# Speaker-conditioning Single-channel Target Speaker Extraction using
Conformer-based Architectures

###### Abstract

Target speaker extraction aims at extracting the target speaker from a mixture
of multiple speakers exploiting auxiliary information about the target
speaker. In this paper, we consider a complete time-domain target speaker
extraction system consisting of a speaker embedder network and a speaker
separator network which are jointly trained in an end-to-end learning process.
We propose two different architectures for the speaker separator network which
are based on the convolutional augmented transformer (conformer). The first
architecture uses stacks of conformer and external feed-forward blocks
(Conformer-FFN), while the second architecture uses stacks of temporal
convolutional network (TCN) and conformer blocks (TCN-Conformer). Experimental
results for 2-speaker mixtures, 3-speaker mixtures, and noisy mixtures of
2-speakers show that among the proposed separator networks, the TCN-Conformer
significantly improves the target speaker extraction performance compared to
the Conformer-FFN and a TCN-based baseline system.

Index Terms—  target speaker extraction, multi-task learning, TCN, attention,
conformer.

##  1 Introduction

Recent advances in deep-neural networks (DNNs) have greatly improved the
performance of speaker extraction systems, yet it remains a challenge when
multiple speakers and background noise are present in the mixture. One
possibility is to first extract all individual speakers from the mixture using
blind source separation [1, 2, 3, 4, 5, 6, 7] and then select the target
speaker from the extracted speakers. Alternatively, a speaker-conditioning
target speaker extraction approach can be utilized, which requires auxiliary
information about the target speaker to guide the DNN towards directly
extracting the target speaker from the mixture [8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19]. Commonly used auxiliary information includes reference speech
of the target speaker [8, 9, 10, 11, 12, 13], visual information [14, 15],
directional information [16], brain signal [18], or speech activity
information of the target speaker [19]. In this paper, we focus on single-
channel target speaker extraction using reference speech as auxiliary
information.

Fig. 1: Block diagram of a time-domain speaker-conditioning target speaker
extraction system.

A typical speaker-conditioning target speaker extraction system consists of a
speaker embedder network and a speaker separator network. The speaker embedder
network generates an embedding from the reference speech of the target
speaker, while the speaker separator network aims at estimating the target
speaker from the mixture guided by the speaker embedding. The speaker embedder
and separator networks can be trained either separately [8, 9, 10] or jointly
[11] to perform the speaker extraction. In [8, 9, 10, 11] different
combinations of network architectures have been investigated for the embedder
and separator networks. An LSTM-based embedder network was used in combination
with a CNN-LSTM based separator network [8] or a TCN-based separator network
[10]. In [9, 11] ResNet-based architectures were used for the embedder network
in combination with an attention-based separator network [9] or a TCN-based
separator network [11]. The separator networks in [8, 9] perform the speaker
extraction in the frequency-domain and inherently suffer from phase estimation
issues, whereas the separator networks in [11, 10] avoid phase estimation
issues by performing the speaker extraction in the time-domain. In this paper,
we consider complete time-domain target speaker extraction systems, where the
embedder and separator networks are jointly trained.

Aiming at improving the performance of the TCN-based baseline system [11] for
target speaker extraction, in this paper we propose two different
architectures for the speaker separator network based on the convolutional
augmented transformer (conformer) [20], which has shown its efficiency in
capturing global feature information using attention and local feature
information using convolution. The first proposed architecture is based on
stacking conformer blocks and external feed-forward blocks (Conformer-FFN),
whereas the second proposed architecture is based on stacking TCN blocks and
conformer blocks (TCN-Conformer). Experimental results for 2-speaker mixtures,
3-speaker mixtures, and noisy mixtures of 2-speakers simulated using the WSJ0
dataset [3] and WHAM dataset [21] show that the proposed TCN-Conformer system
significantly improves the speaker extraction performance for all mixture
types compared to the TCN-based baseline system [11] and proposed Conformer-
FFN system in terms of scale-invariant signal-to-distortion ratio (SI-SDR).

##  2 Time-domain Target Speaker Extraction System

A complete time-domain speaker-conditioning target speaker extraction system
consists of a speaker embedder network and a speaker separator network both
performing in the time-domain. The speaker separator network consists of three
processing stages (see Fig. 1): encoder, separator, and decoder. The encoder
transforms segments of the mixture signal to their intermediate feature
representation, while the decoder reconstructs the target speaker signal from
the masked encoded features. The multiplicative masks are estimated from the
encoded features guided by the speaker embedding which is obtained from the
reference speech of the target speaker using the speaker embedder network.

In [11], the first stage of the speaker separator network uses a multi-scale
encoder with three different filter lengths (low, medium, and high) to obtain
different intermediate feature representations. The different filter lengths
help in obtaining different temporal resolutions. The masks for the target
speaker are estimated from the concatenation of the speaker embedding and the
multi-scale encoded features using a TCN-based separator network for each
filter length. The target speaker signal is reconstructed from the masked
encoded features using a multi-scale decoder for each filter length. A multi-
task learning procedure is utilized to train the speaker embedder and speaker
separator networks jointly, where the separator network is optimized for
speaker extraction using a multi-scale scale-invariant signal-to-noise ratio
(SI-SNR), while the embedder network is optimized for speaker identification.
The multi-scale SI-SNR is the weighted sum of the SI-SNRs estimated for all
target signals obtained using multi-scale decoding. Similarly to [11], in this
paper we consider the same ResNet-based embedder network architecture, but
instead of using a TCN-based separator network we propose two different
architectures based on the conformer.

##  3 Conformer-based Architectures

In this section, we discuss the proposed conformer-based architectures for the
speaker separator network. The conformer [20] is capable of incorporating both
local context as well as global context features. In Section 3.1, we discuss
the Conformer-FFN architecture combining conformer blocks with feed-forward
blocks to exploit local and global context features with the help of attention
and convolutional operations. In Section 3.2, we discuss the TCN-Conformer
architecture combining TCN blocks with Conformer blocks to exploit local and
global context features.

###  3.1 Conformer-FFN architecture

Fig. 2 depicts the proposed Conformer-FFN architecture which is based on
stacking conformer blocks and external feed-forward blocks. The motivation
behind this architecture is to utilize both local and global context features
using conformer blocks, while reducing the overall parameters using external
feed-forward blocks. As proposed in [20] each conformer block consists of four
different blocks: two feed-forward blocks, one multi-head self-attention
block, and a convolutional block. The first feed-forward block is utilized
before applying the multi-head attention. The convolutional block is utilized
after the multi-head attention block just before the second feed-forward
block. In the proposed architecture, each conformer block is followed by an
external feed-forward block. The external feed-forward block consists of two
feed-forward layers, a swish activation [22], and dropout layers (see Fig.
4(a)). The output dimension of the external feed-forward block is half of the
input dimension, leading to two advantages. First, we obtain a fixed
dimensional input for all conformer blocks. Second, the overall number of
parameters for the speaker separator network is reduced. The input to the
first conformer block is the concatenation of the encoded features obtained
from the mixture signal using the multi-scale encoder and the speaker
embedding, while the inputs to the rest of the conformer blocks are the
concatenation of the external feed-forward block output and the speaker
embedding. Similarly to the TCN-based baseline [11], the speaker embedding is
repeatedly concatenated along the feature dimension.

Fig. 2: Proposed Conformer-FFN architecture, where K denotes the number of
stacks of conformer and feed-forward blocks.

Fig. 3: Proposed TCN-Conformer architecture, where K denotes the number of
stacks of TCN and conformer blocks.

(a) Feed-Forward Block

(b) TCN Block

Fig. 4: Feed-forward block and TCN block in speaker separator networks.

###  3.2 TCN-Conformer architecture

Fig. 3 depicts the proposed TCN-Conformer architecture, which is based on
stacking TCN blocks and conformer blocks, i.e., each TCN block is followed by
a conformer block. The motivation behind this architecture is to first utilize
the best local context features using TCN blocks and then exploit both local
and global context features using conformer blocks. Each TCN block [5] (see
Fig. 4(b)) consists of two 1-dimensional convolutional (1D-CNN) layers, two
PReLU activations with global layer normalization (gln), and one dilated
depth-wise separable convolutional layer (DDS-CNN). The conformer block
consists of the same architecture as discussed in Section 3.1. The input to
the first TCN block is the concatenation of the encoded features obtained from
the mixture signal using the multi-scale encoder and the speaker embedding
estimated using the speaker embedder network, while the inputs to the rest of
TCN blocks are the concatenation of the conformer block output and speaker
embedding. Similarly to the TCN-based baseline [11], the speaker embedding is
repeatedly concatenated along the feature dimension.

##  4 Experiments

In this section, we discuss the dataset for training and testing, the used
parameters and hyper-parameters, and the training procedure of the considered
target speaker extraction systems.

###  4.1 Dataset

We have simulated different datasets for 2-speaker mixtures (2-mix), 3-speaker
mixtures (3-mix), and noisy mixtures of 2-speakers (noisy-mix) from the WSJ0
corpus [3] and WHAM [21] at a sampling rate of $16$ kHz. The subset
$si\\_tr\\_s$ is utilized to create the training and development set, while
the subsets $si\\_dt\\_05$ and $si\\_et\\_05$ from the WSJ0 corpus are
utilized to create the test set. The test set contains completely different
speakers than the training and developement sets. To create the 2-mix dataset,
two different speakers are randomly chosen and mixed at an SNR between $0$ and
$5$ dB, where the first speaker is regarded as the target speaker and the
second speaker as the interfering speaker. A different utterance of the target
speaker is selected as the reference speech to obtain the speaker embedding.
We follow the same procedure to create the 3-mix dataset, where both
interfering speakers have the same power, and the mixture with the target
speaker is simulated at an SNR between $0$ and $5$ dB. Similarly, we create
the noisy-mix dataset, where the target and interfering speakers are selected
from the WSJ0 corpus and the noise is selected from the training, development
and test set of WHAM. The noisy-mix data are simulated utilizing the official
scripts of WHAM data simulation for 2-speaker mixtures. All together we have
$47926$ utterances for training, $12792$ utterances for development, and
$7478$ utterances for testing for 2-mix, 3-mix, and noisy-mix.

###  4.2 Training settings

We have used the same speaker embedder network as the baseline system [11] to
generate fixed $256$-dimensional embedding from the reference speech of the
target speaker. The embedder network utilizes a ResNet architecture consisting
of $3$ residual blocks. The input and output dimensions of the residual blocks
are fixed to ($256$, $256$), ($256$, $512$), and ($512$, $512$) respectively.
Each residual block consists of two 1D-CNN layers with a kernel size of $1$,
where each CNN layer is followed with a batch-normalization and a PReLU
activation function. A skip connection is used between the input and the
second batch-normalization output, while a 1D max-pooling with a kernel size
of $3$ is used as the output layer of each residual block. As the baseline
system, we have retrained the TCN-based target speaker extraction system
proposed in [11]. Each TCN block used in this work has similar hyper-parameter
settings as the first TCN block utilized in [11]. The input and convolutional
size of the TCN block are fixed to $512$, and the kernel size is fixed to $3$.
Each conformer block utilizes $8$-head attention, while the convolutional
kernel size is fixed to $31$. The output of the first convolution layer is
expanded with a factor of $3$ in each block, while the output of the linear
layer is set to be $4$ times as input size. The external feed-forward blocks
in the proposed Conformer-FFN architecture (see Fig. 2) utilize an input
dimension of $512$ and an output dimension of $256$ as well as a swish
activation function (like the feed-forward block utilized in the conformer
block).

To train all considered systems, we have utilized a multi-task objective
function, namely multi-scale scale-invariant signal-to-noise ratio (SI-SNR)
for the speaker separator network and cross-entropy for the speaker embedder
network. Each system has been trained for $4$-seconds long segments for $150$
epochs with an early stopping criterion of $6$ epochs using the ADAM optimizer
[23]. The three filter lengths for the multi-scale encoder and decoder are set
as $2.5$ ms, $10$ ms, and $20$ ms. The proposed Conformer-FFN and TCN-
Conformer have been trained for $K\in\\{1,3,4\\}$ number of stacks.

##  5 Results and Discussion

We first train the TCN-based baseline system and proposed Conformer-FFN and
TCN-Conformer systems ($K=3$ stacks) using only the 2-mix dataset. We then
train all considered systems using all datasets (2-mix, 3-mix, noisy-mix)
together. For both training conditions, we evaluate the performance for the
2-mix, 3-mix and noisy-mix test set individually. As performance measure, we
use SI-SDR (dB) [24], which is considered to be more robust than SDR [25] for
single-channel speaker extraction.

For all considered target speaker extraction systems trained only with the
2-mix dataset, Table 1 shows the mean SI-SDR values obtained on the 2-mix,
3-mix, and noisy-mix test sets. First, it can be observed that the proposed
TCN-Conformer system improves the speaker extraction performance by $0.70$ dB
for 2-mix, $0.38$ dB for 3-mix, and $2.06$ dB for noisy-mix compared to the
TCN-based baseline system. Second, it can be observed that the proposed
Conformer-FFN system achieves a lower performance than the baseline system for
all test sets. Third, it can be observed that all considered systems trained
only with the 2-mix dataset achieve high performance for the 2-mix test set,
but perform poorly when either an additional interfering speaker (3-mix) or
additional background noise (noisy-mix) is added.

Systems | K | 2-mix | 3-mix | noisy-mix
---|---|---|---|---
Input mixture | - | 2.51 | -1.27 | -3.21
baseline [11] | - | 16.15 | 4.18 | -2.30
Conformer-FFN | 3 | 15.60 | 4.08 | -3.64
TCN-Conformer | 3 | 16.85 | 4.56 | -0.24
Table 1: SI-SDR (dB) for the input mixture, baseline system and proposed
Conformer-FFN and TCN-Conformer systems trained only with the 2-mix dataset.

For all considered target speaker extraction systems trained with all datasets
together, Table 2 shows the mean SI-SDR values obtained on the 2-mix, 3-mix,
and noisy-mix test sets. First, it can be observed that for all mixtures all
systems achieve a significant performance improvement with respect to the
input mixture. Second, it can be observed that the performance for both
proposed systems improves as the number of stacks $K$ increases. Third, it can
be observed that the best performing Conformer-FFN system ($K=4$) does not
outperform the baseline system but the best performing TCN-Conformer system
($K=4$) improves the speaker extraction performance by $2.64$ dB for 2-mix,
$2.27$ dB for 3-mix, and $1.40$ dB for noisy-mix compared to the baseline
system.

Systems | K | 2-mix | 3-mix | noisy-mix
---|---|---|---|---
Input mixture | - | 2.51 | -1.27 | -3.21
baseline [11] | - | 14.87 | 8.43 | 7.92
Conformer-FFN | 1 | 11.99 | 6.34 | 6.30
Conformer-FFN | 3 | 13.03 | 7.09 | 7.08
Conformer-FFN | 4 | 14.07 | 7.67 | 7.56
TCN-Conformer | 1 | 12.34 | 7.12 | 6.85
TCN-Conformer | 3 | 15.47 | 9.21 | 8.87
TCN-Conformer | 4 | 17.51 | 10.70 | 9.32
Table 2: SI-SDR (dB) for the input mixture, baseline system and proposed
Conformer-FFN and TCN-Conformer systems trained with all considered datasets
(2-mix, 3-mix, noisy-mix) together.

##  6 Conclusion

In this paper, we explored the benefits of using conformer-based architectures
for the separator network in time-domain speaker-conditioning target speaker
extraction systems. Aiming at exploiting both local as well as global context
features, we proposed the Conformer-FFN architecture, stacking conformer
blocks and external feed-forward blocks and the TCN-Conformer architecture,
stacking TCN blocks and conformer blocks. Experimental results for different
mixtures show that the proposed TCN-Conformer outperforms the proposed
Conformer-FFN system and a TCN-based baseline system. In future work, we will
investigate the effect of noisy auxiliary information on the performance of
target speaker extraction systems.

##  7 References

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# A characterization of finite simple set-theoretic solutions of the Yang-
Baxter equation111The author is a member of GNSAGA (INdAM).

Marco CASTELLI<EMAIL_ADDRESS>-<EMAIL_ADDRESS>
###### Abstract

In this paper we present a characterization of finite simple involutive non-
degenerate set-theoretic solutions of the Yang-Baxter equation by means of
left braces and we provide some significant examples.

###### keywords:

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

###### MSC:

[2020] 16T25, 81R50

## Introduction

The quantum Yang-Baxter equation first appeared in theoretical physics, in a
paper by C.N. Yang [24], and in statistical mechanics, in R.J. Baxter’s work
[2]. To date, it is subject of many studies of very current interest in pure
mathematics even beyond theoretical physics. In 1992 Drinfel’d [10] suggested
the study of the set-theoretical version of this equation. Specifically, a
_set-theoretic solution of the Yang-Baxter equation_ on a non-empty set $X$ is
a pair $\left(X,r\right)$, where $r:X\times X\to X\times X$ is a map such that
the relation

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

is satisfied. Writing a solution $(X,r)$ as
$r\left(x,y\right)=\left(\lambda_{x}\left(y\right)\rho_{y}\left(x\right)\right)$,
with $\lambda_{x},\rho_{x}$ maps from $X$ into itself, for every $x\in X$, we
say that $(X,r)$ is _non-degenerate_ if
$\lambda_{x},\rho_{x}\in\operatorname{Sym}_{X}$, for every $x\in X$, and
_involutive_ if $r^{2}=\operatorname{id}_{X\times X}$.
The papers of Gateva-Ivanova and Van Den Bergh [12] and Etingov, Schedler, and
Soloviev [11] attracted several authors to the study of the involutive non-
degenerate set-theoretic solutions (which we simply call _solutions_). In
particular, in [11, Section 2] the class of indecomposable solutions was
introduced. The interest in these solutions is motivated by the fact that they
allow to construct other solutions, not necessarily indecomposable, by
suitable construction-tools, such as dynamical extension and retraction-
process (see [3, 11, 22] for more details). In this context, many theory has
been developed and several interesting results have been obtained (see, for
example, [3, 4, 5, 7, 13, 14, 19, 20, 21] and related references).
Among indecomposable solutions, the _simple_ ones play a special role since
they are the ”fundamental blocks” to construct all the others by dynamical
extensions (see [3, Proposition 2] and [22, Corollary 2.13]). On the other
hand, until a year ago, very little was known about simple solutions. In
particular, there were no methods for building simple solutions and only a few
number of simple solutions was known: the simple solutions having size $\leq
8$, obtained by inspection of the database of small solutions, and the
indecomposable ones having prime size $p$ (where $\lambda_{x}=\rho^{-1}_{x}$
and all the $\lambda_{x}$ are all equal to the same $p$-cycle). In the last
year, Cedó and Okniński provided a construction-methods that allow to obtain
other families of simple solutions [8, 9] for the first time. In particular,
they exhibited simple solutions having size $n^{2}$ and $mn^{2}$, for $n,m>1$.
For these constructions, the imprimitivity action of the permutation group
associated to these solutions has a crucial importance (see [7] for more
details). Following a different approach, in [9] they found a striking link
between simple left braces and simple solutions, showing that a large family
of simple left braces provides simple solutions [9, Theorem 5.1]. Actually, a
natural problem, recently posed by Okniński during a talk in [15], is the
construction of different simple solutions.
The aim of this paper is to give a characterization of finite simple solutions
by means of left braces, algebraic structures introduced by Rump in [18]. This
naturally provides the precise class of left braces that give rise to simple
solutions. The paper is organized as follows. In Section $1$ we introduce the
necessary background on left braces and on solutions, using the language of
cycle sets. In Section $2$ we show the main result, which give a
characterization of simple solutions by means of left braces. In Section $3$
we apply our result to provide some examples of simple solutions, some of
which are different from the ones obtained in [8, 9].

##  1 Basic results

In this section, we mainly recall some basics on solutions, cycle sets and
left braces that are useful throughout the paper.

###  1.1 Solutions of the Yang-Baxter equation and cycle sets

In [16], Rump found a one-to-one correspondence between solutions and an
algebraic structure with a single binary operation, which he called _non-
degenerate cycle sets_. To illustrate this correspondence, let us firstly
recall the following definition.

######  Definition 1 (pag. 45, [16]).

A pair $(X,\cdot)$ is said to be a _cycle set_ if each left multiplication
$\sigma_{x}:X\longrightarrow X,$ $y\mapsto x\cdot y$ is bijective and

$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z),$

for all $x,y,z\in X$. Moreover, a cycle set $(X,\cdot)$ is called non-
degenerate if the squaring map $x\mapsto x\cdot x$ is bijective.

######  Convention.

Even if not specified, all the cycle sets are finite and non-degenerate
throughout the paper.

######  Proposition 1 (Propositions 1-2, [16]).

Let $(X,\cdot)$ be a cycle set. Then the pair $(X,r)$, where
$r(x,y):=(\sigma_{x}^{-1}(y),\sigma_{x}^{-1}(y)\cdot x)$, for all $x,y\in X$,
is a solution of the Yang-Baxter equation which we call the associated
solution to $(X,\cdot)$. Moreover, this correspondence is one-to-one.

A first useful tool to construct new solutions, introduced in [11], is the so-
called retract relation, an equivalence relation on $X$ which we denote by
$\sim_{r}$. Precisely, if $(X,r)$ is a solution, then $x\sim_{r}y$ if and only
if $\lambda_{x}=\lambda_{y}$, for all $x,y\in X$. In this way, $(X,r)$ induces
another solution, having the quotient $X/\sim_{r}$ as underlying set, which is
named retraction of $(X,r)$ and is denoted by $\operatorname{Ret}(X,r)$. As
one can expect, the retraction of a solution corresponds to the retraction of
a non-degenerate cycle set. Specifically, in [16] Rump showed that the binary
relation $\sim_{\sigma}$ on $X$ given by

$x\sim_{\sigma}y:\Longleftrightarrow\sigma_{x}=\sigma_{y}$

for all $x,y\in X$, is a _congruence_ of $(X,\cdot)$, i.e. an equivalence
relation for which $x\sim_{\sigma}y$ and $x^{\prime}\sim_{\sigma}y^{\prime}$
implies $x\cdotp x^{\prime}\sim_{\sigma}y\cdotp y^{\prime},$ for all
$x,x^{\prime},y,y^{\prime}\in X$. Moreover, he proved that the quotient
$X/\sim_{\sigma}$, which we denote by $\operatorname{Ret}(X)$, is a cycle set
and he called it the _retraction_ of $(X,\cdot)$. As the name suggests, if
$(X,\cdot)$ is the cycle set associated to a solution $(X,r)$, then the
retraction $\operatorname{Ret}(X)$ is the cycle set associated to
$\operatorname{Ret}(X,r)$. Besides, a cycle set $X$ is said to be
irretractable if $\operatorname{Ret}(X)=X$, otherwise it is called
retractable.

For a cycle set $X$, the permutation group generated by the set
$\\{\sigma_{x}\,|\,x\in X\\}$ will be denoted by $\mathcal{G}(X)$ and we call
it the associated permutation group. Obviously, in terms of solutions, the
associated permutation group is exactly the permutation group generated by the
set $\\{\lambda_{x}\,|\,x\in X\\}$.

Our attention is mainly posed on cycle sets that are indecomposable.

######  Definition 2.

A cycle set $(X,\cdot)$ is said to be indecomposable if the permutation group
$\mathcal{G}(X)$ acts transitively on $X$.

######  Remark 1.

Note that a solution $(X,r)$ is indecomposable if and only if the associated
cycle set $(X,\cdot)$ is indecomposable. Now, in the rest of the paper, we
will study indecomposable solutions using the language of cycle sets. However,
all the results involving cycle sets can be translated in terms of solutions
by 1.

In a classical way one can define the notion of homomorphism between two cycle
sets $X,Y$, i.e. a function $f$ from $X$ to $Y$ such that $f(x\cdotp
y)=f(x)\cdotp f(y)$ for all $x,y\in X$. A surjective homomorphism is said to
be _epimorphism_ , while a bijective homomorphism is said to be _isomorphism_.
It is easy to show that if $h$ is an epimorphism from a cycle set $Y$ to a
cycle set $Z$, then the binary relation given by
$x\sim_{h}y:\Longleftrightarrow h(x)=h(y)$ is a congruence of $(Y,\cdotp)$ and
conversely every congruence of $Y$ give rise to an epimorphism of cycle sets.
Now, we can give the key-notion of this paper.

######  Definition 3.

A cycle set $X$ is said to be _simple_ if $|X|>1$ and for every epimorphism
$f$ from $X$ to a cycle set $Y$ we have that $|Y|=1$ or $f$ is an isomorphism.

We remark that even if this definition of simplicity, given in [8], is
different from the original one given by Vendramin in [22], by [3, Lemma 1]
both definitions coincide for finite indecomposable cycle sets. However, we
have the following result.

######  Proposition 2.

([8, Lemma 4.1 - Proposition 4.2]) Let $X$ be a simple cycle set. Then, if
$|X|>2$, $X$ is indecomposable. Moreover, if $|X|$ is not a prime number, $X$
is irretractable.

###  1.2 Solutions of the Yang-Baxter equation and left braces

At first, we introduce the following definition that, as observed in [6], is
equivalent to the original introduced by Rump in [18].

######  Definition 4 ([6], Definition 1).

A set $A$ endowed of two operations $+$ and $\circ$ is said to be a left brace
if $(A,+)$ is an abelian group, $(A,\circ)$ a group, and

$a\circ(b+c)+a=a\circ b+a\circ c,$

for all $a,b,c\in A$.

######  Example 3.

If $X$ is a cycle set, then one can show that the permutation group
$\mathcal{G}(X)$ give rise to a left brace $(\mathcal{G}(X),+,\circ)$, where
$\circ$ is the usual composition in $\mathcal{G}(X)$ (see, for example, [1,
Section 2] for more details). From now on, we will refer to
$(\mathcal{G}(X),+,\circ)$ as the _associated left brace_.

Given a left brace $A$ and $a\in A$, let us denote by
$\lambda_{a}:A\longrightarrow A$ the map from $A$ into itself defined by

$\lambda_{a}(b):=-a+a\circ b,$

for all $b\in A$. As shown in [18, Proposition 2] and [6, Lemma 1], these maps
have special properties. We recall them in the following proposition.

######  Proposition 4.

Let $A$ be a left brace. Then, the following are satisfied:

  * 1)

$\lambda_{a}\in\operatorname{Aut}(A,+)$, for every $a\in A$;

  * 2)

the map $\lambda:A\longrightarrow\operatorname{Aut}(A,+)$,
$a\mapsto\lambda_{a}$ is a group homomorphism from $(A,\circ)$ into
$\operatorname{Aut}(A,+)$.

The map $\lambda$ is of crucial importance to construct cycle sets (and hence
solutions of the Yang-Baxter equation) using left braces, as one can see in
the following proposition.

######  Proposition 5 (Lemma 2, [6]).

Let A be a left brace and $\cdotp$ the binary operation on $A$ map given by

$a\cdotp b:=\lambda_{a}^{-1}(b),$

for all $a,b\in A$. Then, $(A,\cdotp)$ is a cycle set.

Therefore, the previous proposition and 3 show that if $X$ is a cycle set,
then we can construct a cycle set on the permutation group $\mathcal{G}(X)$.
For the following definition, we refer the reader to [18, pg. 160] and [6,
Definition 3].

######  Definition 5.

Let $A$ be a left brace. A subset $I$ of $A$ is said to be a left ideal if it
is a subgroup of the multiplicative group and $\lambda_{a}(I)\subseteq I$, for
every $a\in A$. Moreover, a left ideal is an ideal if it is a normal subgroup
of the multiplicative group.

As one can expect, if $I$ is an ideal of a left brace $A$, then the structure
$A/I$ is a left brace called the _quotient left brace_ of $A$ modulo $I$.
Moreover, the ideal $\\{0\\}$ will be called the _trivial_ ideal and and a
left brace $A$ which contains no ideals different from $\\{0\\}$ and $A$ will
be called a _simple_ left brace.
In [18], Rump introduced the special notion of the socle of a left brace that,
in the terms of [6, Section 4], is the following.

######  Definition 6.

Let $A$ be a left brace. Then, the set

$Soc(A):=\\{a\in A\ |\ \forall\,b\in A\quad a+b=a\circ b\\}$

is named _socle_ of $A$.

Clearly, $Soc(A):=\\{a\in A\ |\ \lambda_{a}=\operatorname{id}_{A}\\}$.
Moreover, we have that $Soc(A)$ is an ideal of $A$. If a cycle set $X$ is
irretractable, then the socle of its associated left brace is trivial.

######  Proposition 6.

(Lemma 2.1, [1]) Let $X$ be an irretractable cycle set. Then,
$Soc(\mathcal{G}(X))=\\{id_{X}\\}$.

The socle is useful to establish a link between a left brace $A$ and the
permutation group associated to the cycle set $(A,\cdotp)$.

######  Proposition 7.

(Lemma 2.2, [8]) Let $A$ be a left brace and $\mathcal{G}(A)$ the permutation
group associated to the cycle set $(A,\cdotp)$. Then, as left braces,
$A/Soc(A)\cong\mathcal{G}(A)$.

If $A$ is a left brace, by 4, the maps $\lambda_{a}$ determine an action of
$(A,\circ)$ on $(A,+)$. According to [18, 19], a subset $X$ of $A$ which is a
union of orbits with respect to such an action and generating the additive
group $(A,+)$ is called cycle base. Moreover, if a cycle base is a single
orbit then is said to be a transitive cycle base. The following result, that
is useful for our scopes, is implicitly contained in [8].

######  Proposition 8 (Section 4, [8]).

Let $X$ be an indecomposable and irretractable cycle set and $\mathcal{G}(X)$
be the left brace on the associated permutation group. Then, $X$ can be
regarded as a transitive cycle base of $\mathcal{G}(X)$.

We close this preliminary section showing the Galois-correspondence between
ideals of a left brace $A$ and congruences of a transitive cycle base $X$
founded by Rump in [17, 18]. This is essentially a mixture between [18,
Theorem 1] and [17, Theorem 1], adapted for our aim (we remark that, in this
context, all the cycle sets are finite). At first, recall that if $\sim$ and
$\sim^{\prime}$ are equivalence relations on a set $S$, then $\sim$ is said to
be a _refinement_ of $\sim^{\prime}$ if $x\sim y$ implies $x\sim^{\prime}y$,
for all $x,y\in S$.

######  Theorem 9 (Theorem 1, [18] \- Theorem 1, [17]).

Let $A$ be a finite left brace such that $Soc(A)=\\{0\\}$ and $X$ be a
transtive cycle base. Then, if $I$ is an ideal of $A$, the orbits of $X$ under
the action of $I$ give rise to a congruence of $X$. Conversely, if $\sim$ is a
congruence of $X$, the additive subgroup $I^{\prime}$ of $A$ generated by
$\\{x-y\ |\ x,y\in X,\ x\sim y\\}$ is an ideal and the orbits of $X$ under the
action of $I^{\prime}$ give rise to a congruence of $X$ which is a refinement
of $\sim$.

##  2 Simple solutions and left braces

This section is devoted to provide a characterization of finite simple
solutions. We start with a couple of lemma.

######  Lemma 10.

Let $B$ be a finite left brace, with $|B|>1$, such that $Soc(B)=\\{0\\}$ and
having a transitive cycle base $X$. Then, the left brace $\mathcal{G}(X)$ is
isomorphic to $B$.

###### Proof.

Clearly, the permutation $\sigma_{x}$ in $\mathcal{G}(X)$ is equal to the
element $\lambda_{x}^{-1}$ in $\mathcal{G}(B)$ restricted to the set $X$.
Since the natural embedding from $X$ to $B$ induces the left braces
homomorphism $f$ from $\mathcal{G}(X)$ to $\mathcal{G}(B)$ given by
$f(\sigma_{x}^{-1}):=\lambda_{x}$ for all $x\in X$ (see [9, Section 2] for
more details), by 7 it is sufficient to show that $f$ is bijective. As done in
the first part of the proof of [9, Theorem 5.1] (where $f$ is denoted by
$\bar{i}$), one can show that $f$ is surjective. Now, if $g\in\mathcal{G}(X)$
is such that $g\in Ker(f)$, clearly $g(x)=x$ for all $x\in X$, hence $f$ is
injective and the thesis follows. ∎

######  Lemma 11.

(Proposition 4.3, [8]) Let $X$ be a simple cycle set such that $|X|$ has not
prime size. Then, the left brace $\mathcal{G}(X)$ has a unique minimal ideal
$I$ equal to the additive subgroup generated by $\\{\sigma_{x}-\sigma_{y}\ |\
x,y\in X\\}$. Moreover, $\mathcal{G}(X)/I$ is a trivial cyclic left brace.

Now, we are able to show the main result of the section.

######  Theorem 12.

Let $B$ be a finite left brace, with $|B|>1$, such that $Soc(B)=\\{0\\}$ and
having a transitive cycle base $X$. Then, the following conditions are
equivalent:

  * 1)

$X$ is a simple cycle set;

  * 2)

every non-trivial ideal of $B$ acts transitively on $X$;

  * 3)

$B$ has a unique minimal ideal $I$ that acts transitively on $X$.

###### Proof.

If $X$ is simple and $B$ has a non-trivial ideal $I$ that does not acts
transitively on $X$, then by 9 induces a congruence $\sim_{I}$ on $X$.
Therefore, $x\sim_{I}y$ if and only if $x=y$ and, since $X$ is a transitive
cycle base, it follows that $I\subseteq Soc(B)$, a contradiction. Hence we
showed that $1)$ implies $2)$.
Conversely, suppose that $X$ is not simple. By hypothesis, one can easily show
that $|X|>1$. Therefore, there exist a non-trivial congruence $\sim_{P}$
(here, non-trivial means that $\sim_{P}$ does not induces a trivial partition
of $X$). By 9, the additive subgroup $I$ of $B$ generated by $\\{x-y\in X\
\mid\ x\sim_{P}y\\}$ is an ideal such that $\sim_{I}$ is a refinement of
$\sim_{P}$. Moreover, since $\sim_{P}$ is not a trivial congruence, $I$ can
not be the trivial ideal. Since $I$ acts transitively on $X$, $x\sim_{I}y$ for
all $x,y\in X$ and hence $x\sim_{P}y$ for all $x,y\in X$, a contradiction,
therefore $2)$ implies $1)$.
It remains to show that $1)$ is equivalent to $3)$. Suppose $X$ simple. By
Lemma 10 we have that $B\cong\mathcal{G}(X)$, hence $|X|$ can not be a prime
number and by Lemma 11 $B$ has a unique minimal ideal $I$. Since we showed
that $1)$ is equivalent to $2)$, it follows that $I$ acts transitively on $X$.
Finally, if $B$ has a unique minimal ideal $I$ that acts transitively on $X$,
it follows that every non-trivial ideal $J$ of $B$ acts transitively on $X$,
hence $3)\Rightarrow 2)$, which implies $3)\Rightarrow 1)$. ∎

By the previous result, we have that [9, Theorem 5.1] follows as a corollary.

######  Corollary 13.

[9, Theorem 5.1] Let $B$ be a simple non-trivial left brace having a
transitive cycle base $X$. Then, $X$ is a simple cycle set.

###### Proof.

Since $B$ is simple, the unique ideal different from $\\{0\\}$ is $B$, that
acts transitively on $X$. Then, the thesis follow by 12. ∎

Before giving the announced characterization, which closes the section, we
recall that in [11, Theorem 2.13] Etingof, Schedler and Soloviev proved that
there is, up to isomorphisms, a unique indecomposable cycle set of size $p$
for every prime number $p$.

######  Corollary 14.

Let $X$ be a cycle set. If $X$ has size $2$, then $X$ is simple. If $|X|>2$,
the following conditions are equivalent:

  * 1)

$X$ is a simple cycle set;

  * 2)

$X$ is the unique indecomposable cycle set of size $p$ (for some prime number
$p$) or $X$ is an irretractable and indecomposable cycle set such that every
non-trivial ideal of $\mathcal{G}(X)$ acts transitively on $X$;

  * 3)

$X$ is the unique indecomposable cycle set of size $p$ (for some prime number
$p$) or $X$ is an irretractable and indecomposable cycle set such that
$\mathcal{G}(X)$ has a unique minimal ideal $I$ that acts transitively on $X$.

###### Proof.

The first part trivially follows. Suppose that $X$ is a simple cycle set with
$|X|>2$. Then, by 2 it is indecomposable. If $X$ has not prime size, again by
2 it also is irretractable. Therefore, by 6, 8 and 12 (applied with
$B:=\mathcal{G}(X)$) we obtain $1)\Rightarrow 2)$ and $1)\Rightarrow 3)$.
If $X$ is an indecomposable cycle set of size $p$, for some prime number $p$,
then it is simple by [3, Lemma 1]. While, if $X$ is an irretractable and
indecomposable cycle set such that every non-trivial ideal of $\mathcal{G}(X)$
acts transitively on $X$, again by 6, 8 and 12 (applied with
$B:=\mathcal{G}(X)$) we obtain $2)\Rightarrow 1)$. In a similar way, one can
show $3)\Rightarrow 1)$. ∎

##  3 Examples

In this section, we apply 12 to some left braces (found using a GAP package
[23]) to provide concrete examples of simple cycle sets, some of which are
different from those given in [8, 9].

In order to start with a small example, we show providing a well-known cycle
set of size $4$.

######  Example 15.

Let $B$ be the left brace $B_{8,27}$ of [23]. Then, it is a left brace of size
$8$ having only one non-trivial ideal $I$, which has size $4$, different from
$B$. Moreover, it has a transitive cycle base $X$ having size $4$. In
particular, $X$ is isomorphic to the cycle set given by $C:=\\{1,2,3,4\\}$ and

$\sigma_{1}:=(2,4)$ $\sigma_{2}:=(1,3)$ $\sigma_{3}:=(1,2,3,4)$
$\sigma_{4}:=(1,4,3,2).$

Since $I$ acts transitively on $X$, the cycle set $C$ is simple.

In [9] a families of simple cycle sets having size $4p_{1}^{2}$ or
$p_{1}^{2}\cdotp\dots\cdotp p_{n}^{2}$ , where $p_{1},\dots,p_{n}$ are prime
numbers, and with a simple associated left brace were exhibited. In the
following, we give a simple cycle set having a simple associated left brace
which is smaller than the ones provided in [9].

######  Example 16.

Let $B$ be the left brace $B_{24,94}$ of [23]. Then, it is a simple left brace
of size $24$. It has a transitive cycle base $X$ having size $12$. In
particular, $X$ is isomorphic to the cycle set given by $C:=\\{1,...,12\\}$
and

$\sigma_{1}:=(1,8)(2,10)(3,11)(4,9)(5,6)(7,12)$
$\sigma_{2}:=(1,9)(2,5,6,10)(3,11,12,7)(4,8)$
$\sigma_{3}:=(1,9)(2,10,6,5)(3,7,12,11)(4,8)$
$\sigma_{4}:=(1,2,12)(3,4,6)(5,9,11)(7,10,8)$
$\sigma_{5}:=(1,10,4,5)(2,11)(3,8,12,9)(6,7)$
$\sigma_{6}:=(1,6,12)(2,3,4)(5,9,7)(8,11,10)$
$\sigma_{7}:=(1,10)(2,7)(3,9)(4,5)(6,11)(8,12)$
$\sigma_{8}:=(1,5,4,10)(2,11)(3,9,12,8)(6,7)$
$\sigma_{9}:=(1,3,6)(2,4,12)(5,11,8)(7,9,10)$
$\sigma_{10}:=(1,12,6)(2,4,3)(5,7,9)(8,10,11)$
$\sigma_{11}:=(1,12,2)(3,6,4)(5,11,9)(7,8,10)$
$\sigma_{12}:=(1,6,3)(2,12,4)(5,8,11)(7,10,9)$

By 13, $X$ is simple. Moreover, by Lemma 10, $\mathcal{G}(X)$ is a simple left
brace.

######  Example 17.

Let $B$ be the left brace $B_{48,1532}$ of [23]. Then, it is a left brace of
size $48$ having only one non-trivial ideal $I$, which has size $24$,
different from $B$. Moreover, it has a transitive cycle base $X$ having size
$12$. In particular, $X$ is isomorphic to the cycle set given by
$C:=\\{1,...,12\\}$ and

$\sigma_{1}:=(1,8,11,6)(2,5)(3,4,9,10)(7,12)$
$\sigma_{2}:=(1,4)(2,9,12,3)(5,8,7,6)(10,11)$
$\sigma_{3}:=(1,6,11,8)(2,5)(3,10,9,4)(7,12)$
$\sigma_{4}:=(1,10)(2,3)(4,11)(5,6)(7,8)(9,12)$
$\sigma_{5}:=(1,6)(2,7)(3,4)(5,12)(8,11)(9,10)$
$\sigma_{6}:=(1,4)(2,3,12,9)(5,6,7,8)(10,11)$
$\sigma_{7}:=(1,3,5,11,9,7)(2,10,8,12,4,6)$
$\sigma_{8}:=(1,5,3,11,7,9)(2,6,10,12,8,4)$
$\sigma_{9}:=(1,3,7,11,9,5)(2,4,6,12,10,8)$
$\sigma_{10}:=(1,7,9,11,5,3)(2,6,4,12,8,10)$
$\sigma_{11}:=(1,9,7,11,3,5)(2,4,8,12,10,6)$
$\sigma_{12}:=(1,5,9,11,7,3)(2,8,10,12,6,4)$

Since $I$ acts transitively on $X$, the cycle set $C$ is simple.

######  Example 18.

Let $B$ be the left brace $B_{32,24526}$ of [23]. Then, it is a left brace of
size $32$ having only one non-trivial ideal $I$, which has size $16$,
different from $B$. Moreover, it has a transitive cycle base $X$ having size
$16$. In particular, $X$ is isomorphic to the cycle set given by
$C:=\\{1,...,16\\}$ and

$\sigma_{1}:=(1,13)(2,4)(3,15)(6,12)(8,10)(14,16)$
$\sigma_{2}:=(1,3)(2,14)(4,16)(5,11)(7,9)(13,15)$
$\sigma_{3}:=(1,11,13,7)(2,6,14,10)(3,9,15,5)(4,8,16,12)$
$\sigma_{4}:=(1,5,13,9)(2,12,14,8)(3,7,15,11)(4,10,16,6)$
$\sigma_{5}:=(1,7,13,11)(2,10,14,6)(3,5,15,9)(4,12,16,8)$
$\sigma_{6}:=(1,9,13,5)(2,8,14,12)(3,11,15,7)(4,6,16,10)$
$\sigma_{7}:=(2,16)(4,14)(5,9)(6,8)(7,11)(10,12)$
$\sigma_{8}:=(1,15)(3,13)(5,7)(6,10)(8,12)(9,11)$
$\sigma_{9}:=(1,8,3,6)(2,9,4,11)(5,16,7,14)(10,13,12,15)$
$\sigma_{10}:=(1,10,3,12)(2,7,4,5)(6,15,8,13)(9,14,11,16)$
$\sigma_{11}:=(1,2,15,16)(3,4,13,14)(5,10,11,8)(6,7,12,9)$
$\sigma_{12}:=(1,16,15,2)(3,14,13,4)(5,8,11,10)(6,9,12,7)$
$\sigma_{13}:=(1,14,15,4)(2,3,16,13)(5,6,11,12)(7,8,9,10)$
$\sigma_{14}:=(1,4,15,14)(2,13,16,3)(5,12,11,6)(7,10,9,8)$
$\sigma_{15}:=(1,12,3,10)(2,5,4,7)(6,13,8,15)(9,16,11,14)$
$\sigma_{16}:=(1,6,3,8)(2,11,4,9)(5,14,7,16)(10,15,12,13)$

Since $I$ acts transitively on $X$, the cycle set $C$ is simple.

######  Example 19.

Let $B$ be the left brace $B_{81,705}$ of [23]. Then, it is a left brace of
size $81$ having only one non-trivial ideal $I$, which has size $27$,
different from $B$. Moreover, it has a transitive cycle base $X$ having size
$27$. In particular, $X$ is isomorphic to the cycle set given by
$C:=\\{1,...,27\\}$ and

$\sigma_{1}:=(1,17,18)(2,20,13)(3,23,5)(4,27,16)(6,14,25)(7,22,15)(8,19,24)(9,12,10)(11,26,21)$
$\sigma_{2}:=(1,7,10)(2,23,8)(3,15,11)(4,24,12)(5,6,9)(13,18,14)(16,22,20)(17,19,21)(25,26,27)$
$\sigma_{3}:=(1,8,26)(2,27,7)(3,24,13)(4,14,11)(5,22,21)(6,20,17)(9,16,19)(10,23,25)(12,18,15)$
$\sigma_{4}:=(1,26,11,23,10,12,27,2,13)(3,17,15,4,5,24,18,16,14)(6,7,20,22,8,21,19,25,9)$
$\sigma_{5}:=(1,18,8,23,3,25,27,4,7)(2,5,20,26,16,21,10,17,9)(6,13,24,22,11,14,19,12,15)$
$\sigma_{6}:=(1,9,24)(2,3,19)(4,6,26)(7,8,25)(10,18,22)(11,13,12)(14,23,20)(15,27,21)$
$\sigma_{7}:=(1,11,6)(2,15,17)(3,20,25)(4,21,7)(5,26,24)(8,18,9)(10,14,16)(12,22,23)(13,19,27)$
$\sigma_{8}:=(1,3,7,23,4,8,27,18,25)(2,16,9,26,17,20,10,5,21)(6,11,15,22,12,24,19,13,14)$
$\sigma_{9}:=(1,15,9)(2,10,26)(5,13,25)(6,22,19)(7,16,11)(8,17,12)(14,21,27)(20,23,24)$
$\sigma_{10}:=(1,20,15)(2,4,22)(3,6,10)(5,16,17)(7,25,8)(9,14,27)(18,19,26)(21,24,23)$
$\sigma_{11}:=(1,12,19)(2,24,16)(3,21,8)(4,9,25)(5,10,15)(6,23,13)(7,18,20)(11,22,27)(14,17,26)$
$\sigma_{12}:=(1,4,25,23,18,7,27,3,8)(2,17,21,26,5,9,10,16,20)(6,12,14,22,13,15,19,11,24)$
$\sigma_{13}:=(1,23,27)(2,22,18)(3,26,19)(4,10,6)(5,25,12)(7,13,16)(8,11,17)(14,24,15)$
$\sigma_{14}:=(1,19,5,23,6,16,27,22,17)(2,8,15,26,25,24,10,7,14)(3,13,21,4,11,9,18,12,20)$
$\sigma_{15}:=(1,14,20)(2,26,10)(3,18,4)(5,12,7)(8,16,13)(9,27,24)(11,25,17)(15,21,23)$
$\sigma_{16}:=(1,27,23)(2,19,4)(3,10,22)(5,7,11)(6,18,26)(8,12,16)(9,20,21)(13,17,25)$
$\sigma_{17}:=(1,13,22)(2,14,5)(3,9,7)(4,20,8)(6,27,12)(10,24,17)(11,19,23)(15,16,26)(18,21,25)$
$\sigma_{18}:=(1,21,14)(2,18,6)(3,22,26)(4,19,10)(5,17,16)(9,15,23)(11,12,13)(20,24,27)$
$\sigma_{19}:=(1,10,13,23,2,11,27,26,12)(3,5,14,4,16,15,18,17,24)(6,8,9,22,25,20,19,7,21)$
$\sigma_{20}:=(1,16,3)(2,9,11)(4,23,17)(5,18,27)(6,24,7)(8,22,14)(10,21,13)(12,26,20)(15,25,19)$
$\sigma_{21}:=(1,6,17,23,22,5,27,19,16)(2,25,14,26,7,15,10,8,24)(3,11,20,4,12,21,18,13,9)$
$\sigma_{22}:=(1,25,2)(3,14,12)(4,15,13)(5,19,20)(6,21,16)(7,26,23)(8,10,27)(9,17,22)(11,18,24)$
$\sigma_{23}:=(1,22,16,23,19,17,27,6,5)(2,7,24,26,8,14,10,25,15)(3,12,9,4,13,20,18,11,21)$
$\sigma_{24}:=(1,5,4)(2,21,12)(3,27,17)(6,15,8)(7,19,14)(9,13,26)(10,20,11)(16,18,23)(22,24,25)$
$\sigma_{25}:=(2,6,3)(4,26,22)(5,8,13)(7,12,17)(9,21,20)(10,19,18)(11,16,25)(14,15,24)$
$\sigma_{26}:=(1,2,12,23,26,13,27,10,11)(3,16,24,4,17,14,18,5,15)(6,25,21,22,7,9,19,8,20)$
$\sigma_{27}:=(1,24,21)(3,4,18)(5,11,8)(6,19,22)(7,17,13)(9,23,14)(12,25,16)(15,20,27)$

Since $I$ acts transitively on $X$, the cycle set $C$ is simple.

######  Remark 2.

We conclude the paper remarking that in [8, 9] three types of simple cycle
sets were provided: cycle sets that are transitive cycle bases of simple left
braces, cycle sets of size $n^{2}$ and cycle sets having size $n^{2}m$ and an
element $x$ such that $\sigma_{x}$ is a $n^{2}m$-cycle. Therefore, 17 and 19
are simple cycle sets that are different from the ones obtained in [8, 9].

## Acknowledgment

The author thank Leandro Vendramin for the informations about his GAP Package
and Paola Stefanelli for the discussion about left braces.

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