x)}\kappa\lambda_{x}\left(y\right),$

i.e., 2. holds. Conversely, if 1. and 2. hold then, by following the previous
chain of equivalences we also obtain (Q). ∎

We remark that the previous result extends 1.9, since if $(X,r)$ is an
involutive left non-degenerate solution then $(X,\triangleright_{r})$ is the
trivial rack.

######  Corollary 2.2.

Let $(X,r)$ be a left non-degenerate solution and let us assume that $\kappa$
is a $\lambda$-centralizing map for $(X,r)$. Then, $\kappa\in\mathcal{K}(X,r)$
if and only if $\kappa\in\mathcal{K}(X,r_{\triangleright})$.

###### Proof.

The claim follows by 1.17 and 2.1. ∎

Notice that property $2.$ of 2.1 might suggest that a $\lambda$-centralizing
reflection $\kappa$ of a left non-degenerate solution $(X,r)$ is an
endomorphism of its associated rack $\left(X,\triangleright_{r}\right)$.
However, in general, this is not the case, as one can verify in the following
example.

######  Example 2.3.

If we consider the shelf $\left(X,\triangleright\right)$ in 1.15-2. on
$X=\\{1,\,2,\,3\\}$ with $f:X\to X$ given by $f=122$, then the map
$\kappa:X\to X$ defined by $\kappa=112$ is a reflection that is not a shelf
endomorphism of $\left(X,\triangleright\right)$.

Therefore, one question that arises is whether left multiplications are
reflections for a solution associated with a shelf structure.

######  Proposition 2.4.

Let $(X,\triangleright)$ be a left shelf, $(X,r_{\triangleright})$ its
associated solution, and $a\in X$. Then, the following conditions are
equivalent:

  1. 1.

$L_{a}\in\mathcal{K}(X,r_{\triangleright})$;

  2. 2.

$L_{a}L_{x}=L_{a}L_{L_{a}(x)}$, for every $x\in X$;

  3. 3.

$L_{L_{a}(x)}L_{a}=L_{a}L_{L_{a}(x)}$, for every $x\in X$.

###### Proof.

It directly follows from 2.1 and (1). ∎

The property 2. of 2.1 further simplifies if we assume that $L_{x}$ is
bijective, for all $x\in X$, i.e., $(X,\triangleright_{r})$ is a rack.
According to [11, Corollary 2.16], this is the case of a left non-degenerate
solution which is also bijective.

######  Corollary 2.5.

Let $(X,r)$ be a bijective left non-degenerate solution and $\kappa:X\to X$ a
$\lambda$-centralizing map. Then, $\kappa\in\mathcal{K}(X,r)$ if and only if
$\kappa\in\operatorname{End}\left(X,\triangleright_{r}\right)$ such that

$\forall\,x\in X\quad\kappa L_{x}=\kappa L_{\kappa\left(x\right)}\,.$

In particular, $\kappa$ is an involutive reflection for $(X,r)$ if and only if
$\kappa$ is an involutive automorphism of $\left(X,\triangleright_{r}\right)$.

######  Remark 2.6.

Let $\kappa_{1},\kappa_{2}$ be $\lambda$-centralizing reflections of a
bijective left non-degenerate solution $(X,r)$. If
$\kappa_{1}\kappa_{2}=\kappa_{2}\kappa_{1}$, then $\kappa_{1}\kappa_{2}$ is a
reflection of $(X,r)$. The converse is not true since in 1.5 it holds that
$\kappa_{2}\kappa_{3}=\kappa_{2}$ and $\kappa_{3}\kappa_{2}=\kappa_{4}$.
Moreover, in general, the commutation assumption can not be relaxed since for
a given solution there can exist $\lambda$-centralizing reflections $\kappa$
and $\omega$ whose composition is not again a reflection (see 1.3-3.).

In the case of a non-degenerate left derived solution, 2.1 allows to
characterize all reflections only in terms of the associated rack.

######  Corollary 2.7.

Let $\left(X,\triangleright\right)$ be a left rack and
$(X,r_{\triangleright})$ its associated solution. Then, a
$\kappa\in\mathcal{K}(X,r_{\triangleright})$ if and only if
$\kappa\in\operatorname{End}\left(X,\triangleright_{r}\right)$ such that

$\forall\,x\in X\quad\kappa L_{x}=\kappa L_{\kappa\left(x\right)}.$

The 2.7 allows us to easily find examples of reflections of the solutions
associated with racks.

######  Examples 2.8.

Let $(X,\triangleright)$ be a left rack.

  1. 1.

Any idempotent endomorphism of $(X,\triangleright)$ lies in $\mathcal{K}(X)$.

  2. 2.

If $(X,\triangleright)$ is a quandle, any constant map with a constant value
$c\in X$ is a reflection for $\left(X,r_{\triangleright}\right)$.

  3. 3.

If $(X,\triangleright)$ is the cyclic rack, since one can easily deduce that
$\operatorname{End}(X,\triangleright)=\operatorname{LMlt}(X,\triangleright)$,
then $\mathcal{K}(X)=\operatorname{LMlt}(X,\triangleright)$.

Another consequence of the previous corollaries is that bijective reflections
of a left rack allow us to obtain new solutions of the Yang-Baxter equation,
as we explain in the following.

######  Proposition 2.9.

Let $(X,\triangleright)$ be a left rack and let us consider
$\kappa\in\mathcal{K}_{\text{bij}}(X,r_{\triangleright})$. Then, the map
$r_{\kappa}:X\times X\to X\times X$ defined by

$\displaystyle
r_{\kappa}\left(x,\,y\right)=\left(\kappa\left(y\right),\,\kappa^{-1}\left(y\triangleright
x\right)\right),$

for all $x,y\in X$, is a bijective and non-degenerate solution on $X$. In
addition, it holds that $\omega\in\mathcal{K}(X,r_{\kappa})$ if and only if
$\kappa\omega=\omega\kappa$.

###### Proof.

The proof of the fact that $r_{\kappa}$ is a bijective and non-degenerate
solution is a consequence of [11, Theorem 2.15] and 2.7. Moreover, the rest of
the claim follows from 2.5. ∎

Notice that the solution $(X,r_{\kappa})$ in 2.9 is equivalent to
$(X,r_{\triangleright})$ if and only if $\kappa=\operatorname{id}_{X}$. An
easy calculation shows that $\triangleright=\triangleright_{r_{\kappa}}$ and
therefore $r_{\triangleright}$ and $r_{\kappa}$ share the same left rack
structure. According to [11, Lemma 2.25], it follows that $r_{\triangleright}$
and $r_{\kappa}$ are Drinfel’d-isomorphic, with the explicit
Drinfel’d-isomorphism $\Phi:X\times X\to X\times X$ given by
$\Phi(x,y)=(x,\kappa(y))$, for all $x,y\in X$.

The $\lambda$-centralizing reflections for bijective left non-degenerate
solutions can be directly studied using the _centralizer_ of
$\operatorname{LMlt}\left(X,\triangleright\right)$, as we will show in the
result below. Hereafter, with $C_{M}(S)$ we will denote the centralizer in a
monoid $M$ of a subset $S$.

######  Theorem 2.10.

Let $(X,r)$ be a bijective left non-degenerate solution and $\kappa:X\to X$ a
$\lambda$-centralizing map. Then the following hold:

  1. 1.

If $\kappa\in
C_{\operatorname{End}(X,\triangleright)}\left(\operatorname{LMlt}\left(X,\triangleright\right)\right)$,
then $\kappa\in\mathcal{K}(X,r)$;

  2. 2.

If $\kappa\in\mathcal{K}_{\text{inj}}(X,r)$, then $\kappa\in
C_{\operatorname{End}(X,\triangleright)}\left(\operatorname{LMlt}\left(X,\triangleright\right)\right)$;

  3. 3.

In particular, $\kappa\in\mathcal{K}_{\text{bij}}(X,r)$ if and only if
$\kappa\in
C_{\operatorname{Aut}(X,\triangleright)}\left(\operatorname{LMlt}\left(X,\triangleright\right)\right)$.

###### Proof.

The first statement directly follows by 2.5. Now, assume that $\kappa$ is an
injective reflection. Again, from property $1.$ we can deduce that
$L_{x}=L_{\kappa(x)}$, from which it follows that
$L_{x}\kappa=L_{\kappa(x)}\kappa=\kappa L_{x}$, for all $x\in X$. Clearly, the
last point of the assertion is a consequence of the former two. ∎

######  Corollary 2.11.

If $(X,\triangleright)$ is a left rack, then

$\displaystyle
C_{\operatorname{End}(X,\triangleright)}\left(\operatorname{LMlt}\left(X,\triangleright\right)\right)\subseteq\mathcal{K}(X)\,\quad\text{and}\quad\mathcal{K}_{\text{inj}}(X)\subseteq
C_{\operatorname{End}(X,\triangleright)}\left(\operatorname{LMlt}\left(X,\triangleright\right)\right).$

In particular,
$C_{\operatorname{Aut}(X,\triangleright)}\left(\operatorname{LMlt}\left(X,\triangleright\right)\right)=\mathcal{K}_{\text{bij}}(X)$.

As an easy further consequence, we state the following result.

######  Corollary 2.12.

Let $(X,\triangleright)$ be a left rack and $a\in X$. Then, the map
$L_{a}\in\mathcal{K}(X)$ if and only if
$L_{a}\in\text{Z}\left(\operatorname{LMlt}\left(X,\triangleright\right)\right)$,
the center of $\operatorname{LMlt}(X,\triangleright)$.

######  Example 2.13.

Let $X=\mathbb{Z}_{n}$ and $(X,\triangleright)$ be the dihedral quandle of
order $n\in\mathbb{N}$. In [14, Theorem 2.1] the authors prove that
$\operatorname{Aut}(X)=\operatorname{Aff}(\mathbb{Z}_{n})$, the group of
affine transformations of $\mathbb{Z}_{n}$. With an analogous proof one can
compute $\operatorname{End}(X)$ and show that

$\operatorname{End}(X)=\left\\{f_{b,a}:\mathbb{Z}_{n}\to\mathbb{Z}_{n}\mid
f_{b,a}(x)=b+ax,\,x,a,b\in\mathbb{Z}_{n}\right\\}.$

In light of 2.7, a map $\kappa:X\to X$ is a reflection for
$(X,r_{\triangleright})$ if and only if there exist $a,b\in\mathbb{Z}_{n}$
such that $\kappa=f_{b,a}$ and it satisfies $\kappa L_{x}=\kappa
L_{\kappa(x)}$, for all $x\in\mathbb{Z}_{n}$. Therefore, a straightforward
computation shows that $\kappa\in\mathcal{K}(X,r_{\triangleright})$ if and
only if

$2a(a-1)x+2ab=0\text{ (mod $n$)}\,$

holds, for all $x\in\mathbb{Z}_{n}$. Evidently, this is equivalent to both
requiring that

$\begin{cases}2ab=0&\text{ (mod $n$)}\,,\\\ 2a(a-1)=0&\text{ (mod
$n$)}\,.\end{cases}$

The distinct solutions of such a system will provide all reflections $\kappa$
for $(X,r_{\triangleright})$. For example if $\kappa$ is bijective, i.e. if
$a\in\mathbb{Z}_{n}^{*}$ is invertible, then the following cases occur:

  * -

if $\gcd(n,4)=1$ then $(b,a)=(0,1)$;

  * -

if $\gcd(n,4)=2$ then $(b,a)=(0,1),\left(\frac{n}{2},1\right)$;

  * -

if $\gcd(n,4)=4$ then
$(b,a)=(0,1),\left(0,\frac{n}{2}+1\right),\left(\frac{n}{2},1\right),\left(\frac{n}{2},\frac{n}{2}+1\right)$.

In particular, $|\mathcal{K}_{\text{bij}}(X,\triangleright)|=\gcd(n,4)$ and
all bijective reflections are involutive.
An analogous procedure can be carried on to determine that the only
reflections for the left derived solution associated to the infinite dihedral
quandle are the identity map and the maps of arbitrary constant value.

Let us conclude this section with some considerations on idempotent left non-
degenerate solutions.

######  Proposition 2.14.

Let $(X,r)$ be an idempotent left non-degenerate solution. Then:

  1. 1.

$\kappa:X\to X$ is a $\lambda$-centralizing reflection if and only if
$\kappa^{2}=\kappa$;

  2. 2.

$\kappa:X\to X$ is a $\rho$-invariant reflection if and only if
$\kappa=\operatorname{id}_{X}$.

###### Proof.

Let us initially observe that
$\rho_{y}\left(x\right)=\lambda^{-1}_{\lambda_{x}\left(y\right)}\lambda_{x}\left(y\right)$,
for all $x,y\in X$. Hence, the first statement easily follows from Lemma 1.6.
For the second one, note that (Q) is satisfied if and only if
$\kappa\rho_{y}=\rho_{y}$, for every $y\in X$. Consequently, (T) holds if and
only if $\lambda_{x}\kappa=\lambda_{x}$, for every $x\in X$, and, since $r$ is
left non-degenerate, we get $\kappa=\operatorname{id}_{X}$. ∎

Note that the previous result is similar to that in the bijective case, i.e.,
2.5, since the shelf associated with any idempotent solution is that in
1.15-$2.$ with $f=\operatorname{id}_{X}$, namely, the trivial shelf.

##  3 Reflections for right-non degenerate solutions

In this section, we investigate reflections for right non-degenerate solutions
by shifting the attention to the class of $\rho$-invariant reflections, that
is, $\rho_{x}=\rho_{\kappa(x)}$ holds, for all $x\in X$.

######  Theorem 3.1.

Let $(X,r)$ be a right non-degenerate solution, $(X,\triangleleft_{r})$ its
associated shelf, and $\kappa:X\to X$ a $\rho$-invariant map. Then,
$\kappa\in\mathcal{K}(X,r)$ if and only if $\kappa R_{x}=R_{x}\kappa$, for all
$x\in X$.

###### Proof.

Let $x,y\in X$. Using 1.19, we get $\kappa
q(x,y)=\kappa\rho_{\rho_{y}(x)}\lambda_{x}(y)=\kappa R_{\rho_{y}(x)}(y)$ and,
on the other hand,

$\displaystyle
q(x,\kappa(y))=\rho_{\kappa\rho_{\kappa(y)}(x)}\lambda_{x}\kappa(y)=R_{\rho_{\kappa(y)}(x)}\kappa(y)=R_{\rho_{y}(x)}\kappa(y).$

Moreover, it holds that

$\displaystyle
t(x,y)=\rho^{-1}_{\kappa\rho_{\kappa\rho_{y}(x)}\lambda_{x}(y)}R_{\kappa\rho_{y}(x)\lambda_{x}(y)}\kappa\rho_{y}(x)=\rho^{-1}_{\kappa
q(x,y)}R_{\kappa q(x,y)}\kappa\rho_{y}(x)$

and, similarly,
$t(x,\kappa(y))=\rho^{-1}_{q(x,\kappa(y))}R_{q(x,\kappa(y))}\kappa\rho_{y}(x)$.
Thus, (T) depends on equation (Q). Therefore, to get the claim it is enough to
observe that (Q) is equivalent to requiring that $\kappa$ commutes with every
right multiplication $R_{x}$, for all $x\in X$. ∎

Below, we give a list of consequences of 3.1. First, let us remark that the
previous result extends 1.10 to the class of arbitrary right non-degenerate
solutions which are not necessarily involutive. This follows from the fact
that if $(X,r)$ is a right non-degenerate involutive solution then
$(X,\triangleleft_{r})$ is trivial.

######  Corollary 3.2.

Let $\left(X,r\right)$ be a right non-degenerate solution. If
$\kappa,\omega:X\to X$ are $\rho$-invariant reflections for
$\left(X,r\right)$, then $\kappa\omega\in\mathcal{K}\left(X,r\right)$ and
$\kappa\omega$ is $\rho$-invariant.

######  Corollary 3.3.

Let $(X,r)$ be a right non-degenerate solution and let us assume that $\kappa$
is a $\rho$-invariant map. Then, $\kappa\in\mathcal{K}(X,r)$ if and only if
$\kappa\in\mathcal{K}\left(X,r_{\triangleleft}\right)$.

###### Proof.

The claim follows by 1.17 and 3.1. ∎

As in the case of left non-degenerate solutions, we can formulate the
statement in 3.1 by means of the right multiplication group of the right rack
structure.

######  Corollary 3.4.

Let $(X,r)$ be a bijective right non-degenerate solution,
$(X,\triangleleft_{r})$ its associated right rack, and $\kappa:X\to X$ a
$\rho$-invariant map. Then, the following hold:

  1. 1.

$\kappa\in\mathcal{K}(X,r)\iff\kappa\in
C_{\operatorname{Map}_{X}}\left(\operatorname{RMlt}\left(X,\triangleleft_{r}\right)\right)$.

  2. 2.

$\kappa\in\mathcal{K}_{\text{bij}}(X,r)\iff\kappa\in
C_{\operatorname{Sym}_{X}}\left(\operatorname{RMlt}\left(X,\triangleleft_{r}\right)\right)$.

######  Example 3.5.

Let $X=\mathbb{Z}_{n}$ and consider $(X,\triangleleft)$ the right dihedral
quandle of order $n\in\mathbb{N}$. The 3.4 implies that a map
$\kappa\in\mathcal{K}\left(X,r_{\triangleleft}\right)$ if and only if the
following holds for all $x,y\in X$:

$\kappa(x)+\kappa(2y-x)=2y\,.$ (2)

Such a map $\kappa$ has to satisfy $\kappa(2y)=2y-\kappa(0)$ and
$\kappa(2y-1)=2y-\kappa(1)$, for all $y\in X$, therefore it is parametrized by
its values in $0$ and $1$. To determine their values we notice that
$2\kappa(0)=0$ and $2(\kappa(1)-1)=0$, as a consequence of (2). Therefore, all
reflections for $(X,r_{\triangleleft})$ can be classified as follows:

  * -

if $n$ is odd then $\kappa=\operatorname{id}_{X}$;

  * -

if $n$ is even then $\kappa(0)\in\\{0,\frac{n}{2}\\}$,
$\kappa(1)\in\\{1,\frac{n}{2}+1\\}$ and $\kappa$ is determined as above.

Analogously, one can prove that the right derived solution associated with a
right infinite dihedral quandle admits the identity map as its unique
reflection.

######  Proposition 3.6.

Let $(X,\triangleleft)$ be a right shelf, $(X,r_{\triangleleft})$ its
associated solution, and $a\in X$. Then, the following conditions are
equivalent:

  1. 1.

$R_{a}$ is a reflection for $(X,r_{\triangleleft})$;

  2. 2.

$R_{a}R_{x}=R_{x}R_{a}$, for all $x\in X$;

  3. 3.

$R_{R_{a}(x)}R_{a}=R_{x}R_{a}$, for all $x\in X$.

###### Proof.

The equivalence of $1.$ and $2.$ directly follows from 3.1, while the
equivalence of $2.$ and $3.$ easily stems from the definition of right shelf.
∎

######  Remark 3.7.

As a direct consequence of 3.6, if $(X,\triangleleft)$ is a right rack then
$Z(\operatorname{RMlt}(X,\triangleleft))\subseteq\mathcal{K}(X)$. In
particular, if the right multiplication group is abelian - as in the case of a
cyclic rack - then every right multiplication provides a reflection for the
associated solution.

Let us conclude with an observation on idempotent right non-degenerate
solutions.

######  Remark 3.8.

Note that if $(X,r)$ is an idempotent right non-degenerate solution, one can
check that the unique $\rho$-invariant reflection is the identity
$\operatorname{id}_{X}$. On the other hand, the unique bijective
$\lambda$-centralizing reflection for $(X,r)$ is the identity
$\operatorname{id}_{X}$.

##  4 Reflections for bijective non-degenerate solution

This section is devoted to a global analysis of reflections for bijective non-
degenerate solutions in light of results obtained in the previous two
sections.

Let us start with some considerations on involutive non-degenerate solutions
$(X,r)$. One can easily show that any $\rho$-invariant reflection can be
described in terms of the retract solution of $(X,r)$. Recall that the
equivalence relation of _retraction_ on $X$ is defined by setting $x\sim
y\iff\lambda_{x}=\lambda_{y}$, for all $x,y\in X$. Thus, there is a natural
induced solution $\operatorname{Ret}(X,r)$ on the quotient
$\tilde{X}:=X/\sim$, called the _retract_ of $(X,r)$, which itself is
involutive and non-degenerate. For more information, we refer the reader to
the standard literature (see [15, Section 3]).

######  Remark 4.1.

The retraction relation for an involutive non-degenerate solution $(X,r)$ can
be equivalently defined by requiring that $x\sim y\iff\rho_{x}=\rho_{y}$, for
all $x,y\in X$. This directly follows by the identity
$\rho_{x}=T\lambda^{-1}_{x}T^{-1}$, for all $x\in X$, where $T:X\to X$ is the
bijective map defined by $T(x)=\rho_{x}^{-1}(x)$, for all $x\in X$ (cf. [15,
Proposition 2.2]).

######  Proposition 4.2.

Let $(X,r)$ be an involutive non-degenerate solution. Then, they hold:

  1. 1.

Any map $\kappa:X\to X$ such that $\kappa(x)\in[x]_{\sim}$, for every $x\in
X$, is a $\rho$-invariant reflection for $(X,r)$. In particular, all such
reflections can be constructed in this manner.

  2. 2.

If $\kappa\in\mathcal{K}(X,r)$ is $\rho$-invariant, then it induces a
reflection $\tilde{\kappa}:\tilde{X}\to\tilde{X}$ of the retraction
$\operatorname{Ret}(X,r)$ such that
$\tilde{\kappa}([x]_{\sim})=[\kappa(x)]_{\sim}$, for all $x\in X$.

###### Proof.

The two statements are a direct consequence of 1.10. ∎

######  Remark 4.3.

Let $\left(X,r\right)$ be an involutive non-degenerate solution with $X$
finite and assume that $\kappa:X\to X$ is a $\rho$-invariant map. If $r$ is
irretractable, then $\kappa\in\mathcal{K}_{\text{inj}}(X,r)$. Indeed, by 1.10,
$\kappa$ is a reflection. Moreover, if $x,y\in X$ are such that
$\kappa\left(x\right)=\kappa\left(y\right)$, then
$\rho_{x}=\rho_{\kappa\left(x\right)}=\rho_{\kappa\left(y\right)}=\rho_{y}$,
hence, by irretractability, $x=y$. As a consequence, since $X$ is finite, then
$\kappa\in\mathcal{K}_{\text{bij}}(X,r)$.

Now, we aim to combine the results obtained in the previous sections and
analyze what happens in the special case of bijective non-degenerate
solutions. To do so, in [25, cf. Proposition 3.9], it is shown the existing
link between left and right structure racks.

######  Lemma 4.4.

Let $(X,r)$ be a bijective non-degenerate solution. Then, the left and the
right structure racks of $(X,r)$ are anti-isomorphic via the bijection $T:X\to
X$ defined by $T(x)=\rho^{-1}_{x}(x)$, for all $x\in X$. Concretely, the
relation

$\displaystyle T\left(x\triangleright_{r}y\right)=T(y)\triangleleft_{r}T(x)$

holds, for all $x,y\in X$. Equivalently, $L_{x}=T^{-1}R_{T(x)}T$, where
$T^{-1}(x)=\lambda_{x}^{-1}\left(x\triangleright_{r}x\right)$, for all $x\in
X$.

######  Remark 4.5.

To be thorough, let us observe that if $\left(X,r\right)$ is a bijective non-
degenerate square-free solution, i.e., $r\left(x,x\right)=\left(x,x\right)$,
for all $x\in X$, then $\triangleright_{r}=\triangleleft_{r}^{op}$. The
converse is not necessarily true (consider for example a non-trivial
Lyubashenko’s solution).
However, in general, it is easy to check that two solutions of derived type
$\left(X,r_{\triangleright}\right)$ and $\left(Y,r_{\triangleleft}\right)$ are
isomorphic if and only if $L_{a}=\operatorname{id}_{X}$, for every $a\in X$,
and $R_{u}=\operatorname{id}_{Y}$, for every $u\in Y$. In particular, this
happens only for involutive solutions.

As an easy consequence of Lemma 4.4, since any anti-homomorphism between
$\left(X,\triangleright_{r}\right)$ and $\left(X,\triangleleft_{r}\right)$
satisfies $L_{x}=F^{-1}R_{F\left(x\right)}F$, for every $x\in X$, it is a
routine computation to check the following.

######  Theorem 4.6.

Let $(X,r)$ be a bijective non-degenerate solution, $\kappa:X\to X$ a map,
$F:X\to X$ an anti-homomorphism from $\left(X,\triangleright_{r}\right)$ in
$\left(X,\triangleleft_{r}\right)$, and $\omega,\bar{\omega}:X\to X$ the maps
defined by $\omega:=F\kappa F^{-1}$ and $\bar{\omega}:=F^{-1}\kappa F$. Then,
the following hold:

  1. 1.

if $\kappa$ is $\lambda$-centralizing, then,

$\displaystyle\kappa\in\mathcal{K}(X,r)\Leftrightarrow\ \begin{cases}\
\forall\,x\in X\quad\kappa L_{x}=\kappa L_{\kappa(x)}&\\\ \ \
\kappa\in\operatorname{End}\left(X,\triangleright_{r}\right)\end{cases}\Leftrightarrow\
\begin{cases}\ \forall\,x\in X\quad\omega R_{x}=\omega R_{\omega(x)}&\\\ \ \
\omega\in\operatorname{End}\left(X,\triangleleft_{r}\right)\end{cases}$

  2. 2.

if $\kappa$ is $\rho$-invariant, then

$\displaystyle\kappa\in\mathcal{K}(X,r)\ \Leftrightarrow\ \forall\,x\in
X\quad\kappa R_{x}=R_{x}\kappa\ \Leftrightarrow\ \forall\,x\in
X\quad\bar{\omega}L_{x}=L_{x}\bar{\omega}.$

If we specialize the 4.6 to a solution $(X,r)$ of derived type, it does not
yield a novel result, as we explain below. One can analogously reason if
$(X,r)$ is a solution of right derived type.

######  Remark 4.7.

If $(X,r)$ is a solution whose left derived solution is associated with a left
rack $(X,\triangleright)$, then its right derived solution
$(X,r_{\triangleleft})$ has the form
$r_{\triangleleft}(x,y)=(y\triangleleft_{r}x,x)=(x\triangleright y,x)$, for
all $x,y\in X$, whence $\triangleleft_{r}=\triangleright^{\text{op}}$.
Therefore, the map $T$ from Lemma 4.4 actually is an automorphism of
$(X,\triangleright)$. If we notice that
$\operatorname{Aut}(X,r)=\operatorname{Aut}(X,\triangleright)$ then 4.6 simply
recovers the observation made in 1.2.

######  Corollary 4.8.

Let $(X,r)$ be a bijective non-degenerate solution and let $\kappa:X\to X$ be
a $\rho$-centralizing and $\rho$-invariant map. Then, the following holds

$\displaystyle\kappa\in\mathcal{K}(X,r)\iff\kappa\text{ is
$\lambda$-centralizing}\,.$

###### Proof.

First assume that $\kappa\in\mathcal{K}(X,r)$. Then, since by assumption
$\kappa$ is $\rho$-invariant, 4.6-2 ensures that $\kappa$ commutes with all
$R_{x}$. As a consequence, for all $x,y\in X$ the following holds:

$\displaystyle\kappa\lambda_{x}(y)$
$\displaystyle=\kappa\rho^{-1}_{\rho_{y}(x)}R_{\rho_{y}(x)}(y)=\rho^{-1}_{\rho_{y}(x)}R_{\rho_{y}(x)}\kappa(y)=\lambda_{x}\kappa(y)\,,$

proving the first part. Conversely, the assertion easily follows from 4.6-2.
since the hypothesis yields that $\kappa$ satisfies $\kappa
R_{x}=R_{x}\kappa$, for every $x\in X$. ∎

In a general setting and inspired by [26, Proposition 3.5], we show how to
obtain reflections for arbitrary solutions starting from a reflection and maps
that are $\lambda,\rho$-centralizing and $\lambda,\rho$-invariant.

######  Theorem 4.9.

Let $\left(X,r\right)$ be a solution, $\kappa\in\mathcal{K}\left(X,r\right)$,
and $\varphi,\psi:X\to X$ maps that are $\lambda,\rho$-centralizing and
$\lambda,\rho$-invariant. Then, the map
$\omega:=\varphi\kappa\psi\in\mathcal{K}\left(X,r\right)$.

###### Proof.

Initially, observe that

  * -

$\varphi$ is $\lambda$-centralizing and $\rho$-invariant if and only if
$r\left(\operatorname{id}_{X}\times\varphi\right)=\left(\varphi\times\operatorname{id}_{X}\right)r$;

  * -

$\varphi$ is $\rho$-centralizing and $\lambda$-invariant if and only if
$\left(\operatorname{id}_{X}\times\varphi\right)r=r\left(\varphi\times\operatorname{id}_{X}\right)$.

Similarly, the equivalences hold for the map $\psi$. By these equivalences, we
get

$\displaystyle
r\left(\operatorname{id}_{X}\times\omega\right)r\left(\operatorname{id}_{X}\times\omega\right)$
$\displaystyle=\left(\varphi\times\varphi\right)r\left(\operatorname{id}_{X}\times\kappa\right)r\left(\operatorname{id}_{X}\times\kappa\right)\left(\psi\times\psi\right)$
$\displaystyle\left(\operatorname{id}_{X}\times\omega\right)r\left(\operatorname{id}_{X}\times\omega\right)r$
$\displaystyle=\left(\varphi\times\varphi\right)\left(\operatorname{id}_{X}\times\kappa\right)r\left(\operatorname{id}_{X}\times\kappa\right)r\left(\psi\times\psi\right).$

Therefore, since $\kappa$ is a reflection, then using (RE), $\omega$ is as
well. ∎

######  Remark 4.10.

Notice that if $(X,r)$ is an involutive solution, then, for the maps
$\varphi,\psi$, the assumptions of being $\lambda$-invariant and
$\rho$-centralizing are redundant. Hence, the result in [26, Proposition 3.5]
is a special case of 4.9.

The following is an example of a bijective non-degenerate solution admitting
reflections which are $\lambda$-centralizing, $\rho$-invariant and neither of
both.

######  Example 4.11.

Let $X=\\{1,\dots,8\\}$ and consider the bijective non-degenerate solution
$(X,r)$ given by

$\displaystyle\lambda_{1}=\lambda_{2}=\lambda_{7}=\lambda_{8}=\operatorname{id}_{X}\,,\quad$
$\displaystyle\lambda_{3}=\lambda_{4}=\lambda_{5}=\lambda_{6}=(35)(46)(78)\,,$
$\displaystyle\rho_{1}=\rho_{2}=\operatorname{id}_{X}\,,\quad\rho_{3}=\rho_{4}=(3645)\,,\quad$
$\displaystyle\rho_{5}=\rho_{6}=(3546)\,,\quad\rho_{7}=\rho_{8}=(34)(56)\,.$

With the aid of the GAP software [18], we determined that there are 128
reflections for $(X,r)$. For example, among these we have the following:

  * -

$\kappa_{1}=21222211$ is a $\lambda$-centralizing reflection which is not
$\rho$-invariant;

  * -

$\kappa_{2}=11346578$ is $\rho$-invariant reflection which is not
$\lambda$-centralizing;

  * -

$\kappa_{3}=21436578$ is a $\lambda$-centralizing and $\rho$-invariant
reflection;

  * -

$\kappa_{4}=12556611$ is a reflection which is neither $\lambda$-centralizing
nor $\rho$-invariant.

For the first three examples we employed 4.6 to verify that they actually are
reflections. Moreover, there are 64 which are neither $\lambda$-centralizing
nor $\rho$-invariant. However, by applying 4.9, we can recover each of them in
the form $\varphi\kappa\psi$ for some $\lambda,\rho$-centralizing and
$\lambda,\rho$-invariant maps $\varphi$, $\psi$ (different from the identity)
and for some $\kappa\in\mathcal{K}(X,r)$. For instance, if $\kappa_{4}$ is as
in the example above, then $\kappa_{4}=\varphi\kappa\psi$ where
$\varphi=21436587=\psi$ and $\kappa=12556611$.

Finally, we provide a class of bijective non-degenerate solutions, including
the solution in 2.9, whose reflections are not yet necessarily
$\lambda$-centralizing or $\rho$-invariant.

######  Example 4.12.

Let $(X,\triangleright)$ a left rack and
$\varphi\in\operatorname{Aut}(X,\triangleright)$. Then, [11, Theorem 2.15]
ensures that the map $r(x,y)=\left(\varphi(y),L_{y}\varphi^{-1}(x)\right)\,$,
for all $x,y\in X$, is a bijective non-degenerate solution. An easy
application of Lemma 1.6 allows to determine that

$\kappa\in\mathcal{K}(X,r)\iff\ \begin{cases}\ \forall\,x\in X\quad\kappa
L_{x}=\kappa L_{\kappa(x)}&\\\ \ \
\kappa\in\operatorname{End}\left(X,\triangleright\right)\end{cases}$

In particular, observe that $\kappa$ is $\lambda$-centralizing if and only if
$\kappa\varphi=\varphi\kappa$. On the other hand, $\kappa$ is $\rho$-invariant
if and only if $\kappa$ is $L$-invariant. For instance, let
$(X,\triangleright)$ be the dihedral quandle of order $n\neq 1,2$ on
$\mathbb{Z}_{n}$, $\kappa$ the constant map of constant value
$0\in\mathbb{Z}_{n}$, and notice that $\kappa\in\mathcal{K}(X)$ is neither
$\lambda$-centralizing nor $\rho$-invariant.

For other observations, we refer the reader to the appendix Section A of this
document.

##  5 Constructions of reflections

One can find in the literature some techniques to construct solutions from
already known solutions, such as the _twisted unions_ of involutive non-
degenerate solutions [15], the _matched product_ [2], and the _strong
semilattice_ of solutions [3]. In this context, this section is dedicated to
providing methods for constructing reflections of these solutions starting
from reflections on the known solutions.

###  5.1 The reflection of the matched product of two solutions

Let us begin by recalling the essential notions contained in [2]. If
$(S,r_{S})$ and $(T,r_{T})$ are two solutions,
$\alpha:T\to\operatorname{Sym}\left(S\right)$ and
$\beta:S\to\operatorname{Sym}\left(T\right)$ are maps, set
$\alpha_{u}:=\alpha\left(u\right)$, for every $u\in T$, and
$\beta_{a}:=\beta\left(a\right)$, for every $a\in S$. Then, the quadruple
$\left(r_{S},r_{T},\alpha,\beta\right)$ is said to be a _matched product
system of solutions_ if the following conditions hold

$\displaystyle\alpha_{u}\alpha_{v}=\alpha_{\lambda_{u}\left(v\right)}\alpha_{\rho_{v}\left(u\right)}$
(s1)

$\displaystyle\beta_{a}\beta_{b}=\beta_{\lambda_{a}\left(b\right)}\beta_{\rho_{b}\left(a\right)}$
(s2)

$\displaystyle\rho_{\alpha^{-1}_{u}\\!\left(b\right)}\alpha^{-1}_{\beta_{a}\left(u\right)}\left(a\right)=\alpha^{-1}_{\beta_{\rho_{b}\left(a\right)}\beta^{-1}_{b}\left(u\right)}\rho_{b}\left(a\right)$
(s3)

$\displaystyle\rho_{\beta^{-1}_{a}\\!\left(v\right)}\beta^{-1}_{\alpha_{u}\left(a\right)}\left(u\right)=\beta^{-1}_{\alpha_{\rho_{v}\left(u\right)}\alpha^{-1}_{v}\left(a\right)}\rho_{v}\left(u\right)$
(s4)

$\displaystyle\lambda_{a}\alpha_{\beta^{-1}_{a}\left(u\right)}{}=\alpha_{u}{\lambda_{\alpha^{-1}_{u}{\left(a\right)}}{}}$
(s5)

$\displaystyle\lambda_{u}{\beta_{\alpha^{-1}_{u}{\left(a\right)}}{}}=\beta_{a}{\lambda_{\beta^{-1}_{a}\left(u\right)}{}}$
(s6)

for all $a,b\in S$ and $u,v\in T$.
As shown in [2, Theorem 2], any matched product system of solutions
$\left(r_{S},\,r_{T},\,\alpha,\,\beta\right)$ determines a new solution
$r_{S}\bowtie r_{T}:(S{\times}T)^{2}\to(S{\times}T)^{2}$ defined by

$\displaystyle r_{S}\bowtie
r_{T}\left(\left(a,u\right),\left(b,v\right)\right):=\left(\left(\alpha_{u}{\lambda_{\bar{a}}{\left(b\right)}},\,\beta_{a}\lambda_{\bar{u}}{\left(v\right)}\right),\
\left(\alpha^{-1}_{\overline{U}}{\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}},\,\beta^{-1}_{\overline{A}}\rho_{\beta_{\bar{a}}\left(v\right)}{\left(u\right)}\right)\right),$

where we set

$\bar{a}:=\alpha^{-1}_{u}{\left(a\right)}$,
$\bar{u}:=\beta^{-1}_{a}\left(u\right)$,
$A:=\alpha_{u}{\lambda_{\bar{a}}{\left(b\right)}}$,
$U:=\beta_{a}\lambda_{\bar{u}}{\left(v\right)}$,
$\overline{A}:=\alpha^{-1}_{U}{\left(A\right)}$,
$\overline{U}:=\beta^{-1}_{A}\left(U\right)$,

for all $\left(a,u\right),\left(b,v\right)\in S\times T$, called the _matched
product of the solutions_ $r_{S}$ and $r_{T}$ (via $\alpha$ and $\beta$).
In particular, if the solutions $(S,r_{S})$ and $(T,r_{T})$ we start from are
bijective with finite order, then $r_{S}\bowtie r_{T}$ is bijective with
finite order. In addition, if $(S,r_{S})$ and $(T,r_{T})$ are non-degenerate,
then $r_{S}\bowtie r_{T}$ is left non-degenerate. Thus, in the finite case,
$r_{S}\bowtie r_{T}$ is also right non-degenerate.

######  Lemma 5.1.

Let $\left(r_{S},r_{T},\alpha,\beta\right)$ be a matched product system of
solutions, $\kappa:S\to S$ and $\omega:T\to T$ maps such that

$\displaystyle\alpha_{u}\kappa$
$\displaystyle=\kappa\alpha_{u}\qquad\text{and}\qquad\beta_{a}\omega=\omega\beta_{a},$
(M)

for all $u\in T$ and $a\in S$. Set
$\mathcal{A}:=\lambda_{a}{\alpha_{\bar{u}}{\kappa\left(b\right)}}$ and
$\mathcal{U}:=\lambda_{u}{\beta_{\bar{a}}{\omega\left(v\right)}}$, for all
$a,b\in S$ and $u,v\in T$, then the following statements hold:

  1. 1.

$t\left(\left(a,u\right),\left(\kappa\left(b\right),\omega\left(v\right)\right)\right)=\left(\lambda_{\mathcal{A}}{\kappa\rho_{\alpha_{\bar{u}}{\kappa\left(b\right)}}{\left(a\right)}},\lambda_{\mathcal{U}}{\omega\rho_{\beta_{\bar{a}}{\omega\left(v\right)}}{\left(u\right)}}\right)$,

  2. 2.

$t\left(\left(a,u\right),\left(b,v\right)\right)=\left(\lambda_{A}{\kappa\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}},\lambda_{U}{\omega\rho_{\beta_{\bar{a}}{\left(v\right)}}{\left(u\right)}}\right),$

for all $\left(a,u\right),\left(b,v\right)\in S\times T$.

###### Proof.

Let $(a,u),(b,v)\in S\times T$. We show the proof for the first components of
the identities in $1.$ and $2.$, since the proof for the second ones will be
similar, by exchanging the roles of $a$ and $u$, $b$ and $v$, $\mathcal{A}$
and $\mathcal{U}$, $\alpha$ and $\beta$, and $\kappa$ and $\omega$, and by
using $\left(s6\right)$ instead of $\left(s5\right)$. The first component of
$t\left(\left(a,u\right),\left(\kappa\left(b\right),\omega\left(v\right)\right)\right)$
is given by

$\displaystyle\alpha_{\mathcal{U}}{\lambda_{\overline{\mathcal{A}}}{\kappa\alpha^{-1}_{\overline{\mathcal{U}}}{\rho_{\alpha_{\bar{u}}{\kappa\left(b\right)}}{\left(a\right)}}}}\underset{\eqref{cond_match_ref}}{=}\alpha_{\mathcal{U}}{\lambda_{\overline{\mathcal{A}}}{\alpha^{-1}_{\overline{\mathcal{U}}}{\kappa\rho_{\alpha_{\bar{u}}{\kappa\left(b\right)}}{\left(a\right)}}}}\underset{\left(s5\right)}{=}\lambda_{\mathcal{A}}{\kappa\rho_{\alpha_{\bar{u}}{\kappa\left(b\right)}}{\left(a\right)}}$

and the first component of $\left(\left(a,u\right),\left(b,v\right)\right)$ is

$\displaystyle\alpha_{U}{\lambda_{\overline{A}}{\kappa\alpha^{-1}_{\overline{U}}{\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}}}}\underset{\eqref{cond_match_ref}}{=}\alpha_{U}{\lambda_{\overline{A}}{\alpha^{-1}_{\overline{U}}{\kappa\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}}}}\underset{\left(s5\right)}{=}\lambda_{A}{\kappa\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}},$

which complete the proof. ∎

######  Lemma 5.2.

Let $\left(r_{S},r_{T},\alpha,\beta\right)$ be a matched product system of
solutions, $\kappa:S\to S$ and $\omega:T\to T$ maps such that (M) holds. Set

$\displaystyle\mathcal{A^{\prime}}:=\alpha_{\mathcal{U}}{\lambda_{\overline{\mathcal{A}}}{\left(\kappa\alpha^{-1}_{\overline{\mathcal{U}}}{\rho_{\alpha_{\bar{u}}{\kappa\left(b\right)}}{\left(a\right)}}\right)}}\qquad\mathcal{U^{\prime}}:=\beta_{\mathcal{A}}{\lambda_{\overline{\mathcal{U}}}{\left(\omega\beta^{-1}_{\overline{\mathcal{A}}}{\rho_{\beta_{\bar{a}}{\omega\left(v\right)}}{\left(u\right)}}\right)}}$
$\displaystyle\mathcal{A^{\prime\prime}}:=\alpha_{U}{\lambda_{\overline{A}}{\left(\kappa\alpha^{-1}_{\overline{U}}{\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}}\right)}}\qquad\mathcal{U^{\prime\prime}}:=\beta_{A}{\lambda_{\overline{U}}{\left(\omega\beta^{-1}_{\overline{A}}{\rho_{\beta_{\bar{a}}{\left(v\right)}}{\left(u\right)}}\right)}},$

for all $a\in S$ and $u\in T$. Then, the following statements hold:

  1. 1.

$q\left(\left(a,u\right),\left(\kappa\left(b\right),\omega\left(v\right)\right)\right)=\left(\alpha^{-1}_{\overline{\mathcal{U^{\prime}}}}{\rho_{\kappa\rho_{\kappa\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}}{\lambda_{a}{\kappa\alpha_{\bar{u}}{\left(b\right)}}}},\beta^{-1}_{\overline{\mathcal{A^{\prime}}}}{\rho_{\omega\rho_{\omega\beta_{\bar{a}}{\left(v\right)}}{\left(u\right)}}{\lambda_{u}{\omega\beta_{\bar{a}}{\left(v\right)}}}}\right),$

  2. 2.

$\left(\kappa\times\omega\right)q\left(\left(a,u\right),\left(b,v\right)\right)=\left(\alpha^{-1}_{\overline{\mathcal{U^{\prime\prime}}}}{\kappa\rho_{\kappa\alpha_{\bar{u}}{\left(b\right)}}{\lambda_{a}{\alpha_{\bar{u}}{\left(b\right)}}}},\beta^{-1}_{\overline{\mathcal{A^{\prime\prime}}}}{\omega\rho_{\omega\rho_{\beta_{\bar{a}}{\left(v\right)}}{\left(u\right)}}{\lambda_{u}{\beta_{\bar{a}}{\left(v\right)}}}}\right)$,

for all $\left(a,u\right),\left(b,v\right)\in S\times T$.

###### Proof.

Similar to the proof of Lemma 5.1, applying the conditions (M) and
$\left(s5\right)$-$\left(s6\right)$. ∎

######  Theorem 5.3.

Let $\left(r_{S},r_{T},\alpha,\beta\right)$ be a matched product system of
solutions, $\kappa:S\to S$ and $\omega:T\to T$ maps such that

$\displaystyle\alpha_{u}\kappa=\kappa\alpha_{u}\qquad\text{and}\qquad\beta_{a}\omega=\omega\beta_{a}$
(M)

for all $a\in S$ and $u\in T$. Then, the following statements are equivalent:

  1. (i)

$\kappa\in\mathcal{K}\left(S,r_{S}\right)$ and
$\omega\in\mathcal{K}\left(T,r_{T}\right)$;

  2. (ii)

$\kappa\times\omega\in\mathcal{K}\left(S\bowtie T,r_{S}\bowtie r_{T}\right)$.

###### Proof.

Initially, suppose that $\kappa\in\mathcal{K}(S,r_{S})$ and
$\omega\in\mathcal{K}(T,r_{T})$. If $(a,u),(b,v)\in S\times T$, we obtain

$\displaystyle
t\left(\left(a,u\right),\left(\kappa\left(b\right),\omega\left(v\right)\right)\right)=\left(\lambda_{\mathcal{A}}{\kappa\rho_{\alpha_{\bar{u}}{\kappa\left(b\right)}}{\left(a\right)}},\lambda_{\mathcal{U}}{h\rho_{\beta_{\bar{a}}{h\left(v\right)}}{\left(u\right)}}\right)$
by Lemma 5.1-$1.$
$\displaystyle=\left(\lambda_{\lambda_{a}{\kappa\alpha_{\bar{u}}{\left(b\right)}}}{\kappa\rho_{\kappa\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}},\lambda_{\lambda_{u}{\omega\beta_{\bar{a}}{\left(v\right)}}}{\omega\rho_{\omega\beta_{\bar{a}}{\left(v\right)}}{\left(u\right)}}\right)$
by (M)
$\displaystyle=\left(t\left(a,\kappa\alpha_{\bar{u}}\left(b\right)\right),t\left(u,\omega\beta_{\bar{a}}\left(v\right)\right)\right)$
$\displaystyle=\left(t\left(a,\alpha_{\bar{u}}\left(b\right)\right),t\left(u,\beta_{\bar{a}}\left(v\right)\right)\right)$
by (T)-(Q)
$\displaystyle=\left(\lambda_{A}{\kappa\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}},\lambda_{U}{\omega\rho_{\beta_{\bar{a}}{\left(v\right)}}{\left(u\right)}}\right)$
$\displaystyle=t\left(\left(a,u\right),\left(b,v\right)\right)$ by Lemma
5.1-$2.$

Moreover, it follows from the last computations and (M) that

$\displaystyle\mathcal{A^{\prime}}$
$\displaystyle=\alpha_{\mathcal{U}}{\lambda_{\overline{A}}{\left(\kappa\alpha^{-1}_{\overline{\mathcal{U}}}{\rho_{\alpha_{\bar{u}}{\kappa\left(b\right)}}{\left(a\right)}}\right)}}=\lambda_{A}{\kappa\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}}=\alpha_{U}{\lambda_{\overline{A}}{\left(\kappa\alpha^{-1}_{\overline{U}}{\rho_{\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}}\right)}}=\mathcal{A^{\prime\prime}}$

and similarly $\mathcal{U^{\prime}}=\mathcal{U^{\prime\prime}}$. Hence, we
have

$\displaystyle
q\left(\left(a,u\right),\left(\kappa\left(b\right),\omega\left(v\right)\right)\right)$
$\displaystyle=\left(\alpha^{-1}_{\overline{\mathcal{U^{\prime}}}}{\rho_{\kappa\rho_{\kappa\alpha_{\bar{u}}{\left(b\right)}}{\left(a\right)}}{\lambda_{a}{\kappa\alpha_{\bar{u}}{\left(b\right)}}}},\beta^{-1}_{\overline{\mathcal{A^{\prime}}}}{\rho_{\omega\rho_{\omega\beta_{\bar{a}}{\left(v\right)}}{\left(u\right)}}{\lambda_{u}{\omega\beta_{\bar{a}}{\left(v\right)}}}}\right)$
by Lemma 5.2-$1.$
$\displaystyle=\left(\alpha^{-1}_{\overline{\mathcal{U^{\prime}}}}q\left(a,\kappa\alpha_{\bar{u}}{\left(b\right)}\right),\beta^{-1}_{\overline{\mathcal{A^{\prime}}}}{q}\left(u,\omega\beta_{\bar{a}}{\left(v\right)}\right)\right)$
$\displaystyle=\left(\alpha^{-1}_{\overline{\mathcal{U^{\prime}}}}\kappa\left(q\left(a,\alpha_{\bar{u}}{\left(b\right)}\right)\right),\beta^{-1}_{\overline{\mathcal{A^{\prime}}}}{\omega}\left(q\left(u,\beta_{\bar{a}}{\left(v\right)}\right)\right)\right)$
by (T)-(Q)
$\displaystyle=\left(\alpha^{-1}_{\overline{\mathcal{U^{\prime\prime}}}}\kappa\left(q\left(a,\alpha_{\bar{u}}{\left(b\right)}\right)\right),\beta^{-1}_{\overline{\mathcal{A^{\prime\prime}}}}{\omega}\left(q\left(u,\beta_{\bar{a}}{\left(v\right)}\right)\right)\right)$
since $\mathcal{A^{\prime}}=\mathcal{A^{\prime\prime}}$ and
$\mathcal{U^{\prime}}=\mathcal{U^{\prime\prime}}$
$\displaystyle=\left(\alpha^{-1}_{\overline{\mathcal{U^{\prime\prime}}}}{\kappa\rho_{\kappa\alpha_{\bar{u}}{\left(b\right)}}{\lambda_{a}{\alpha_{\bar{u}}{\left(b\right)}}}},\beta^{-1}_{\overline{\mathcal{A^{\prime\prime}}}}{h\rho_{h\rho_{\beta_{\bar{a}}{\left(v\right)}}{\left(u\right)}}{\lambda_{u}{\beta_{\bar{a}}{\left(v\right)}}}}\right)$
$\displaystyle=\left(\kappa\times\omega\right)q\left(\left(a,u\right),\left(b,v\right)\right)$
by Lemma 5.2-$2.$

Thus, $\kappa\times\omega\in\mathcal{K}\left(S\bowtie T,r_{S}\bowtie
r_{T}\right)$.
Conversely, we suppose that $\kappa\times\omega$ is a reflection of
$r_{S}\bowtie r_{T}$ and we show that $\kappa\in\mathcal{K}(S,r_{S})$. The
proof for the map $\omega$ will be omitted since it is analogous.
Let $(a,u),(b,v)\in S\times T$ and

$\displaystyle\mathcal{A}:=\lambda_{a}{\alpha_{\bar{u}}{\kappa\left(\alpha^{-1}_{\bar{u}}{\left(b\right)}\right)}}\qquad
A:=\alpha_{u}{\lambda_{\bar{a}}{\left(\alpha^{-1}_{\bar{u}}{\left(b\right)}\right)}},$

thus from the previous computations, since $\kappa\times\omega$ is a
reflection, we get

$\displaystyle\lambda_{\mathcal{A}}{\kappa\rho_{\alpha_{\bar{u}}\kappa\left(\alpha^{-1}_{\bar{u}}{\left(b\right)}\right)}{\left(a\right)}}=\lambda_{A}{\kappa\rho_{\alpha_{\bar{u}}\alpha^{-1}_{\bar{u}}{\left(b\right)}}{\left(a\right)}},$

namely
$\lambda_{\mathcal{A}}{\kappa\rho_{\kappa\left(b\right)}\left(a\right)}=\lambda_{A}{\kappa\rho_{{\left(b\right)}}{\left(a\right)}}.$
Thus, since by (M) and $\left(s_{5}\right)$,

$\mathcal{A}=\lambda_{a}{\alpha_{\bar{u}}{\kappa\left(\alpha^{-1}_{\bar{u}}{\left(b\right)}\right)}}=\lambda_{a}{\kappa\left(b\right)}$
$A=\alpha_{u}{\lambda_{\bar{a}}{\left(\alpha^{-1}_{\bar{u}}{\left(b\right)}\right)}}=\lambda_{a}{\left(b\right)}$,

we have that

$\displaystyle
t\left(a,\kappa\left(b\right)\right)=\lambda_{\lambda_{a}{\kappa\left(b\right)}}{\kappa\rho_{\kappa\left(b\right)}{\left(a\right)}}=\lambda_{\mathcal{A}}{\kappa\rho_{\kappa\left(b\right)}{\left(a\right)}}=\lambda_{A}{\kappa\rho_{b}{\left(a\right)}}=\lambda_{\lambda_{a}{\left(b\right)}}{\kappa\rho_{b}{\left(a\right)}}=t\left(a,b\right).$

Moreover, since

$\displaystyle\rho_{\kappa\rho_{\kappa\alpha_{\bar{u}}{\left(\alpha^{-1}_{\bar{u}}\left(b\right)\right)}}{\left(a\right)}}{\lambda_{a}{\kappa\alpha_{\bar{u}}{\left(\alpha^{-1}_{\bar{u}}\left(b\right)\right)}}}=\kappa\rho_{\kappa\alpha_{\bar{u}}{\left(\alpha^{-1}_{\bar{u}}\left(b\right)\right)}}{\lambda_{a}{\alpha_{\bar{u}}{\left(\alpha^{-1}_{\bar{u}}\left(b\right)\right)}}},$

we obtain $q(a,\kappa(b))=\kappa q(a,b)$, i.e., $\kappa$ is a reflection for
the solution $r_{S}$. ∎

According to [2, Corollary 6], in the case of left non-degenerate and
involutive solutions, the requirements to have a matched product system of
solutions become easier. In particular, a quadruple
$\left(r_{S},\,r_{T},\,\alpha,\,\beta\right)$ is a matched product system of
solutions if and only if they hold

$\displaystyle\alpha_{u}\alpha_{\lambda_{u}^{-1}(v)}$
$\displaystyle=\alpha_{v}\alpha_{\lambda_{v}^{-1}(u)}$

$\displaystyle\beta_{a}\beta_{\lambda_{a}^{-1}(b)}=\beta_{b}\beta_{\lambda_{b}^{-1}(a)}$

$\displaystyle\lambda_{a}\alpha_{\overline{u}}$
$\displaystyle=\alpha_{u}\lambda_{\overline{a}}$

$\displaystyle\lambda_{u}\beta_{\overline{a}}=\beta_{a}\lambda_{\overline{u}}$

for all $a,b\in S$ and $u,v\in T$. Note that one can easily choose
$\alpha_{u}=\lambda_{u}$, for every $u\in T$, and $\beta_{a}=\lambda_{a}$, for
every $a\in S$. Hence, we can specialize the 5.3 for reflections of involutive
left non-degenerate solutions that are $\lambda$-centralizing.

######  Corollary 5.4.

Let $(S,r_{S})$ and $(T,r_{T})$ be involutive left non-degenerate solutions
and let $\left(r_{S},r_{T},\alpha,\beta\right)$ be a matched product system of
solutions. Let us assume that $\kappa:S\to S$ and $\omega:T\to T$ are
$\lambda$-centralizing reflections for $r_{S}$ and $r_{T}$, respectively,
satisfying (M). Then, $\kappa\times\omega$ is a $\lambda$-centralizing
reflection for $\left(S\bowtie T,r_{S}\bowtie r_{T}\right)$.

###### Proof.

It is straightforward. ∎

######  Remark 5.5.

Note that, under the same assumptions of 5.4, if we choose
$\alpha_{u}=\lambda_{u}$, for every $u\in T$, and $\beta_{a}=\lambda_{a}$, for
every $a\in S$, then condition (M) is automatically satisfied.

Notice that if $\left(r_{S},r_{T},\alpha,\beta\right)$ is a matched product
system of solutions that are involutive right non-degenerate and $\kappa$ and
$\omega$ are $\rho$-invariant reflections for $r_{S}$ and $r_{T}$,
respectively, satisfying (M), the reflection $\kappa\times\omega$ for the
solution $r_{S}\bowtie r_{T}$ may not have the same behaviour. It is
sufficient to consider the semidirect product of two solutions, as we show in
the next example.

######  Example 5.6.

Consider the trivial solutions $(S,r_{S})$ and $(T,r_{T})$ given by
$r_{S}(a,b)=(b,a)$ and $r_{T}(u,v)=(v,u)$, for all $a,b\in S$ and $u,v\in T$,
$\kappa=\operatorname{id}_{S}$, and let $\omega:T\to T$ be an arbitrary map.
Clearly, $\omega\in\mathcal{K}(T,r_{T})$ is $\rho$-invariant. Moreover, let
$\beta_{a}=\operatorname{id}_{T}$, for every $a\in S$, and let
$\alpha_{u}:S\to S$ be maps such that
$\alpha_{u}\alpha_{v}=\alpha_{v}\alpha_{u}$, for all $u,v\in T$. Then,
$\left(r_{S},r_{T},\alpha,\beta\right)$ is a matched product system of
solutions. Besides, by 5.3, $\kappa\times\omega$ is a reflection for
$r_{S}\bowtie r_{T}$ and

$\displaystyle\kappa\times\omega\,\ \text{is $\rho$-invariant}\ \iff\
\alpha_{u}=\alpha_{\omega(u)}.$

###  5.2 The reflection of the strong semilattice of solutions

Let us start by recalling the construction of strong semilattice of solutions
provided in [3, Theorem 4.1]. Let $Y$ be a (lower) semilattice,
$\left\\{X_{\alpha}\ \left|\ \alpha\in Y\right.\right\\}$ a family of disjoint
sets indexed by $Y$, and $\left(X_{\alpha},r_{\alpha}\right)$ a solution, for
every $\alpha\in Y$. For all $\alpha,\beta\in Y$ such that $\alpha\geq\beta$,
let $\phi_{\alpha,\beta}:X_{\alpha}\to X_{\beta}$ be a map such that

  1. 1.

$\phi_{\alpha,\alpha}$ is the identity map of $X_{\alpha}$, for every
$\alpha\in Y$,

  2. 2.

$\phi_{\beta,\gamma}\phi_{\alpha,\beta}=\phi_{\alpha,\gamma}$, for all
$\alpha,\beta,\gamma\in Y$ such that $\alpha\geq\beta\geq\gamma$,

  3. 3.

$\left(\phi_{\alpha,\beta}\times\phi_{\alpha,\beta}\right)r_{\alpha}=r_{\beta}\left(\phi_{\alpha,\beta}\times\phi_{\alpha,\beta}\right)$,
for all $\alpha,\beta\in Y$ such that $\alpha\geq\beta$.

Set $X=\bigcup\left\\{X_{\alpha}\ \left|\ \alpha\in Y\right.\right\\}$, then
the map $r:X\times X\longrightarrow X\times X$ defined by

$\displaystyle
r\left(x,y\right)=r_{\alpha\beta}\left(\phi_{\alpha,\alpha\beta}\left(x\right),\phi_{\beta,\alpha\beta}\left(y\right)\right),$

for all $x\in X_{\alpha}$ and $y\in X_{\beta}$, is a solution on $X$. We call
the pair $\left(X,r\right)$ the _strong semilattice $Y$ of solutions
$\left(X_{\alpha},r_{\alpha}\right)$_.
In general, a strong semilattice of solutions, with $|Y|>1$, is not bijective
and is degenerate, even in the case we involve bijective non-degenerate
solutions, cf. [3, p. 13].

######  Theorem 5.7.

Let $\left(X,r\right)$ be a strong semilattice $Y$ of solutions
$\left(X_{\alpha},r_{\alpha}\right)$ and assume that $\kappa_{\alpha}$ is a
reflection for $\left(X_{\alpha},r_{\alpha}\right)$, for every $\alpha\in Y$.
Then, the map $\kappa:X\to X$ given by $\kappa(a)=\kappa_{\alpha}(a)$, for
every $a\in X_{\alpha}$, is a reflection of the solution $(X,r)$ if and only
if

$\displaystyle\phi_{\beta,\alpha\beta}\kappa_{\beta}=\kappa_{\alpha\beta}\phi_{\beta,\alpha\beta}$
(S)

is satisfied, for all $\alpha,\beta\in Y$ such that $\alpha\geq\beta$.

###### Proof.

Let $a\in X_{\alpha}$ and $b\in X_{\beta}$. Then, we get

$\displaystyle t(a,\kappa(b))$
$\displaystyle=\lambda_{\lambda_{\phi_{\alpha,\alpha\beta}\left(a\right)}\phi_{\beta,\alpha\beta}\kappa_{\beta}\left(b\right)}\kappa_{\alpha\beta}\rho_{\phi_{\beta,\alpha\beta}\kappa_{\beta}\left(b\right)}\phi_{\alpha,\alpha\beta}\left(a\right)$
$\displaystyle=\lambda_{\lambda_{\phi_{\alpha,\alpha\beta}\left(a\right)}\kappa_{\alpha\beta}\phi_{\beta,\alpha\beta}\left(b\right)}\kappa_{\alpha\beta}\rho_{\kappa_{\alpha\beta}\phi_{\beta,\alpha\beta}\left(b\right)}\phi_{\alpha,\alpha\beta}\left(a\right)$
by (S)
$\displaystyle=t\left(\phi_{\alpha,\alpha\beta}\left(a\right),\kappa_{\alpha\beta}\phi_{\beta,\alpha\beta}\left(b\right)\right)$
$\displaystyle=t\left((\phi_{\alpha,\alpha\beta}\left(a\right),\phi_{\beta,\alpha\beta}\left(b\right)\right)$
$\kappa_{\alpha\beta}$ is a reflection
$\displaystyle=\lambda_{\lambda_{\phi_{\alpha,\alpha\beta}\left(a\right)}\phi_{\beta,\alpha\beta}\left(b\right)}\kappa_{\alpha\beta}\rho_{\phi_{\beta,\alpha\beta}\left(b\right)}\phi_{\alpha,\alpha\beta}\left(a\right)$
$\displaystyle=t(a,b)$

and

$\displaystyle q(a,\kappa(b))$
$\displaystyle=\rho_{\kappa_{\alpha\beta}\rho_{\phi_{\beta,\alpha\beta}\kappa_{\beta}\left(b\right)}{\phi_{\alpha,\alpha\beta}\left(a\right)}}\lambda_{\phi_{\alpha,\alpha\beta}\left(a\right)}\phi_{\beta,\alpha\beta}\kappa_{\beta}\left(b\right)$
$\displaystyle=\rho_{\kappa_{\alpha\beta}\rho_{\kappa_{\alpha\beta}\phi_{\beta,\alpha\beta}\left(b\right)}{\phi_{\alpha,\alpha\beta}\left(a\right)}}\lambda_{\phi_{\alpha,\alpha\beta}\left(a\right)}\kappa_{\alpha\beta}\phi_{\beta,\alpha\beta}\left(b\right)$
by (S)
$\displaystyle=q\left(\phi_{\alpha,\alpha\beta}\left(a\right),\kappa_{\alpha\beta}\phi_{\beta,\alpha\beta}\left(b\right)\right)$
$\displaystyle=\kappa_{\alpha\beta}\left(q\left(\phi_{\alpha,\alpha\beta}\left(a\right),\phi_{\beta,\alpha\beta}\left(b\right)\right)\right)$
$\kappa_{\alpha\beta}$ is a reflection
$\displaystyle=\kappa_{\alpha\beta}\rho_{\kappa_{\alpha\beta}\rho_{\phi_{\beta,\alpha\beta}\left(b\right)}\phi_{\alpha,\alpha\beta}\left(a\right)}\lambda_{\phi_{\alpha,\alpha\beta}\left(a\right)}\phi_{\beta,\alpha\beta}\left(b\right)$
$\displaystyle=\kappa(q(a,b)),$

hence $\kappa$ is a reflection of the solution $r$. ∎

######  Corollary 5.8.

Let $\left(X,r\right)$ be a strong semilattice $Y$ of solutions
$\left(X_{\alpha},r_{\alpha}\right)$ and assume that
$\kappa_{\alpha}\in\mathcal{K}\left(X_{\alpha},r_{\alpha}\right)$, for every
$\alpha\in Y$, such that $\eqref{cond_strong}$ is satisfied. Consider the
reflection $\kappa:X\to X$ given by $\kappa(a)=\kappa_{\alpha}(a)$, for every
$a\in X_{\alpha}$. Then, the following hold:

  1. 1.

if $\kappa_{\alpha}$ is $\lambda^{[\alpha]}$-centralizing for the solution
$\left(X_{\alpha},r_{\alpha}\right)$, for every $\alpha\in Y$, then $\kappa$
is $\lambda$-centralizing for $(X,r)$;

  2. 2.

if the map $\kappa_{\alpha}$ is $\rho^{[\alpha]}$-invariant, for $\alpha\in
Y$, then $\kappa$ is $\rho$-invariant.

###### Proof.

It is straightforward. ∎

###  5.3 The reflection of the twisted unions

In [15, Section 3.4], Etingof, Schedler, and Soloviev provided a technique to
construct involutive non-degenerate solutions in the finite case, called the
_twisted union_. A computer program generated all solutions, up to
isomorphism, on sets of cardinality $|X|\leq 8$, and showed that almost all
are twisted union of involutive non-degenerate solutions on smaller sets. We
extend such a construction to arbitrary solutions.

######  Proposition 5.9.

Let $(X,u)$ and $(Y,v)$ be two solutions. Set
$Z=X\mathbin{\mathchoice{\ooalign{$\displaystyle\cup$\cr$\displaystyle\cdot$}}{\ooalign{$\textstyle\cup$\cr$\textstyle\cdot$}}{\ooalign{$\scriptstyle\cup$\cr$\scriptstyle\cdot$}}{\ooalign{$\scriptscriptstyle\cup$\cr$\scriptscriptstyle\cdot$}}}Y$
and consider the map $r:Z\times Z\to Z\times Z$ defined by

$r\left(a,\,b\right)=\begin{cases}\ u\left(a,\,b\right)&\ \text{if}\quad
a,\,b\in X\\\ \ v\left(a,\,b\right)&\ \text{if}\quad a,\,b\in Y\\\ \
\left(g\left(b\right),\,f\left(a\right)\right)&\ \text{if}\quad a\in X,b\in
Y\\\ \ \left(\alpha\left(b\right),\,\beta\left(a\right)\right)&\
\text{if}\quad a\in Y,b\in X\end{cases}$

for all $a,b\in Z$, with $f,\alpha:X\to X$ and $g,\beta:Y\to Y$ maps. Then,
$(Z,\,r)$ is a solution if and only if for all $x\in X$ and $y\in Y$ the
following hold:

$\displaystyle\alpha f=f\alpha$

$\displaystyle g\beta=\beta g$

$\displaystyle\left(f\times f\right)u=u\left(f\times f\right)$

$\displaystyle\left(g\times g\right)v=v\left(g\times g\right)$

$\displaystyle\left(\alpha\times\alpha\right)u=u\left(\alpha\times\alpha\right)$

$\displaystyle\left(\beta\times\beta\right)v=v\left(\beta\times\beta\right)$

$\displaystyle\alpha\lambda_{f\left(x\right)}=\lambda_{x}\alpha$

$\displaystyle\beta\rho_{g\left(y\right)}=\rho_{y}\beta$

$\displaystyle f\rho_{\alpha(x)}=\rho_{x}f$

$\displaystyle g\rho_{\alpha(y)}=\rho_{y}g$

The pair $(Z,r)$ is called the _twisted union of $(X,u)$ and $(Y,v)$ via
$f,\alpha$ and $g,\beta$_.

###### Proof.

To get the claim it is enough to verify that

$\displaystyle\eqref{first}$
$\displaystyle\Leftrightarrow\alpha\lambda_{x}=\lambda_{\alpha\left(x\right)}\alpha,\
\ g\lambda_{y}=\lambda_{g\left(y\right)}g,\ \
\alpha\lambda_{f\left(x\right)}=\lambda_{x}\alpha,\ \text{and }\
g\lambda_{\beta\left(y\right)}=\lambda_{y}g,$ $\displaystyle\eqref{second}$
$\displaystyle\Leftrightarrow f\alpha=\alpha f,\ \ g\beta=\beta g,\ \
f\lambda_{x}=\lambda_{f\left(x\right)}f,\ \
\beta\lambda_{y}=\lambda_{\beta\left(y\right)}\beta,\ $
$\displaystyle\,\quad\alpha\rho_{x}=\rho_{\alpha\left(x\right)}\alpha,\ \
\text{and }\ g\rho_{y}=\rho_{g\left(y\right)}g,$ $\displaystyle\eqref{third}$
$\displaystyle\Leftrightarrow f\rho_{x}=\rho_{f\left(x\right)}f,\ \
\beta\rho_{y}=\rho_{\beta\left(y\right)}\beta,\ \
f\rho_{\alpha\left(x\right)}=\rho_{x}f,\ \text{and }\
\beta\rho_{g\left(y\right)}=\rho_{y}\beta,$

for all $x\in X$ and $y\in Y$, respectively. ∎

######  Example 5.10.

Let $\left(X,u\right)$ and $\left(Y,v\right)$ be the solutions given by
$u\left(x,y\right)=\left(f\left(y\right),\alpha\left(x\right)\right)$ and
$v\left(x,y\right)=\left(g\left(y\right),\beta\left(x\right)\right)$ with
$f,\alpha:X\to X$ maps such that $f\alpha=\alpha f$ and $g,\beta:Y\to Y$ maps
such that $g\beta=\beta g$. Then, the twisted union $(Z,r)$ of
$\left(X,u\right)$ and $\left(Y,v\right)$ via $f,\alpha$ and $g,\beta$ is
solution that does not belong to the class of Lyubashenko’s solution.

######  Corollary 5.11.

Let $(X,u)$ and $(Y,v)$ be bijective non-degenerate solutions,
$f,\alpha\in\operatorname{Sym}_{X}$ and $g,\beta\in\operatorname{Sym}_{Y}$.
Then, the twisted union $(Z,r)$ of $(X,u)$ and $(Y,v)$ via $f,\alpha$ and
$g,\beta$ is a bijective non-degenerate solution.

######  Remark 5.12.

It is clear that 5.11 includes the construction of twisted union provided in
[15] if we consider two involutive non-degenerate solutions and aim to
construct solutions of the same type. Indeed, this is the case with
$\alpha=f^{-1}$ and $\beta=g^{-1}$.

More generally, by choosing suitable maps $f,\alpha,g$, and $\beta$, two
involutive non-degenerate solutions can give rise to a bijective non-
degenerate solution not necessarily involutive, as we show in the next
example.

######  Example 5.13.

Let $(X,u)$ and $(Y,v)$ be the twist maps on $X$ and $Y$, respectively. As
observed in 1.4, all maps $\kappa:X\to X$ and $\omega:Y\to T$ are reflections
for $(X,u)$ and $(Y,v)$, respectively. Hence, by choosing $f=\alpha=\kappa$
and $g=\beta=\omega$, conditions in 5.9 are satisfied. Furthermore, if
$\kappa$ or $\omega$ are not involutive, then $r$ is not involutive.

In the following, we show that, in the construction of twisted union, one can
choose as maps $f$, $\alpha$, $g$, and $\beta$, the reflections of the
starting solutions if these are of derived type.

######  Example 5.14.

Let $\left(X,u\right)$ be the solution associated to a left rack
$\left(X,\triangleright\right)$ and $(Y,v)$ the solution associated with a
right rack $\left(Y,\triangleleft\right)$. Moreover, let
$\kappa\in\mathcal{K}(X,u)$ and $\omega\in\mathcal{K}(Y,v)$. Set
$f=\alpha=\kappa$, $g=\beta=\omega$, and
$Z=X\mathbin{\mathchoice{\ooalign{$\displaystyle\cup$\cr$\displaystyle\cdot$}}{\ooalign{$\textstyle\cup$\cr$\textstyle\cdot$}}{\ooalign{$\scriptstyle\cup$\cr$\scriptstyle\cdot$}}{\ooalign{$\scriptscriptstyle\cup$\cr$\scriptscriptstyle\cdot$}}}Y$,
by 2.11 and 5.9, we obtain that the map $r:Z\times Z\to Z\times Z$ defined by

$\displaystyle r\left(a,b\right)=\begin{cases}\ \left(b,b\triangleright
a\right)&\ \text{if}\quad a,\,b\in X\\\ \ \left(b\triangleleft a,a\right)&\
\text{if}\quad a,\,b\in Y\\\ \
\left(\omega\left(b\right),\,\kappa\left(a\right)\right)&\ \text{if}\quad a\in
X,b\in Y\\\ \ \left(\kappa\left(b\right),\,\omega\left(a\right)\right)&\
\text{if}\quad a\in Y,b\in X\end{cases}$

is a non-degenerate bijective solution.

######  Theorem 5.15.

Let $(Z,r)$ be the twisted union of two solutions $(X,u)$ and $(Y,v)$,
consider $\kappa\in\mathcal{K}(X,u)$ and $\kappa^{\prime}\in\mathcal{K}(Y,v)$.
Then, the map $\omega:Z\to Z$ defined by

$\omega(a)=\begin{cases}\kappa(a)&\text{if}\quad a\in X\\\
\kappa^{\prime}(a)&\text{if}\quad a\in Y\end{cases}$

is a reflection for $(Z,\,r)$ if and only if $\kappa(f\alpha)=(f\alpha)\kappa$
and $\kappa^{\prime}(g\beta)=(g\beta)\kappa^{\prime}$.

###### Proof.

It is straightforward. ∎

##  A Reflections for solutions associated with skew braces

We gather numerical data computing all reflections for some bijective non-
degenerate solutions. To this aim, we focus on the solutions associated with
the algebraic structure of _skew braces_. For more details about these
structures, we refer the reader to [19]. In particular, we implement the GAP
software and devise an elementary algorithmic approach to compute all
reflections for the solutions associated to the first skew braces in the
database furnished by the YangBaxter package [29].

The algorithm proceeds as explained below.

######  Algorithm.

  1. Input:

A positive integer $n$, a list of positive integers $T$;

  2. 1.

For each $m\in T$, consider the skew brace $B$ of order $n$ and type $m$ in
the database, compute the associated solution and determine all its
reflections with Lemma 1.6;

  3. 2.

Determine which reflections are $\lambda$-centralizing or $\rho$-invariant;

  4. Output:

A list of the reflections for each type labelled as $\lambda$-centralizing or
$\rho$-invariant.

The following table records the number of _all_ reflections for solutions
associated with skew braces until the order $8$. Specifically, for each case,
we enumerate the reflections which are only $\lambda$-centralizing, only
$\rho$-invariant or both. Lastly, for a better analysis, we also highlight the
trivial and almost-trivial skew braces writing the type of the former in
boldface and underlining the type of the latter.

$\\#$ | Type | $(B,+)$ | $(B,\circ)$ | Tot. |  $\lambda$-cen. |  $\rho$-inv. | Both
---|---|---|---|---|---|---|---
2 | 1 | $C_{2}$ | $C_{2}$ | 4 | 0 | 0 | 4
3 | 1 | $C_{3}$ | $C_{3}$ | 27 | 0 | 0 | 27
4 | 1 | $C_{4}$ | $C_{4}$ | 256 | 0 | 0 | 256
2 | $C_{4}$ | $C_{2}\times C_{2}$ | 24 | 8 | 8 | 8
3 | $C_{2}\times C_{2}$ | $C_{2}\times C_{2}$ | 256 | 0 | 0 | 256
4 | $C_{2}\times C_{2}$ | $C_{4}$ | 24 | 8 | 8 | 8
5 | 1 | $C_{5}$ | $C_{5}$ | 3125 | 0 | 0 | 3125
6 | 1 | $S_{3}$ | $S_{3}$ | 49 | 48 | 0 | 1
2 | $S_{3}$ | $S_{3}$ | 2 | 0 | 0 | 2
3 | $S_{3}$ | $C_{6}$ | 11 | 8 | 0 | 1
4 | $S_{3}$ | $C_{6}$ | 28 | 27 | 0 | 1
5 | $C_{6}$ | $S_{3}$ | 864 | 135 | 720 | 9
6 | $C_{6}$ | $C_{6}$ | 46656 | 0 | 0 | 46656
7 | 1 | $C_{7}$ | $C_{7}$ | 823543 | 0 | 0 | 823543
Table 1: $\lambda$-cen. and $\rho$-inv. reflections for solutions associated with skew braces until order $7$. Type | $(B,+)$ | $(B,\circ)$ | Tot. |  $\lambda$-cen. |  $\rho$-inv. | Both
---|---|---|---|---|---|---
1 | $C_{8}$ | $C_{8}$ | 77824 | 12288 | 61440 | 4096
2 | $C_{8}$ | $D_{8}$ | 67328 | 1792 | 65280 | 256
3 | $C_{8}$ | $Q_{8}$ | 67328 | 1792 | 65280 | 256
4 | $C_{8}$ | $C_{8}$ | 16777216 | 0 | 0 | 16777216
5 | $C_{8}$ | $C_{4}\times C_{2}$ | 4352 | 112 | 240 | 16
6 | $C_{4}\times C_{2}$ | $C_{2}\times C_{2}\times C_{2}$ | 77824 | 12288 | 61440 | 4096
7 | $C_{4}\times C_{2}$ | $C_{4}\times C_{2}$ | 12480 | 224 | 224 | 32
8 | $C_{4}\times C_{2}$ | $C_{2}\times C_{2}\times C_{2}$ | 4800 | 224 | 224 | 32
9 | $C_{4}\times C_{2}$ | $C_{4}\times C_{2}$ | 16777216 | 0 | 0 | 16777216
10 | $C_{4}\times C_{2}$ | $C_{4}\times C_{2}$ | 77824 | 12288 | 61440 | 4096
11 | $C_{4}\times C_{2}$ | $C_{4}\times C_{2}$ | 4352 | 112 | 240 | 16
12 | $C_{4}\times C_{2}$ | $D_{8}$ | 81664 | 16128 | 65280 | 256
13 | $C_{4}\times C_{2}$ | $D_{8}$ | 81664 | 16128 | 65280 | 256
14 | $C_{4}\times C_{2}$ | $C_{4}\times C_{2}$ | 77824 | 12288 | 61440 | 4096
15 | $C_{4}\times C_{2}$ | $Q_{8}$ | 81664 | 16128 | 65280 | 256
16 | $C_{4}\times C_{2}$ | $C_{4}\times C_{2}$ | 4800 | 224 | 224 | 32
17 | $C_{4}\times C_{2}$ | $D_{8}$ | 960 | 224 | 224 | 32
18 | $C_{4}\times C_{2}$ | $D_{8}$ | 1152 | 63 | 0 | 1
19 | $C_{4}\times C_{2}$ | $D_{8}$ | 5312 | 2016 | 224 | 32
20 | $D_{8}$ | $Q_{8}$ | 1184 | 1024 | 0 | 32
21 | $D_{8}$ | $D_{8}$ | 3040 | 3008 | 0 | 32
22 | $D_{8}$ | $D_{8}$ | 64 | 0 | 0 | 64
23 | $D_{8}$ | $Q_{8}$ | 192 | 32 | 0 | 32
24 | $D_{8}$ | $C_{8}$ | 288 | 256 | 16 | 16
25 | $D_{8}$ | $C_{4}\times C_{2}$ | 1120 | 1024 | 0 | 32
26 | $D_{8}$ | $C_{4}\times C_{2}$ | 1120 | 1024 | 0 | 32
27 | $D_{8}$ | $C_{8}$ | 128 | 32 | 16 | 16
28 | $D_{8}$ | $C_{2}\times C_{2}\times C_{2}$ | 1120 | 1024 | 0 | 32
29 | $D_{8}$ | $D_{8}$ | 192 | 32 | 0 | 32
30 | $D_{8}$ | $C_{4}\times C_{2}$ | 288 | 32 | 0 | 32
31 | $D_{8}$ | $D_{8}$ | 1184 | 1024 | 0 | 32
32 | $Q_{8}$ | $Q_{8}$ | 3040 | 3008 | 0 | 32
33 | $Q_{8}$ | $Q_{8}$ | 64 | 0 | 0 | 64
34 | $Q_{8}$ | $C_{2}\times C_{2}\times C_{2}$ | 288 | 32 | 0 | 32
35 | $D_{8}$ | $Q_{8}$ | 192 | 32 | 0 | 32
36 | $Q_{8}$ | $D_{8}$ | 1184 | 1024 | 0 | 32
37 | $Q_{8}$ | $C_{4}\times C_{2}$ | 1120 | 1024 | 0 | 32
38 | $Q_{8}$ | $C_{8}$ | 128 | 32 | 16 | 16
39 | $Q_{8}$ | $C_{8}$ | 288 | 256 | 16 | 16
40 | $C_{2}\times C_{2}\times C_{2}$ | $C_{2}\times C_{2}\times C_{2}$ | 16777216 | 0 | 0 | 16777216
41 | $C_{2}\times C_{2}\times C_{2}$ | $C_{2}\times C_{2}\times C_{2}$ | 12480 | 224 | 224 | 32
42 | $C_{2}\times C_{2}\times C_{2}$ | $Q_{8}$ | 960 | 224 | 224 | 32
43 | $C_{2}\times C_{2}\times C_{2}$ | $C_{4}\times C_{2}$ | 4800 | 224 | 224 | 32
44 | $C_{2}\times C_{2}\times C_{2}$ | $C_{4}\times C_{2}$ | 77824 | 12288 | 61440 | 4096
45 | $C_{2}\times C_{2}\times C_{2}$ | $C_{4}\times C_{2}$ | 4352 | 112 | 240 | 16
46 | $C_{2}\times C_{2}\times C_{2}$ | $D_{8}$ | 81664 | 16128 | 65280 | 256
47 | $C_{2}\times C_{2}\times C_{2}$ | $D_{8}$ | 1278 | 63 | 0 | 1
Table 2: $\lambda$-cen. and $\rho$-inv. reflections for solutions associated
with skew braces of order $8$.

Finally, for the sake of completeness, the following table collects all
reflections for solutions associated with some skew braces which are only
$\lambda$-invariant or only $\rho$-centralizing. Additionally, in the last
column, we enumerate the reflections that belong to all four classes.

$\\#$ | Type | $(B,+)$ | $(B,\circ)$ |  $\lambda$-inv. |  $\rho$-cen. | All
---|---|---|---|---|---|---
2 | 1 | $C_{2}$ | $C_{2}$ | 0 | 0 | 4
3 | 1 | $C_{3}$ | $C_{3}$ | 0 | 0 | 27
4 | 1 | $C_{4}$ | $C_{4}$ | 0 | 0 | 256
2 | $C_{4}$ | $C_{2}\times C_{2}$ | 0 | 0 | 8
3 | $C_{2}\times C_{2}$ | $C_{2}\times C_{2}$ | 0 | 0 | 256
4 | $C_{2}\times C_{2}$ | $C_{4}$ | 0 | 0 | 8
5 | 1 | $C_{5}$ | $C_{5}$ | 0 | 0 | 3125
6 | 1 | $S_{3}$ | $S_{3}$ | 0 | 0 | 1
2 | $S_{3}$ | $S_{3}$ | 0 | 0 | 1
3 | $S_{3}$ | $C_{6}$ | 2 | 0 | 1
4 | $S_{3}$ | $C_{6}$ | 0 | 0 | 1
5 | $C_{6}$ | $S_{3}$ | 0 | 0 | 9
6 | $C_{6}$ | $C_{6}$ | 0 | 0 | 46656
7 | 1 | $C_{7}$ | $C_{7}$ | 0 | 0 | 823543
Table 3: $\lambda$-inv. and $\rho$-cen. reflections for solutions associated
with skew braces until order $7$.

## Data availability

The GAP code used for the computation is available from the authors on
request.

## Acknowledgments

The authors are all members of GNSAGA (INdAM). This work was supported by the
Dipartimento di Matematica e Fisica “Ennio De Giorgi” - Università del
Salento.

A. Albano was supported by a scholarship financed by the Ministerial Decree
no. 118/2023, based on the NRRP - funded by the European Union -
NextGenerationEU - Mission 4.

M. Mazzotta was supported by the “HaMMon (Hazard Mapping and vulnerability
Monitoring) - Spoke 2” project.

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  * Andrea ALBANO (ORCID: 0009-0007-5397-5505)
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  * Marzia MAZZOTTA (ORCID: 0000-0001-6179-9862)
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  * Paola STEFANELLI (ORCID: 0000-0003-3899-3151)
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