11institutetext: Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
11email<EMAIL_ADDRESS>22institutetext:
Technische Universität Dresden, 01069 Dresden, Germany

# Acceptance Rates of Invertible Neural Networks on Electron Spectra from
Near-Critical Laser-Plasmas: A Comparison

Thomas Miethlinger 1122 Nico Hoffmann 11 Thomas Kluge 11

###### Abstract

While the interaction of ultra-intense ultra-short laser pulses with near- and
overcritical plasmas cannot be directly observed, experimentally accessible
quantities (observables) often only indirectly give information about the
underlying plasma dynamics. Furthermore, the information provided by
observables is incomplete, making the inverse problem highly ambiguous.
Therefore, in order to infer plasma dynamics as well as experimental
parameter, the full distribution over parameters given an observation needs to
considered, requiring that models are flexible and account for the information
lost in the forward process. Invertible Neural Networks (INNs) have been
designed to efficiently model both the forward and inverse process, providing
the full conditional posterior given a specific measurement. In this work, we
benchmark INNs and standard statistical methods on synthetic electron spectra.
First, we provide experimental results with respect to the acceptance rate,
where our results show increases in acceptance rates up to a factor of 10.
Additionally, we show that this increased acceptance rate also results in an
increased speed-up for INNs to the same extent. Lastly, we propose a composite
algorithm that utilizes INNs and promises low runtimes while preserving high
accuracy.

###### Keywords:

Invertible Neural Networks Inverse Problems
Machine Learning Particle-in-Cell Laser-Plasma Physics

##  1 Introduction

Relativistic plasmas driven by ultra-intense ultra-short laser pulses are
currently increasingly investigated due to various prospective applications in
e.g. medicine, materials science and laboratory astrophysics. While the
dynamics of underdense plasmas, i.e. plasmas with electron density
$n_{\mathrm{e}}$ smaller than the critical plasma density $n_{\mathrm{c}}$,
can in principle be studied with optical methods as incoming light there is
mostly transmitted, the situation is much more difficult for near-critical
(mostly absorption, $n_{\mathrm{e}}\approx n_{\mathrm{c}}$) and overdense
(mostly reflection, $n_{\mathrm{e}}>n_{\mathrm{c}}$) plasmas. Indeed,
inferring experimental parameter values and consequently determining the
relevant underlying plasma dynamics is highly elaborate, heavily depending on
comparisons with observables computed from plasma simulations where typically
the Particle-in-Cell (PIC) method is employed [3, 10]. Furthermore, the
information provided by observables is incomplete in the sense that multiple
experimental parameter and plasma dynamics can cause the same values for
observables, but also retrieving information is regarded non-trivial since
this process usually depends on fitting (scalar) quantities to analytical
expressions which have been derived under strong assumptions.

This is, depending on the context, also the case for the electron spectrum,
which counts the number of electrons $\differential{N}_{\mathrm{e}}$ in an
energy interval $\differential{E}$:

$f_{\mathrm{e}}(E):=\frac{\differential{N}_{\mathrm{e}}}{\differential{E}}.$
(1)

In this work, we study laser-driven ion acceleration [18, 14]. In this
research field, where objectives concern ion-related properties, the electron
spectrum is a secondary quantity that is sometimes measured in conjunction
with the ion spectrum. Being a high-dimensional vector, however, the electron
spectrum is difficult to interpret, infer parameters and draw conclusions
from. Typically, one resorts to computing the mean (kinetic) energy of the
laser-driven electrons111Sometimes (unfortunately) also called electron
temperature.:

$T_{\mathrm{e}}:=\frac{\int_{E_{\mathrm{e}}^{\mathrm{laser}}}^{\infty}{Ef_{\mathrm{e}}(E)\differential{E}}}{\int_{E_{\mathrm{e}}^{\mathrm{laser}}}^{\infty}{f_{\mathrm{e}}(E)\differential{E}}},$
(2)

where the lower integration boundary $E_{\mathrm{e}}^{\mathrm{laser}}$ is
introduced to distinguish between electrons in thermal equilibrium and laser-
driven electrons exhibiting an exponential distribution for high energies. One
can then show, using analytical considerations, that the ion cutoff
energy222Or, equivalently, the maximum ion energy in a laser-driven ion
spectrum. scales linearly with the mean kinetic energy of the electrons,
$E_{\mathrm{i}}^{\mathrm{max}}\propto T_{\mathrm{e}}$, and that the mean
kinetic energy of the electrons itself mostly depends on the laser intensity
$I$: $T_{\mathrm{e}}=T_{\mathrm{e}}(I)$ [15, 13].

However, retrieving information from the electron spectrum beyond
$T_{\mathrm{e}}$ in general requires an automatized, data-driven approach.
This is all the more the case because PIC simulations are computationally
(potentially very) expensive, which motivates researchers to this day to
improve PIC simulation codes, for example, algorithmically or by improved
hardware utilization [4, 5]. Therefore, employing machine learning (ML)
algorithms and ML-based surrogate models is essential to decrease the overall
computational effort which would otherwise be needed due to the necessity of
performing an excessive amount of simulations. For example, Djordjević et al.
used deep learning to predict the time evolution of ion cutoff energies and
electron mean kinetic energies in overdense laser-ion acceleration [8].

##  2 PIC Simulation Setup and Data Generation

In this work, we employ the PIC-code Smilei [5] to generate data for our ML
models. Since the predominant acceleration mechanism in laser-ion acceleration
is target-normal sheath-acceleration (TNSA), which is a one-dimensional
physical effect, and in order to significantly reduce the computational effort
for this study, we use a narrow simulation box with
$240$\mathrm{\SIUnitSymbolMicro m}$\times 0.2$\mathrm{\SIUnitSymbolMicro m}$$
and impose periodic boundary conditions in the $y$-direction [18]. The cell
size is $\Delta x=\Delta
y=\frac{\lambda_{\mathrm{0}}}{\mathrm{resolution}}=\frac{800$\mathrm{n}\mathrm{m}$}{64}=12.5$\mathrm{n}\mathrm{m}$$,
and the time step is $\Delta t=0.995\frac{1}{\sqrt{2}}\frac{\Delta x}{c}$
corresponding to a Courant–Friedrichs–Lewy (CFL) value of $0.995$. We
initialize our plasma with 50 particles per cell. The target is a pre-expanded
hydrogen foil with thickness $D$. Pre-plasma with exponential scale length
$\ell$ is included at the front side of the target such that the density
reaches a maximum $n_{0}$ at $x_{\mathrm{f}}=100$\mathrm{\SIUnitSymbolMicro
m}$$. Moreover, the pre-plasma is cut-off where the density is less than
$0.01n_{\mathrm{c}}$, i.e. $\forall x:n(x)<0.01n_{\mathrm{c}}\implies n(x)=0$.
The back side is not pre-expanded, i.e. the density is step-function-like
shaped. The laser pulse is a Gaussian with full width at half maximum (FWHM)
$\tau$ and normalized vector potential
$a_{0}=E/E_{0}=E/(e^{-1}m_{\mathrm{e}}c\omega_{0})$, where $E$ is the
corresponding electric field, $e$ is the elementary charge, $m_{\mathrm{e}}$
the electron mass, $c$ the speed of light and
$\omega_{0}=2\pi\frac{c}{\lambda_{0}}$ the angular frequency corresponding to
the laser’s central wavelength $\lambda_{0}=800$\mathrm{n}\mathrm{m}$$.

In this work, we performed 5000 simulations in total, varying the five
parameters $a_{0},\tau,n_{0},D$ and $\ell$. An overview of the parameter space
that we studied in this work is given in Table 1. Thus, the laser intensity is
in the range between $10^{20}\leq I(a_{0})/\mathrm{W}$\mathrm{cm}$^{-2}\leq
10^{21}$.

Table 1: Parameter space for PIC simulations. Quantity | Symbol | Unit | Min | Max | Scaling
---|---|---|---|---|---
Normalized vector potential | $a_{0}$ | 1 | 6.8 | 21.5 | linear
Full width at half maximum | $\tau$ | $\mathrm{fs}$ | 25 | 50 | linear
Number density (bulk) | $n_{0}$ | $n_{\mathrm{c}}$ | 15 | 60 | linear
Target thickness | $D$ | $\mathrm{\SIUnitSymbolMicro m}$ | 0.25 | 5 | linear
Pre-plasma scale length | $\ell$ | $\mathrm{\SIUnitSymbolMicro m}$ | 0.01 | 1 | square

Since ML usually strongly benefits from using normalized and/or standardized
values, we designed our experiments as follows:

  * •

In an effort to have our simulations as space-filling as possible in parameter
space, we obtained our parameter vectors
$\mathbf{x}=\left[x_{1},...,x_{5}\right]^{\mathsf{T}}$ from a low-discrepency
sequence. In particular, we used the Halton sequence, which is a common low-
discrepancy sequence used in Monte Carlo integration and design of experiment,
with dimension $n_{\mathbf{x}}=5$ and support
$\hat{x}_{k}\in\left[0,1\right]\forall k\in\\{1,...,5\\}$ [16].

  * •

Since $\ell$ spans two orders of magnitude, we account for that by using a
nonlinear transformation to obtain the parameter values $x_{k}$ as used in our
simulations. This can be expressed as follows:

$x_{k}=\hat{x}_{k}^{\mathrm{s}}(x_{k}^{\mathrm{max}}-x_{k}^{\mathrm{min}})+x_{k}^{\mathrm{min}},$
(3)

where $x_{k}^{\mathrm{min}}$ and $x_{k}^{\mathrm{max}}$ refer to the parameter
range of the $k^{\mathrm{th}}$ parameter $x_{k}$, as defined in Table 1, and
where the scaling exponent $\mathrm{s}$ refers to $\mathrm{s}=1$ for $k\leq 4$
(linear), and $\mathrm{s}=2$ for $k=5$ (squared scaling for $\ell$).

In each simulation, we measured the electron spectra
$500$\mathrm{f}\mathrm{s}$$ after the laser maximum reaches the target. The
electron spectra were computed by binning the weights of electron
macroparticles onto $200$ bins with energies between 0 and
$40$\mathrm{M}\mathrm{e}\mathrm{V}$$. Furthermore, aiming to bridge the more
than five orders of magnitude present in the raw electron spectra, we made a
nonlinear transformation
$\tilde{f}_{\mathrm{e}}(E)=\left(\log{(f_{\mathrm{e}}/10^{-3}+1)}*g\right)(E)$,
which ensures that $\min_{E}{\tilde{f}_{\mathrm{e}}(E)}\geq 0$. Here, $*$
relates to the convolution operation, i.e. we smoothed our spectra with a
Gaussian filter $g(E)=\frac{1}{\sqrt{2\pi}}e^{-E^{2}/2}$ to make the data more
robust. A comparison of the raw electron spectra to the transformed spectra is
provided in Fig. 1.

Figure 1: Examples of electron spectra. Left: Raw spectra $f_{\mathrm{e}}(E)$
as measured in the PIC simulations (note the logarithmic scale). Right:
Nonlinearly transformed spectra $\tilde{f}_{\mathrm{e}}(E)$.

Given the limited data size, especially in relation to the relatively high
number of bins, we experienced in our initial attempts of training a ML model
from parameter vectors333Note that we use, for the sake of better of
readability, henceforth the symbol $\mathbf{x}$ both for our simulation
parameter $\mathbf{x}$ as well as normalized ML parameter $\hat{\mathbf{x}}$.
$\mathbf{x}$ to electron spectra $\tilde{f}_{\mathrm{e}}(E)$ that the training
process is rather difficult and sensitive to hyperparameter. Therefore, in
order to further simplify the training process, we expressed our transformed
spectra in terms of a linear regression model. In particular, we performed
principle component regression (PCR) using $n_{\mathbf{y}}=6$ principle
components $b_{k}(E)$:

$\tilde{f}_{\mathrm{e}}(E)\approx\sum_{k=1}^{6}{c_{k}b_{k}(E)}+\bar{\tilde{f}}_{\mathrm{e}},$
(4)

where the $c_{k}$’s are the coefficients corresponding to the basis functions
$b_{k}(E)$, and $\bar{\tilde{f}}_{\mathrm{e}}$ is the mean transformed
spectrum. Thus, altogether we train our ML models to learn mappings between
$\mathbf{x}$ and $\mathbf{y}:=\left[c_{1},...,c_{6}\right]^{\mathsf{T}}$.
Further information about the ML models used and studied in this work and
their training is provided in Sections 3 and 4, respectively.

##  3 Invertible Neural Networks

In this work, we employ invertible neural networks (INNs) as described by
Ardizzone et al. in [1]. They have been designed having in mind a common
problem in natural sciences, namely that there exists a forward process
(experiment, simulation, …) $f$ that maps a parameter vector
$\mathbf{x}\in\mathbb{R}^{n_{\mathbf{x}}}$ onto experimentally accessible
quantities which we call observables
$\mathbf{y}=f(\mathbf{x})\in\mathbb{R}^{n_{\mathbf{y}}}$. Typically, this
forward process, at least formally, is well understood in the sense that there
exists a (often highly sophisticated) theory that supports this mapping.
However, one is most often interested in the inverse process, i.e. to gain
information about experimentally inaccessible parameter given an experimental
result. Furthermore, since the forward process intrinsically has in general
accompanying information loss, the inverse direction can only be expressed
probabilistically as this loss of information renders the inverse process
ambiguous. In other words, multiple parameter vectors may correspond to the
same observable. Therefore, we are interested in the complete set of solutions
$\\{\mathbf{x}\in\mathbb{R}^{n_{\mathbf{x}}}\,\lvert\,f(\mathbf{x})=\mathbf{y}\\}$,
i.e. the full conditional posterior distribution
$p(\mathbf{x}\lvert\mathbf{y})$ has to be determined.

In order to account for the information loss, INNs introduce a latent space of
dimension $n_{\mathbf{z}}$ and elements
$\mathbf{z}\sim\mathcal{N}(\mathbf{z};\mathbf{0},\mathbf{I})$. Then, the
latent vectors are concatenated with the observables as
$[\mathbf{y},\mathbf{z}]$. Note that invertibility requires that
$n_{\mathbf{x}}=n_{\mathbf{y}}+n_{\mathbf{z}}$, which can be realized by
including zero-padding as required. In our case, since already
$n_{\mathbf{x}}=5<n_{\mathbf{y}}=6$, we fulfilled the aforementioned condition
by padding our parameter vectors as $[\mathbf{x},\mathbf{0}]$, where
$\mathbf{0}$ stands for a
$(n_{\mathbf{y}}+n_{\mathbf{z}}-n_{\mathbf{x}})$-dimensional zero vector.
Then, INNs attain invertibility by composition of affine coupling blocks
(ACBs), which are invertible themselves. While various different architectures
for ACBs have been developed, in this work we use the Glow architecture444Not
including ActNorm, invertible 1x1 convolutions, etc. relevant for their
specific application, but only the coupling part itself., that is very similar
on the RealNVP design [12, 7]. In each ACB, the input is split into two parts
$\mathbf{u}=[\mathbf{u}_{1},\mathbf{u}_{2}]$ of equal size which are then
transformed by an affine function using element-wise multiplication ($\odot$)
and vector addition to an output $\mathbf{v}=[\mathbf{v}_{1},\mathbf{v}_{2}]$:

$\begin{split}\mathbf{v}_{1}&=\mathbf{u}_{1}\odot\exp(s_{2}(\mathbf{u}_{2}))+t_{2}(\mathbf{u}_{2}),\\\
\mathbf{v}_{2}&=\mathbf{u}_{2}\odot\exp(s_{1}(\mathbf{u}_{1}))+t_{1}(\mathbf{u}_{1}).\end{split}$
(5)

Then, given the output $\mathbf{v}=[\mathbf{v}_{1},\mathbf{v}_{2}]$, we can
easily retrieve $\mathbf{u}=[\mathbf{u}_{1},\mathbf{u}_{2}]$ as follows:

$\begin{split}\mathbf{u}_{2}&=(\mathbf{v}_{2}-t_{1}(\mathbf{v}_{1}))\odot\exp(-s_{1}(\mathbf{v}_{1})),\\\
\mathbf{u}_{1}&=(\mathbf{v}_{1}-t_{2}(\mathbf{u}_{2}))\odot\exp(-s_{2}(\mathbf{u}_{2})).\end{split}$
(6)

The functions $[s_{i}(\cdot),t_{i}(\cdot)]$, which are typically implemented
as feedforward neural networks and hence called subnetworks, can be
arbitrarily complicated functions that need not be invertible themselves. We
further elaborate on the design of the subnetworks in Section 4.

INNs are bi-directionally trained with losses
$\mathcal{L}_{\mathbf{x}},\mathcal{L}_{\mathbf{y}}$ and
$\mathcal{L}_{\mathbf{z}}$ defined for $\mathbf{x},\mathbf{y}$ and
$\mathbf{z}$, respectively. While $\mathcal{L}_{\mathbf{y}}$ in general can be
any supervised loss, we use the mean-squared loss (MSE) loss,
$\mathcal{L}_{\mathbf{y}}=\mathbb{E}[(\mathbf{y}-f_{\mathbf{y}}(\mathbf{x}))^{2}]$.
For $\mathcal{L}_{\mathbf{x}}$ and $\mathcal{L}_{\mathbf{z}}$ we use maximum
mean discrepancy (MMD), which is a kernel-based, unsupervised loss on the
space of probability distributions and which is based on reproducing kernel
Hilbert spaces [9]. For our study, we used a multiscale inverse multiquadratic
kernels as follows
$k(\mathbf{x},\mathbf{x}^{\prime})=\sum_{h}{1/(1+\|(\mathbf{x}-\mathbf{x}^{\prime})/h\|_{2}^{2})}$,
where the bandwidth parameter $h\in\\{0.04,0.16,0.64\\}$ are similar to the
ones employed by Ardizzone et al. [1].

##  4 Results

We ran our experiments on the Taurus cluster at ZIH/TU Dresden. We used nodes
of type Haswell, each node having two Intel Xeon E5-2680v3 @ 2.50 GHz
processors with 30 MB L3 cache and 12 cores each, amounting to 24 cores per
node. Each observation (1000 altogether), i.e. electron spectrum, was analyzed
with one core. Each core has $2\cdot 32$KB L1 cache and 256 KB L2 cache. Each
program is written in Python 3.9.12 and imports NumPy 1.21.5 and PyTorch
1.10.2.

We performed two different experiments: (1) we made a comparison of acceptance
rates between different methods for solving the inverse problem and (2) then
measured the actual time needed to find one accepted solution. These
experiments were performed on hyperparameter optimized models as follows:

####  4.0.1 ML Models and Training

In this study, we both employ a multilayer perceptron (MLP) that we use as our
reference model for the forward process $f(\cdot)$ only, and an INN for
solving the inverse problem. For the training of the MLP, we again use MSE
loss, corresponding to the $\mathcal{L}_{\mathbf{y}}$ loss of the INN. For
both models we splitted our data into $80\%$ train and $20\%$ test set555I.e.,
4000 and 1000 data points for the train and test set, respectively., and we
used in both cases the Adam optimizer with learning rate $\alpha=0.001$ and
betas $\beta_{1}=0.9$, $\beta_{2}=0.999$ for training [11]. Furthermore, we
performed a hyperparameter optimization for the MLP with regards to:

  1. 1.

the activation function $\sigma(\cdot)$: ReLU($\cdot$), Tanh($\cdot$),

  2. 2.

widths of hidden layers: 12, 16, 20, 24, 30,

  3. 3.

number of layers: 3, 4, 5,

where we found that the setting MLP: {Tanh($\cdot$), 16, 4} shows the lowest
loss for the test set. For the INN, we extend the hyperparameter optimization
with regards to the dimension of the latent space $n_{\mathbf{z}}$, and the
number of affine coupling blocks (ACBs):

  1. 1.

the activation function $\sigma(\cdot)$: ReLU($\cdot$), Tanh($\cdot$),

  2. 2.

widths of layers in subnetworks: 12, 16, 20, 24, 30,

  3. 3.

number of layers in subnetworks: 2, 3, 4,

  4. 4.

dimension $n_{\mathbf{z}}$: 2, 3, 4, 5, 6,

  5. 5.

number of ACBs: 2, 3, 4, 5, 6,

where the best results, in terms of
$\mathcal{L}_{\mathbf{x}}+\mathcal{L}_{\mathbf{y}}$, were obtained with the
setting INN: {Tanh($\cdot$), 20, 3, 6, 5}. Note that the optimal number of
ACBs in our case is larger than proposed by Dinh who generally suggests to use
four ACBs [6].

###  4.1 Acceptance Rate

In the first experiment we compared, in terms of their acceptance rates,
different methods suitable for (approximately) solving the inverse problem,
i.e. to find a set of samples $\\{\mathbf{x}\\}$ that is representative for
the conditional posterior $p(\mathbf{x}\lvert\mathbf{y}^{\star})$ conditioned
on a specific measurement $\mathbf{y}^{\star}$. We call a parameter vector
$\mathbf{x}$ to be accepted if the acceptance condition:

$d(\mathbf{y}^{\star},f(\mathbf{x}))\leq\epsilon,$ (7)

is fulfilled, where $f(\cdot)$ is, in this study, the hyperparameter optimized
MLP, $d(\cdot,\cdot)$ is a suitable distance function and $\epsilon$ is a non-
negative threshold.

Figure 2: Conditional probability $p(a_{0}\lvert T_{\mathrm{e}})$ used as
prior in the modified ABC routine.

####  4.1.1 Approximate Bayesian Computation

If a surrogate model for the forward process is available, then one standard
method to find an approximate solution for the inverse problem is approximate
Bayesian computation (ABC) [2]. In ABC, the forward process $f$ is embedded in
a rejection-sampling scheme, i.e. the forward model is employed by randomly
sampling $\mathbf{x}$ from the parameter space and subsequently checking for
the acceptance condition.

In practice, depending on the problem’s complexity as well as
$\mathbf{y}^{\star}$ and $\epsilon$, finding an appropriately sized set of
solutions might require the evaluation of $f$ for millions of times.
Furthermore, since the algorithm is, per construction, subject to randomness,
one can simply get ”unlucky”, consequently spending excessive amounts of
compute time. Therefore, in order to speed up the expected convergence, we
also performed an experiment where we again used quasi-random numbers produced
from the Halton sequence instead of purely randomly generated numbers.

Moreover, since in naive ABC we don’t include prior knowledge as we draw
samples (quasi)randomly from the parameter space and therefore implicitly
assume a multivariate uniform distribution as our prior, we don’t sample
optimally and thus increase the computational effort. Consequently, in order
to study the effect of a non-uniform prior, we also conducted an experiment in
which we draw $a_{0}$ based on a probability density function $p(a_{0}\lvert
T_{\mathrm{e}})$, since $T_{\mathrm{e}}=T_{\mathrm{e}}(I(a_{0}))$. The
corresponding probability distribution is illustrated in Fig. 2 and was
numerically computed by applying Bayes’ rule on the train set data.

####  4.1.2 Hill-Climbing

On the other hand, instead of randomly trying different parameter vectors
$\mathbf{x}$ as in ABC, local search algorithms such as hill-climbing (HC) and
related methods try to find solutions by incrementally improving the current
state [17]. While typically gradient-based approaches are preferred, hill-
climbing can be used also if only a black-box model is available. Arguably
it’s simplest form, and also as implemented in this work, is first-choice
hill-climbing, where the current solution is updated directly as soon as a
better candidate solution has been found. Considering that ABC produces a set
of uncorrelated samples, however, it is necessary in HC, once having found the
first solution, to restart the search for the next solution at a randomly
chosen location in order also obtain a statistically uncorrelated sample. A
pseudocode of our implementation of first-choice hill-climbing is provided in
the procedure FirstChoiceHillClimbing in Algorithm 1. For our experiments, we
used a learning rate of $\alpha=10^{-3}$, as commonly used in ML, and a
learning rate of $\alpha=10^{-2}$ for comparison.

####  4.1.3 Comparison

Figure 3: Acceptance rates of different approaches for obtaining
$p(\mathbf{x}\lvert\mathbf{y}^{\star})$ in dependence of acceptance threshold
$\epsilon$, averaged over 1000 different electron spectra.

A comparison of the three different methods (ABC, HC and INN) and their
specific settings is provided in Fig. 3. First, we chose our test set of 1000
electron spectra $f_{\mathrm{e}}(E)$ and computed their PCR coefficients as
described in Section 2. Second, with the intention of obtaining a relative
measure of similarity, we define our distance function $d(\cdot,\cdot)$ based
on the $L^{2}$ distance as follows:

$d[f(E),g(E)]=\frac{\sqrt{\int{(f(E)-g(E))^{2}\differential{E}}}}{\sqrt{\int{g^{2}(E)\differential{E}}}},$
(8)

where $f(E)$ represents an electron spectrum containing errors, e.g. as
proposed by one of our inverse solver, and $g(E)$ is the reference ground
truth spectrum function. In this work, $g(E)$ corresponds to the transformed
electron spectrum of the reference model, i.e. after the PCR procedure, and
the lower and upper limits of integration are again
$0$\mathrm{M}\mathrm{e}\mathrm{V}$$ and $40$\mathrm{M}\mathrm{e}\mathrm{V}$$,
respectively. At last, using 10 iterations with $m_{\mathrm{trial}}=10^{5}$,
we compute the acceptance rate as the ratio of accepted solutions to
all($=10\cdot m_{\mathrm{trial}}=10^{6}$) tested solutions.

From Fig. 3 can see that the efficiency of a method heavily depends on the
acceptance threshold $\epsilon$. Not surprisingly, naive ABC using random
numbers shows a rather low acceptance rate regardless of $\epsilon$.
Furthermore, ABC using quasi-random numbers behaves the same as naive ABC for
larger $\epsilon$, since then the law of large numbers becomes relevant, and
shows significantly better performance for smaller thresholds where
$\epsilon\leq 5\cdot 10^{-3}$. A further improvement can be achieved by using
an informed prior for $a_{0}$: On average, the acceptance rate increases
approximately by a factor of 2 when compared to uninformed ABC. However,
interestingly, it can be seen that the acceptance rate is more than two orders
of magnitude smaller than simply using ABC with the Halton sequence for
$\epsilon=10^{-3}$. This can be understood by noting that in this case
parameters are again sampled randomly, and thus don’t exhibit the enhanced
space-filling property as in the Halton case. The largest acceptance rates for
small $\epsilon$ were obtained by the hill-climbing methods, which are higher
by around one order of magnitude. We can deduce, from the big drop in the
acceptance rate of HC for large $\epsilon$, that HC first needs many steps to
approach a region of reasonably small distance after which, however, it
apparently only takes minimal effort to further optimize the solution. While
HC with learning rate $\alpha=10^{-2}$ always beats HC with learning rate of
$\alpha=10^{-3}$, we can also see that the difference significantly decreases
for decreasing $\epsilon$. On the other hand, the largest acceptance rates for
$\epsilon\geq 10^{-2}$ were obtained by the INN. For $\epsilon\leq 5\cdot
10^{-3}$, the INN shows worse performance than both configurations of HC.
Then, around $\epsilon\approx 3\cdot 10^{-3}$, the INN also exhibits lower
acceptance rates than Halton-based ABC. Again, the reason is that latent
vectors $\mathbf{z}$ are sampled randomly from the multivariate normal
distribution, and not from a quasi-random sequence. Therefore, for the full
range of $\epsilon$, the INN always surpasses the acceptance rate of ABC
Random and ABC $T_{\mathrm{e}}$, since they are also both based on random
numbers rather than a quasi-random sequence.

###  4.2 Runtimes

While the acceptance rate is more interesting for theoretical analysis, in
practice we are interested in the actual computational cost, e.g. in terms of
the runtime. We performed the same experiment as before, but instead we
measured the total runtime relative to the number of accepted samples,
$t_{\mathrm{total}}/m_{\mathrm{acc}}$. Again, we average our results over the
same 1000 electron spectra from the test set. The result is depicted in Fig 4.

Figure 4: Time to solution, i.e. of a single acceptance, of different
approaches for obtaining $p(\mathbf{x}\lvert\mathbf{y}^{\star})$ in dependence
of acceptance threshold $\epsilon$, averaged over 1000 different electron
spectra.

We can see that general trends are preserved, e.g. that the INN is the fastest
method up until $\epsilon\geq 3\cdot 10^{-3}$ and that ABC with informed prior
is always faster than naive ABC. HC is still a fast method for very small
thresholds. However, the relative runtimes difference between HC and all other
methods decreased by around two order of magnitude when compared to the
relative difference in acceptance rate. This can be understood as HC uses
loops and needs to call $f(\cdot)$ many times, while e.g. ABC calls the
forward function only once for all proposal vectors and therefore takes
advantage of optimized matrix operations. Hence, to ensure efficient
computation, HC should as well be embedded in parallel procedures. We can also
see that using the INN does not cause any significant overhead, since the
relative speedup is approximately conserved with respect to the relative
increase in acceptance rate.

Thus, in order find uncorrelated samples for the inverse problem, the method
should be chosen having the required accuracy in mind. An algorithm based on
the combination of an INN and HC, which is designed to also work for very
small thresholds, is provided in Algorithm 1.

Algorithm 1 INN-HC: Inverse solver optimized for low acceptance thresholds.

1:procedure InverseSolver($\mathbf{y}^{\star}$, $m$, $f$, $\mathrm{inn}$, $d$,
$\epsilon$, $\alpha$)

2:
$\mathbf{Y}^{\star}\leftarrow\mathrm{vstack}(\mathbf{y}^{\star},m)$$\triangleright$
Vertically stack $\mathbf{y}^{\star}$, i.e.
$\mathbf{Y}^{\star}\in\mathbb{R}^{m\crossproduct n_{\mathbf{y}}}$

3:
$\mathbf{Z}\leftarrow\mathrm{rand}(\mathcal{N}(0,1),(m,n_{\mathbf{z}}))$$\triangleright$
$\mathbf{Z}\in\mathbb{R}^{m\crossproduct n_{\mathbf{z}}}$

4: $\mathbf{X}\leftarrow\mathrm{inn}^{-1}([\mathbf{Y}^{\star},\mathbf{Z}])$

5: for $i$ in $1,...,m$ do

6: $\mathbf{x}\leftarrow\mathbf{X}_{i,\bm{\cdot}}$

7: if $d(\mathbf{y}^{\star},f(\mathbf{x}))>\epsilon$ then

8:
$\mathbf{x}\leftarrow\mathrm{FirstChoiceHillClimbing}(\mathbf{y}^{\star},f,d,\epsilon,\mathbf{x},\alpha)$

9: $\mathbf{X}_{i,\bm{\cdot}}\leftarrow\mathbf{x}$

10: return $\mathbf{X}$

11:

12:procedure FirstChoiceHillClimbing($\mathbf{y}^{\star}$, $f$, $d$,
$\epsilon$, $\mathbf{x}_{0}$, $\alpha$)

13: $\mathbf{x}\leftarrow\mathbf{x}_{0}$

14: while $d(\mathbf{y}^{\star},f(\mathbf{x}))>\epsilon$ do

15:
$\bm{\xi}\leftarrow\mathrm{rand}(\mathcal{U}([-1,1]),(n_{\mathbf{x}}))$$\triangleright$
Generate vector with random direction

16: $\bm{\xi}\leftarrow\bm{\xi}/\lvert\bm{\xi}\rvert$$\triangleright$
Normalize to unit length

17: $\tilde{\mathbf{x}}\leftarrow\mathbf{x}+\alpha\bm{\xi}$

18: if $d(\mathbf{y}^{\star},f(\mathbf{\tilde{x}}))\leq
d(\mathbf{y}^{\star},f(\mathbf{x}))$ then

19: $\mathbf{x}\leftarrow\tilde{\mathbf{x}}$.

20: return $\mathbf{x}$

##  5 Conclusion

In this work, we have studied INNs on synthetic electron spectra in the
context of near-critical laser-plasma physics. In particular, we compared INNs
with other standard statistical methods for solving the inverse process. We
found that INNs perform, both in terms of acceptance rates as well as
runtimes, better than all other methods up to a small threshold distance.
Furthermore, we show that naive ABC based on random numbers has lower
acceptance rates and larger runtimes than our INN model by a factor of
approximately 10 for any threshold. INNs also surpass informed ABC, where we
used a modified prior $p(a_{0}\lvert T_{\mathrm{e}})$ that we motivated due to
physical considerations, by a significant amount. Moreover, we demonstrate the
importance of quasi-random numbers and recommend to use them as well in
conjunction with INNs. On the other hand, our results suggest that iterative
approaches, in our case hill-climbing, surpass INNs for small thresholds
$\epsilon\leq 3\cdot 10^{-3}$, especially in terms of the acceptance rate.
Therefore, due to the relative strengths of the different algorithms, we
propose a composite algorithm for obtaining the conditional posterior that
combines both hill-climbing and INNs.

## References

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# Skew bracoids and the Yang-Baxter equation

Ilaria Colazzo  University of Exeter
Department of Mathematics and Statistics
Exeter
EX4 4QF
UK University of Leeds
School of Mathematics
Leeds
LS2 9JT
UK<EMAIL_ADDRESS>,  Alan Koch  Department of Mathematics, Agnes Scott
College, 141 E. College Ave., Decatur, GA 30030 USA<EMAIL_ADDRESS>,
Isabel Martin-Lyons  School of Computing and Mathematics
Keele University
Staffordshire
ST5 5BG
UK<EMAIL_ADDRESS>and  Paul J. Truman  School of Computing and
Mathematics
Keele University
Staffordshire
ST5 5BG
UK<EMAIL_ADDRESS>
###### Abstract.

Skew braces provide an algebraic framework for studying bijective
nondegenerate solutions of the set-theoretic Yang-Baxter equation. We show
that left skew bracoids, recently introduced by two of the authors, can be
used to obtain right nondegenerate solutions, and give examples arising from
various methods of constructing left skew bracoids. We compare the solutions
we obtain with the left nondegenerate solutions obtained via (left
cancellative) left semibraces, and establish a correspondence between left
semibraces and a class of left skew bracoids.

###### Key words and phrases:

Yang-Baxter equation, Skew braces, skew bracoids, semibraces

###### 2020 Mathematics Subject Classification:

Primary 16T25; Secondary 20N99

##  1\. Introduction

The Yang-Baxter equation is a fundamental equation of theoretical physics and
statistical mechanics [11], [12]. We will be concerned with its set theoretic
analogue, introduced by Drinfeld [7]. A solution of the set theoretic Yang-
Baxter equation on a nonempty set $G$ (hereafter, simply a solution on $G$) is
a map $r:G\times G\rightarrow G\times G$ such that

$(r\times\mathrm{id})(\mathrm{id}\times
r)(r\times\mathrm{id})=(\mathrm{id}\times
r)(r\times\mathrm{id})(\mathrm{id}\times r)$

as functions on $G\times G\times G$. We say that solution is bijective if $r$
is a bijective function, and involutive if $r^{2}=\mathrm{id}$. For $x,y\in G$
we write

$r(x,y)=(\lambda_{x}(y),\rho_{y}(x));$

we say that a solution is left nondegenerate if $\lambda_{x}$ is bijective for
each $x\in G$, right nondegenerate if $\rho_{y}$ is bijective for each $y\in
G$, and nondegenerate if it is both left and right nondegenerate.

A wide variety of techniques have been brought to bear on the problems of
determining and studying solutions with various properties. A particularly
fruitful algebraic framework is provided by the theory of skew braces; these
objects were introduced by Guanieri and Vendramin [9], generalizing Rump’s
earlier definition of braces [20]. A skew brace is a set $G$ together with two
binary operations, each making $G$ into a group, that satisfy a certain
compatibility relation (see Section 2 for more details). Skew braces yield
bijective nondegenerate solutions, and have also been found to have
connections with numerous other topics including braids, knots, regular
subgroups of permutation groups, and Hopf-Galois theory. In [19] two of the
authors introduce skew bracoids; these consist of two groups $G$ and $N$,
together with a transitive action of the former on the latter that interacts
with the binary operation on $N$ in a manner analogous to the compatibility
relation for skew braces. Skew bracoids can be used to broaden the connection
between skew braces and Hopf-Galois theory [19, Section 5]. It is also natural
to ask what connections they might have with the set theoretic Yang-Baxter
equation; the purpose of this paper is to address this question.

Our main result (Theorem 3.2) is that left skew bracoids $(G,N)$ satisfying a
mild technical hypothesis yield right nondegenerate solutions on $G$. Our
approach employs a result of Lu, Yan, and Zhu for constructing solutions from
groups with certain additional data; we summarize this result, along with some
of the theory of skew braces, in Section 2. In Section 3 we recall some
fundamental definitions and results concerning skew bracoids, specify the
family of skew bracoids we will study (Hypothesis 3.1), and state and prove
Theorem 3.2. We construct examples of skew bracoids satisfying Hypothesis 3.1
in Section 4, drawing upon a variety of constructions in the literature; we
also exhibit a family of skew bracoids that do not satisfy this Hypothesis.
Finally, in Section 5 we compare the right nondegenerate solutions arising
from our left skew bracoids with the left nondegenerate solutions arising from
left semibraces found in [3].

### Acknowledgements

We gratefully acknowledge the support of the Engineering and Physical Sciences
Research Council, project references EP/V005995/1 (Ilaria Colazzo) and
EP/W012154/1 (Paul J. Truman), and the London Mathematical Society, joint
research group 32312, which enabled parts of this collaboration to take place.

##  2\. Preliminaries: solutions arising from groups, and skew braces

In [17] Lu, Yan, and Zhu prove that given a group $G$, a homomorphism
$\lambda:G\rightarrow\operatorname{\mbox{Perm}}(G)$, and an antihomomorphism
$\rho:G\rightarrow\operatorname{\mbox{Perm}}(G)$ such that
$\lambda_{x}(y)\rho_{y}(x)=xy$ for all $x,y\in G$, the map $r:G\times
G\rightarrow G\times G$ defined by $r(x,y)=(\lambda_{x}(y),\rho_{y}(x))$ for
all $x,y\in G$ is a bijective nondegenerate solution on $G$. We shall require
the following mild generalization of their result; we sketch a proof for the
convenience of the reader.

######  Proposition 2.1.

Suppose that $\lambda,\rho:G\rightarrow\mathrm{Map}(G,G)$ satisfy the
following, for all $x,y\in G$:

$\displaystyle\lambda_{xy}$ $\displaystyle=$
$\displaystyle\lambda_{x}\lambda_{y};$ (1) $\displaystyle\rho_{xy}$
$\displaystyle=$ $\displaystyle\rho_{y}\rho_{x};$ (2) $\displaystyle xy$
$\displaystyle=$ $\displaystyle\lambda_{x}(y)\rho_{y}(x).$ (3)

Then the map $r:G\times G\rightarrow G\times G$ defined by
$r(x,y)=(\lambda_{x}(y),\rho_{y}(x))$ for all $x,y\in G$ is a solution on $G$.

###### Proof.

Let $x,y,z\in G$. By straightforward computation we have

$(r\times\mathrm{id})(\mathrm{id}\times
r)(r\times\mathrm{id})(x,y,z)=(\lambda_{\lambda_{x}(y)}(\lambda_{\rho_{y}(x)}(z)),\rho_{\lambda_{\rho_{y}(x)}(z)}(\lambda_{x}(y)),\rho_{z}(\rho_{y}(z)))$
(4)

and

$(\mathrm{id}\times r)(r\times\mathrm{id})(\mathrm{id}\times
r)(x,y,z)=(\lambda_{x}(\lambda_{y}(z)),\lambda_{\rho_{\lambda_{y}(z)}}(\rho_{z}(y)),\rho_{\rho_{z}(y)}(\rho_{\lambda_{y}(z)}(x))).$
(5)

Hypotheses (1) and (3) imply that the first components of the right hand sides
of (4) and (5) agree, and hypotheses (2) and (3) imply that the third
components agree. Hypothesis (3) also implies that the products of the three
components agree, and so we conclude that the second components also agree. ∎

Following [22] we shall call a pair of maps $\lambda,\rho$ as in Proposition
2.1 a Lu-Yan-Zhu pair on $G$.

We record for future use that if $\lambda,\rho$ form a Lu-Yan-Zhu pair on $G$
then by repeated application of (3) for all $x,y,z\in G$ we have

$\displaystyle\lambda_{x}(yz)\rho_{yz}(x)=\;$ $\displaystyle xyz$
$\displaystyle=\;$ $\displaystyle\lambda_{x}(y)\rho_{y}(x)z$
$\displaystyle=\;$
$\displaystyle\lambda_{x}(y)\lambda_{\rho_{y}(x)}(z)\rho_{z}(\rho_{y}(x))$
$\displaystyle=\;$
$\displaystyle\lambda_{x}(y)\lambda_{\rho_{y}(x)}(z)\rho_{yz}(x)$ (by (2))

which yields the useful formula

$\lambda_{x}(yz)=\lambda_{x}(y)\lambda_{\rho_{y}(x)}(z).$ (6)

Lu-Yan-Zhu pairs leading to bijective nondegenerate solutions can be obtained
from skew braces (in fact, these notions are equivalent: see [17, Theorem 2]
and [9, Proposition 1.11]). A skew (left) brace is a triple $(G,\star,\cdot)$
in which $(G,\star)$ and $(G,\cdot)$ are groups (called the additive and
multiplicative groups of the skew brace) and the equation

$x\cdot(y\star z)=(x\cdot y)\star\overline{x}\star(x\star z)$ (7)

holds for all $x,y,z\in G$, where $\overline{x}$ denotes the inverse of $x$
with respect to $\star$ [9, Definition 1.1]. For brevity, we shall refer to
skew braces simply as braces; we note, however, that elsewhere in the
literature this term is often used to refer to skew braces with abelian
additive group, as originally introduced by Rump in [20].

Equation (7) implies that for each $x\in G$ the function
$\gamma_{x}:G\rightarrow G$ defined by $\gamma_{x}(y)=\overline{x}\star(x\cdot
y)$ is an automorphism of $(G,\star)$, and that the resulting function
$\gamma:G\rightarrow\operatorname{\mbox{Perm}}(G)$ is a homomorphism
(sometimes called the $\gamma$-function of the brace) [9, Proposition 1.9]; it
can be shown ([9, Theorem 3.1] or [5]) that if we set
$\lambda_{x}(y)=\gamma_{x}(y)$ and define $\rho_{y}(x)$ by
$\lambda_{x}(y)^{-1}\cdot x\cdot y$ then $\lambda,\rho$ form a Yu-Lan-Zhu pair
and hence, via Proposition 2.1, a solution on $G$. This solution is always
bijective and nondegenerate. It is involutive if and only if the group
$(G,\star)$ is abelian; if $(G,\star)$ is nonabelian then the inverse of the
solution arising from the brace $(G,\star,\cdot)$ is the solution arising from
the opposite brace $(G,\star^{op},\cdot)$: see [15].

##  3\. Skew bracoids containing a brace

A (left) skew bracoid is a quintuple $(G,\cdot,N,\star,\odot)$ in which
$(G,\cdot)$ and $(N,\star)$ are groups and $\odot$ is a transitive (left)
action of $(G,\cdot)$ on $N$ such that the equation

$x\odot(\eta\star\mu)=(x\odot\eta)\star(x\odot e_{N})^{-1}\star(x\odot\mu)$
(8)

holds for all $x\in G$ and $\eta,\mu\in N$ (see [19]). In analogy with braces,
we call $(G,\cdot)$ the multiplicative group, and $(N,\star)$ the additive
group, of the skew bracoid $(G,\cdot,N,\star,\odot)$. We suppress the notation
for the binary operations in $G$ and $N$ whenever possible. We retain the
symbol $\odot$ for the action of $G$ on $N$ in our arguments, but often
suppress it when specifying skew bracoids, viz $(G,N)$. For brevity, we shall
refer to skew bracoids simply as bracoids.

In this section we show that a large family of bracoids with multiplicative
group $G$ yield Lu-Yan-Zhu pairs $\lambda,\rho$ on $G$ in which $\rho_{y}$ is
bijective for all $y\in G$, and so, via Proposition 2.1, right nondegenerate
solutions on $G$.

In order to specify the family of bracoids we shall study, we recall some
definitions from [19]. First, we say that a bracoid $(G,N)$ in which the
subgroup $S=\operatorname{\mbox{Stab}}(e_{N})$ of $G$ is trivial is
essentially a brace (see [19, Example 2.2]), since in this case we may define
a second group operation on $G$ via the rule

$(x\star_{G}y)\odot e_{N}=(x\odot e_{N})\star(y\odot e_{N})$ (9)

such that $(G,\star_{G})\cong(N,\star)$ and $(G,\star_{G},\cdot)$ is a brace.
Second, a subbracoid of a bracoid $(G,N,\odot)$ consists of a subgroup $H$ of
$G$ and a subgroup $M$ of $N$ such that $(H,M,\odot)$ is a bracoid [19,
Definition 3.1]. The family of bracoids we shall study are those satisfying

######  Hypothesis 3.1.

For $(G,N)$ a bracoid, there is a subgroup $H$ of $G$ such that the subbracoid
$(H,N)$ is essentially a brace. Abusing terminology, we shall say that the
bracoid $(G,N)$ contains a brace.

We shall construct examples of bracoids containing a brace in Section 4. If
$(G,N)$ is a such a bracoid then the map $a:H\rightarrow N$ given by
$a(h)=h\odot e_{N}$ is a bijection; we denote its inverse by $b:N\rightarrow
H$, so that $b(\eta)\odot e_{N}=\eta$ for all $\eta\in N$. Using the bijection
$b$ we can reinterpret Equation (9) as

$x\star_{H}y=b(a(x)\star a(y))\mbox{ for all }x,y\in H.$ (10)

Finally, recall that the $\gamma$-function of a bracoid $(G,N)$ is the
homomorphism $\gamma:G\rightarrow\operatorname{\mbox{Aut}}(N)$ defined by

$\gamma_{x}(\eta)=(x\odot e_{N})^{-1}(x\odot\eta)\mbox{ for all }x\in G\mbox{
and }\eta\in N.$

(see [19, Theorem 2.8 and Definition 2.10]).

We can now state our main result:

######  Theorem 3.2.

Suppose that $(G,N)$ is a bracoid containing a brace $(H,N)$. For $x,y\in G$
define

$\lambda_{x}(y)=b(\gamma_{x}(y\odot e_{N}))$

and

$\rho_{y}(x)=\lambda_{x}(y)^{-1}xy.$

Then $\lambda,\rho$ form a Lu-Yan-Zhu pair in which $\lambda_{x}(G)=H$ for all
$x\in G$ and $\rho_{y}$ is bijective for each $y\in G$.

Applying Proposition 2.1 we obtain immediately

######  Corollary 3.3.

The map $r:G\times G\rightarrow G\times G$ defined by
$r(x,y)=(\lambda_{x}(y),\rho_{y}(x))$ for all $x,y\in G$ is a right
nondegenerate, non-bijective solution.

###### Proof of Theorem 3.2..

Let $x,y,z\in G$. First we show that

$\lambda_{xy}=\lambda_{x}\lambda_{y}.$ (11)

We have

$\displaystyle\lambda_{x}\lambda_{y}(z)=\;$
$\displaystyle\lambda_{x}(b(\gamma_{y}(z\odot e_{N})))$ $\displaystyle=\;$
$\displaystyle b(\gamma_{x}(b(\gamma_{y}(z\odot e_{N}))\odot e_{N}))$
$\displaystyle=\;$ $\displaystyle b(\gamma_{x}\gamma_{y}(z\odot e_{N}))$
(definition of $b$) $\displaystyle=\;$ $\displaystyle b(\gamma_{xy}(z\odot
e_{N}))$ ($\gamma$ is a homomorphism) $\displaystyle=\;$
$\displaystyle\lambda_{xy}(z).$

In particular, Equation (11) implies that
$\lambda_{x}(z)=\lambda_{e}(\lambda_{x}(z))$, so that
$\lambda_{x}(G)\subseteq\lambda_{e}(G)$, and that
$\lambda_{e}(z)=\lambda_{x}(\lambda_{x^{-1}}(z))$, so that
$\lambda_{e}(G)\subseteq\lambda_{x}(G)$. Thus $\lambda_{x}(G)=\lambda_{e}(G)$,
which is easily seen to be equal to $H$.

Next we show that

$\rho_{xy}=\rho_{y}\rho_{x}.$ (12)

We have

$\rho_{xy}(z)=\lambda_{z}(xy)^{-1}zxy,$ (13)

whereas

$\displaystyle\rho_{y}\rho_{x}(z)$ $\displaystyle=$
$\displaystyle\rho_{y}\left(\lambda_{z}(x)^{-1}zx\right)$ (14)
$\displaystyle=$
$\displaystyle\lambda_{\lambda_{z}(x)^{-1}zx}(y)^{-1}\lambda_{z}(x)^{-1}zxy.$

Using (11), we see that (13) and (14) agree if and only if

$\lambda_{\lambda_{z}(x)^{-1}}\lambda_{zx}(y)=\lambda_{z}(x)^{-1}\lambda_{z}(xy).$
(15)

To show that (15) holds we note that each side lies in $H$, so it is
sufficient to show that their images under the bijection $a:H\rightarrow N$
agree.

To simplify notation, write $h=\lambda_{z}(x)\in H$. Then the image of the
right hand side of (15) under $a$ is

$(h^{-1}\lambda_{z}(xy))\odot e_{N}=h^{-1}\odot(\gamma_{z}(xy\odot
e_{N}))\mbox{ (definitions of $\lambda$ and $b$).}$ (16)

On the other hand, the image of the left hand side of (15) under $a$ is

$\displaystyle\lambda_{h^{-1}}(\lambda_{zx}(y))\odot e_{N}$ $\displaystyle=\;$
$\displaystyle\gamma_{h^{-1}}(\gamma_{zx}(y\odot e_{N}))$ $\displaystyle=\;$
$\displaystyle(h^{-1}\odot e_{N})^{-1}(h^{-1}\odot\gamma_{zx}(y\odot e_{N}))$
$\displaystyle=\;$ $\displaystyle(h^{-1}\odot
e_{N})^{-1}(h^{-1}\odot\gamma_{z}\gamma_{x}(y\odot e_{N}))$ $\displaystyle=\;$
$\displaystyle(h^{-1}\odot e_{N})^{-1}(h^{-1}\odot\gamma_{z}((x\odot
e_{N})^{-1}(xy\odot e_{N})))$ $\displaystyle=\;$ $\displaystyle(h^{-1}\odot
e_{N})^{-1}(h^{-1}\odot\gamma_{z}(x\odot e_{N})^{-1}\gamma_{z}(xy\odot
e_{N}))$ $\displaystyle=\;$ $\displaystyle(h^{-1}\odot
e_{N})^{-1}(h^{-1}\odot\gamma_{z}(x\odot e_{N})^{-1})(h^{-1}\odot
e_{N})^{-1}(h^{-1}\odot\gamma_{z}(xy\odot e_{N}))$ $\displaystyle=\;$
$\displaystyle(h^{-1}\odot\gamma_{z}(x\odot
e_{N}))^{-1}(h^{-1}\odot\gamma_{z}(xy\odot e_{N}))$ $\displaystyle=\;$
$\displaystyle(h^{-1}\odot(h\odot e_{N}))^{-1}(h^{-1}\odot\gamma_{z}(xy\odot
e_{N}))$ $\displaystyle=\;$ $\displaystyle(h^{-1}\odot\gamma_{z}(xy\odot
e_{N}))$ (17)

Since (16) and (17) agree, (15) holds, and so (12) holds. In particular, this
implies that $\rho_{y}\rho_{y^{-1}}=\rho_{y^{-1}}\rho_{y}=\rho_{e}$. Since
$\rho_{e}(z)=\lambda_{z}(e)^{-1}ze=z$, we see that $\rho_{y}$ is bijective
(with inverse $\rho_{y^{-1}}$), as claimed.

Finally, it is clear from the definition of $\rho_{y}$ that we have
$\lambda_{x}(y)\rho_{y}(x)=xy$. This completes the proof. ∎

Next we study some of the properties of solutions arising via Theorem 3.2. The
following alternative characterization of Hypothesis 3.1 is useful:

######  Lemma 3.4.

A bracoid $(G,N)$ contains a brace if and only if the subgroup
$S=\operatorname{\mbox{Stab}}(e_{N})$ has a complement $H$ in $G$.

###### Proof.

If $(G,N)$ contains a brace then there is a subgroup $H$ of $G$ such that
$(H,N)$ is essentially a brace. In particular, the stabilizer of $e_{N}$ in
$H$ is trivial, and so $H\cap S$ is trivial. Given $x\in G$ there exists a
unique element $h\in H$ such that $x\odot e_{N}=h\odot e_{N}$, so
$s=h^{-1}x\in S$, and so $x=hs$. Hence $H$ is a complement to $S$ in $G$.

Conversely, if $S$ has a complement $H$ in $G$ then given $x\in G$ we may
write $x=hs$ with $h\in H$ and $s\in S$, and then we have $x\odot
e_{N}=hs\odot e_{N}=h\odot e_{N}$; thus $H$ acts transitively on $N$. Since
$H\cap S$ is trivial, the stabilizer of $e_{N}$ in $H$ is trivial, and so
$(H,N)$ is essentially a brace. Hence $(G,N)$ contains a brace. ∎

######  Remark 3.5.

If $(G,N)$ contains a brace $(H,N)$ then we also have an exact factorization
$G=SH$. The functions $\lambda,\rho$ defined in Theorem 3.2 allow us to
translate between these two exact factorizations: if $h\in H$ and $s\in S$
then

$sh=\lambda_{s}(h)\rho_{h}(s);$ (18)

we have $\lambda_{s}(h)\in H$ by definition, and

$\displaystyle\lambda_{s}(h)\odot e_{N}$ $\displaystyle=(s\odot e_{N})(sh\odot
e_{N})$ $\displaystyle=sh\odot e_{N}$ (since $s\in S$)
$\displaystyle=\lambda_{s}(h)\rho_{h}(s)\odot e_{N}$ (using Equation (18))

Thus $\rho_{h}(s)\in S$, and so $\lambda_{s}(h)\rho_{h}(s)$ is the unique
representation of the element $sh$ using the exact factorization $HS$.

We recall that if $(G,N)$ is a bracoid containing a brace $(H,N)$ then the
bijection $b:N\rightarrow H$ can be used to define a new binary operation
$\star_{H}$ on $H$ such that $(H,\star_{H},\cdot)$ is a brace. We can identify
the solution arising from this brace as a subsolution of the one arising from
the bracoid.

######  Proposition 3.6.

Suppose that $(G,N)$ is a bracoid containing a brace $(H,N)$ and let
$r(x,y)=(\lambda_{x}(y),\rho_{y}(x))$ be the corresponding solution on $G$.
Then $r$ restricts to a solution $r_{H}$ on $H$, which coincides with the
solution arising from the brace $(H,\star_{H},\cdot)$, and $r$ restricts to
the solution $r_{S}$ on $S$ given by $r_{S}(x,y)=(e_{G},xy)$ for all $x,y\in
S$.

###### Proof.

First suppose that $x,y\in H$. We have $\lambda_{x}(y)\in H$ by definition,
and $\rho_{y}(x)=\lambda_{x}(y)^{-1}xy\in H$; hence $r(x,y)\in H\times H$. A
short computation shows that the $\gamma$-function of the brace
$(H,\star_{H},\cdot)$ is precisely $\lambda_{x}(y)$, so the restriction of $r$
to $H$ coincides with the solution arising from this brace.

Now suppose that $x,y\in S$. Then we have

$\displaystyle\lambda_{x}(y)$ $\displaystyle=$ $\displaystyle b((x\odot
e_{N})(xy\odot e_{N}))$ $\displaystyle=$ $\displaystyle b(e_{N}e_{N})$
$\displaystyle=$ $\displaystyle e_{G},$

and $\rho_{y}(x)=xy\in S$; hence $r(x,y)=(e_{G},xy)\in S\times S$. ∎

It is natural to ask whether the original solution $r$ on $G$ can be
reconstructed from $r_{H}$ and $r_{S}$. In [4, Section 2] the notion of the
matched product of solutions is introduced: given sets $X,Y$, solutions
$r_{X},r_{Y}$ on these sets, and maps
$\alpha:Y\rightarrow\operatorname{\mbox{Perm}}(X)$ and
$\beta:X\rightarrow\operatorname{\mbox{Perm}}(Y)$, all satisfying various
compatibility conditions, we can construct a solution

$r_{X}\bowtie r_{Y}:(X\times Y)\times(X\times Y)\rightarrow(X\times
Y)\times(X\times Y),$

called the matched product of the solutions $r_{X}$ and $r_{Y}$ (via $\alpha$
and $\beta$).

We claim that the solution $r(x,y)=(\lambda_{x}(y),\rho_{y}(x))$ on $G$
obtained via Theorem 3.2 is isomorphic to a matched product of the solutions
$r_{S}$ and $r_{H}$ defined in Proposition 3.6. Define
$\alpha:H\to\operatorname{\mbox{Perm}}(S)$ by
$\alpha_{h}(s)=(\rho_{h^{-1}}(s^{-1}))^{-1}$ and
$\beta:S\to\operatorname{\mbox{Perm}}(H)$ by $\beta_{s}(h)=\lambda_{s}(h)$;
lengthy but routine calculations show that the conditions of [4, Definition 1]
are satisfied, and that the resulting matched product of solutions is given by

$\displaystyle(r_{S}\bowtie r_{H})((a,u),(b,v))$
$\displaystyle=((e,\beta_{a}\lambda_{\overline{u}}(v)),(\alpha^{-1}_{U}\rho_{\alpha_{\overline{u}}(b)}(a),\rho_{\beta_{\overline{a}}(v)}(u)))$
$\displaystyle=(\lambda_{(a,u)}(b,v),\rho_{(b,v)}(a,u))$

say, where $\overline{a}=\alpha^{-1}_{u}(a)$,
$\overline{u}=\beta^{-1}_{a}(u)$, $A=\alpha_{u}\lambda_{\overline{a}}(b)$,
$U=\beta_{a}\lambda_{\overline{u}}(v)$, $\overline{A}=\alpha^{-1}_{U}(A)$ and
$\overline{U}=\beta^{-1}_{A}(U)$.

Now define $f:S\times H\to G$ by $f(a,u)=u(\rho_{u}(a^{-1}))^{-1}$; this is a
bijection since $G=HS$ exactly and $\rho_{u}$ is bijective for all $u$. Then
employing the useful fact that $\lambda_{x}(hs)=\lambda_{x}(h)$ for any $x\in
G,h\in H,$ and $s\in S$, we have

$\displaystyle\lambda_{f(a,u)}f(b,v)$
$\displaystyle=\lambda_{u(\rho_{u}(a^{-1}))^{-1}}(v(\rho_{v}(b^{-1}))^{-1})$
$\displaystyle=\lambda_{a\lambda_{a^{-1}}(u)}(v)$ ($\rho_{v}(b^{-1})\in S$ by
Remark 3.5) $\displaystyle=\lambda_{a}\lambda_{\lambda_{a^{-1}}(u)}(v)$
$\displaystyle=\beta_{a}\lambda_{\beta_{a}^{-1}(u)}(v)$ (definition of
$\beta$) $\displaystyle=f(\lambda_{(a,u)}(b,v)).$

A similar but longer calculation establishes that
$f(\rho_{(b,v)}(a,u))=\rho_{f(b,v)}f(a,u)$. Therefore the solution of $G$
arising via Theorem 3.2 is isomorphic to the matched product of solutions
$r_{S}\bowtie r_{H}$.

##  4\. Examples of bracoids containing a brace

In this section we construct examples of bracoids satisfying Hypothesis 3.1,
often by employing Lemma 3.4.

A large family of bracoids arises from a certain quotienting procedure on
braces. Recall that a strong left ideal of a brace $(G,\star,\cdot)$ is a
subset $S$ of $G$ such that $(S,\star)\mathrel{\unlhd}(G,\star)$ and
$\gamma_{x}(s)\in S$ for all $x\in G$ and $s\in S$ (these conditions imply
that $(S,\cdot)\leq(G,\cdot)$). If $S$ is a strong left ideal of $G$ then
$(G,\cdot)$ acts transitively on the group $(G/S,\star)$ by left translation
of cosets and $(G,\cdot,G/S,\star,\odot)$ is a bracoid in which
$\operatorname{\mbox{Stab}}_{G}(eS)=S$ [19, Proposition 2.4]. If in addition
$S$ has a complement $H$ in $G$ then we can apply Theorem 3.2 to obtain a
right nondegenerate solution on $G$.

######  Example 4.1.

Suppose that $H,S$ are groups and that there is a homomorphism
$\alpha:S\rightarrow\operatorname{\mbox{Aut}}(H)$. Then the set $H\times S$,
together with the operations

$(h,s)\star(h^{\prime},s^{\prime})=(hh^{\prime},ss^{\prime}),\mbox{ and
}(h,s)\cdot(h^{\prime},s^{\prime})=(h\alpha_{s}(h^{\prime}),ss^{\prime})$

forms a brace (see [21, Example 1.6]). It is straightforward to check that the
$\gamma$-function of this brace is
$\gamma_{(h,s)}(h^{\prime},s^{\prime})=(\alpha_{s}(h^{\prime}),s^{\prime})$,
and it follows quickly that $\\{e\\}\times S$ is a strong left ideal. Since
$\\{e\\}\times S$ has a complement $H\times\\{e\\}$ in $(H\times S,\cdot)$,
Theorem 3.2 applies.

We note that in this example $H\times\\{e\\}$ is a normal complement to
$\\{e\\}\times S$ in $(H\times S,\cdot)$. Bracoids $(G,N)$ such that
$\operatorname{\mbox{Stab}}(e_{N})$ has a normal complement in $G$ have
important applications in Hopf-Galois theory; in the framework of [19, Section
5] they correspond to Hopf-Galois structures on so-called almost classically
Galois field extensions, which occupy a distinguished place in the theory: see
[8, Section 4], [16], [2], for example. It is therefore interesting to note
that Lemma 3.4 and Theorem 3.2 imply that bracoids $(G,N)$ such that
$\operatorname{\mbox{Stab}}(e_{N})$ has a normal complement in $G$ produce
solutions on $G$. See [18] for further explorations of these ideas.

In [13] it is shown that given a group $(G,\cdot)$ and an endomorphism $\psi$
of $(G,\cdot)$ with abelian image (an abelian map) we may define a new binary
operation $\star=\star_{\psi}$ on $G$ such that $(G,\star,\cdot)$ is a brace.
The strong left ideals in braces of this form are characterized in [14], as
follows: writing $\phi(x)=x\psi(x)^{-1}$ for all $x\in G$, a subgroup
$(S,\cdot)$ of $(G,\cdot)$ is a strong left ideal if and only if
$[G,\phi(S)]\leq S$ (where the commutators are computed with respect to
$\cdot$. If this is the case then we may apply [19, Proposition 2.4] and form
the bracoid $(G,G/S)$, as described above. It is possible for bracoids of this
form to contain a brace:

######  Example 4.2.

Let $p$ and $q$ be distinct odd prime numbers, and let

$G=\langle x,y,z\mid
x^{pq}=y^{2}=z^{2}=e,\;yxy^{-1}=zxz^{-1}=x^{-1},\;yz=zy\rangle\cong
C_{pq}\rtimes(C_{2}\times C_{2}).$

The map $\psi:G\rightarrow G$ defined by $\psi(x^{i}y^{j}z^{k})=y^{j}z^{k}$ is
an abelian map, and we have $\phi(x^{i}y^{j}z^{k})=x^{i}$. Now let $S=\langle
x^{q},z\rangle$; then $\psi(S)=\langle x^{q}\rangle$ and
$[G,\phi(S)]\subseteq\langle x^{q}\rangle\subseteq S$, so we may form the
bracoid $(G,G/S)$. Finally, note that the subgroup $S=\langle x^{q},z\rangle$
has a complement $H=\langle x^{p},y\rangle$ in $G$ (neither of $H,S$ is normal
in $G$). Therefore by Lemma 3.4 the bracoid $(G,G/S)$ contains a brace, and
Theorem 3.2 applies.

We can also construct examples of bracoids containing a brace that do not
arise as the quotient of a brace by a strong left ideal as in Example 4.1.
Recall that the holomorph of a group $N$ is the group
$\operatorname{\mbox{Hol}}(N)=N\rtimes\operatorname{\mbox{Aut}}(N)$, which
acts on $N$ via the formula $(\eta,\alpha)(\mu)=\eta\alpha(\mu)$ for all
$\eta,\mu\in N$ and $\alpha\in\operatorname{\mbox{Aut}}(N)$. If $J$ is a
transitive subgroup of $\operatorname{\mbox{Hol}}(N)$ then $(J,N)$ is a
bracoid in which $S=\operatorname{\mbox{Stab}}(e_{N})=\\{(\eta,\alpha)\in
J\mid\eta=e\\}$ [19, Theorem 2.8].

######  Example 4.3.

Let $N$ be an elementary abelian group of order $8$. By [1, Theorem 3.6] the
holomorph of $N$ contains a transitive subgroup $J$ isomorphic to the simple
group $\mathrm{GL}_{3}(\mathbb{F}_{2})$ of order $168$. The resulting bracoid
$(J,N)$ does not arise as the quotient of a brace by a strong left ideal: see
[19, Example 2.22]. However, writing $S=\operatorname{\mbox{Stab}}(e_{N})$ as
usual we have $|S|=21$ by the Orbit-Stabilizer Theorem, so taking $H$ to be a
Sylow $2$-subgroup of $J$ we have an exact factorization $J=HS$, and Lemma 3.4
implies that we may apply Theorem 3.2.

We can also use the holomorph framework to construct examples of bracoids that
do not contain a brace. We seek a group $N$ and a transitive subgroup $J$ of
$\operatorname{\mbox{Hol}}(N)$ such that $S=\operatorname{\mbox{Stab}}(e_{N})$
does not have a complement in $J$. Using results of Darlington [6, Section
3.1] we have

######  Example 4.4.

Let $p$ and $q$ be prime numbers with $p\equiv 1\pmod{q^{2}}$, and let $N$ be
a cyclic group of order $pq$, presented as

$N=\langle\sigma,\tau\mid\sigma^{p}=\tau^{q}=1,\;\sigma\tau=\tau\sigma\rangle.$

Let $\alpha$ be an automorphism of $N$ that has order $q^{2}$ and fixes $\tau$
(for example, let $\alpha$ be the $(p-1)/q^{2}$ power of a generator of
$\operatorname{\mbox{Aut}}(\langle\sigma\rangle)$), and let

$J=\langle(\sigma,\mathrm{id}),(\tau,\alpha)\rangle\leq\operatorname{\mbox{Hol}}(N).$

Then $J$ is a transitive subgroup of $\operatorname{\mbox{Hol}}(N)$ of order
$pq^{2}$, and $(J,N)$ is a bracoid.

We claim that $S=\operatorname{\mbox{Stab}}(e_{N})$ does not have a complement
in $J$. First we determine $S$. Since $\alpha$ fixes $\tau$ we have
$(\tau,\alpha)^{i}=(\tau^{i},\alpha^{i})$ for each $i$; it follows that
$S=\langle(e_{N},\alpha^{q})\rangle$ and that $|S|=q$. If $H$ is a complement
to $S$ in $J$ then in particular there exist $h\in H$ and $s\in S$ such that
$hs=(\tau,\alpha)$; hence $(\tau,\alpha^{i})\in H$ for some $i\equiv
1\pmod{q}$. An element of this form has order $q^{2}$, but $H$ must have order
$pq$, so this is impossible. Hence $S$ does not have a complement in $J$.

More generally, given a bracoid $(G,N,\odot)$ the image of the permutation
representation $\lambda_{\odot}:G\rightarrow\operatorname{\mbox{Perm}}(N)$ is
contained in $\operatorname{\mbox{Hol}}(N)$ [19, Theorem 2.8]. If
$S=\operatorname{\mbox{Stab}}(e_{N})$ has a complement $H$ in $G$ then the
transitive subgroup $\lambda_{\odot}(G)\leq\operatorname{\mbox{Hol}}(N)$
contains a regular subgroup (namely $\lambda_{\odot}(H)$). If
$\lambda_{\odot}$ is injective (that is, if the bracoid is reduced [19,
Definition 2.14]) then the converse is true. We expect the situation of
Example 4.4 to be relatively rare.

##  5\. Connections with solutions obtained from other algebraic structures

In [3] it is shown that left nondegenerate solutions may be obtained from left
semibraces. In this section we obtain a correspondence between left bracoids
containing a brace and left semibraces, and explain how the right
nondegenerate solution arising from the bracoid and the left nondegenerate
solution arising from the corresponding left semibrace are related to one
another.

We begin by recalling some definitions and properties concerning semibraces.
In [3] a left semibrace is defined to be a triple $(G,+,\cdot)$ in which
$(G,\cdot)$ is a group, $(G,+)$ is a left cancellative semigroup, and the
equation

$x\cdot(y+z)=x\cdot y+x\cdot(x^{-1}+y)$ (19)

holds for all $x,y,z\in G$.

In [10], Jespers and Van Antwerpen give an extended the definition of
semibraces that does not require the additive semigroup to be left
cancellative. Left semibraces here defined are what they call left
cancellative left semibrace. For simplicity’s sake we will use the term left
semibrace to refer to left cancellative left semibraces.

As when working with bracoids, we suppress the notation $\cdot$ where
possible.

If $(G,+,\cdot)$ is a left semibrace then for $x,y\in G$ we define

$\mathcal{L}_{x}(y)=x(x^{-1}+y)$

and

$\mathcal{R}_{y}(x)=\mathcal{L}_{x}(y)^{-1}xy=(x^{-1}+y)^{-1}y;$

the maps $\mathcal{L},\mathcal{R}$ then form a Lu-Yan-Zhu pair in which
$\mathcal{L}_{x}$ is bijective for each $x\in G$ and
$\mathcal{R}_{y}(G)=(G,+)$ for each $y\in G$, and so by Proposition 2.1 the
map $s:G\times G\rightarrow G\times G$ defined by
$s(x,y)=(\mathcal{L}_{x}(y),\mathcal{R}_{y}(x))$ for all $x,y\in G$ is a left
nondegenerate solution on $G$.

We note that we may rewrite Equation (19) as

$x\cdot(y+z)=x\cdot y+\mathcal{L}_{x}(y).$

The left semigroup $(G,+)$ admits a decomposition $(G+e,+)\oplus(E,+)$ in
which $(G+e,+)$ is a group and $E$ denotes the set of idempotents of $(G,+)$.
The identity element of $e$ of $(G,\cdot)$ lies in $E$, and an element $x\in
G$ lies in $E$ if and only if $x+e=e$. If $x\in E$ then $x+y=y$ for all $y\in
G$.

To connect semibraces with bracoids containing a brace, it is convenient to
work with bracoids using a particular presentation. Recall that if $(G,N)$ is
a bracoid containing a brace and $H$ is a subgroup of $G$ such that $(H,N)$ is
essentially a brace then we may use the bijection $b:N\rightarrow H$ to define
a second group operation on $H$ via the rule

$h\star k=b((h\odot e_{N})(k\odot e_{N}))\mbox{ for all }h,k\in H.$

We may also transport the transitive action of $G$ of $N$ to a transitive
action of $G$ on $H$ via the rule

$x\odot h=b(x\odot(h\odot e_{N}))=b(xh\odot e_{N})\mbox{ for all }x\in G\mbox{
and }h\in H;$

we then find that $(G,H)$ is a bracoid, and the pair
$(\mathrm{id},b):(G,N)\rightarrow(G,H)$ is an isomorphism of bracoids (see
[19, Section 4]).

If $(G,H)$ is a bracoid of this form containing a brace then we have $h\odot
e=h$ for all $h\in H$, and $h\odot k=hk$ for all $h,k\in H$. In particular, we
have $\gamma_{x}(y)\odot e=\gamma_{x}(y)$ for all $x,y\in G$, and so we have
$\lambda_{x}(y)=\gamma_{x}(y\odot e)$ in Theorem 3.2.

######  Theorem 5.1.

Let $G=(G,\cdot)$ be a group with an exact factorization $HS$. There is a
bijection between

  1. 1)

binary operations $\star$ on $H$ and transitive actions $\odot$ of $G$ on $H$
such that $(G,\cdot,H,\star,\odot)$ is a left bracoid containing a brace and
with $\operatorname{\mbox{Stab}}_{G}(e)=S$;

  2. 2)

binary operations $+$ on $G$ such that $(G,+,\cdot)$ is a left semibrace in
which $G+e=H$ and $E=S$.

###### Proof.

First suppose that $(G,\cdot,H,\star,\odot)$ is a left bracoid containing a
brace and with $\operatorname{\mbox{Stab}}(e)=S$. For $x,y\in G$ let
$\lambda_{x}(y)=\gamma_{x}(y\odot e)$ and $\rho_{y}(x)=\lambda_{x}(y)^{-1}xy$,
and define

$x+y=y\lambda_{y^{-1}}(x).$

We claim that $(G,+,\cdot)$ is a left semibrace. First we show that $+$ is
associative on $G$. For $x,y,z\in G$ we have

$\displaystyle x+(y+z)=\;$ $\displaystyle x+z\lambda_{z^{-1}}(y)$
$\displaystyle=\;$ $\displaystyle
z\lambda_{z^{-1}}(y)\lambda_{(z\lambda_{z^{-1}}(y))^{-1}}(x)$
$\displaystyle=\;$ $\displaystyle
z\lambda_{z^{-1}}(y)\lambda_{\rho_{y}(z^{-1})y^{-1}}(x)$ (since
$\lambda_{z^{-1}}(y)\rho_{y}(z^{-1})=z^{-1}y$,)

and

$\displaystyle(x+y)+z=\;$ $\displaystyle y\lambda_{y^{-1}}(x)+z$
$\displaystyle=\;$ $\displaystyle z\lambda_{z^{-1}}(y\lambda_{y^{-1}}(x))$
$\displaystyle=\;$ $\displaystyle
z\lambda_{z^{-1}}(y)\lambda_{\rho_{y}(z^{-1})}(\lambda_{y^{-1}}(x))$ (by (6) )
$\displaystyle=\;$ $\displaystyle
z\lambda_{z^{-1}}(y)\lambda_{\rho_{y}(z^{-1})y^{-1}}(x)$ ( since $\lambda$ is
a homomorphism.)

Thus $+$ is associative, and so $(G,+)$ is a semigroup.

Next we show that $(G,+)$ is left cancellative. For $x,y,z\in G$ we have

$\displaystyle x+y=x+z$ $\displaystyle\Rightarrow\;$ $\displaystyle
y\lambda_{y^{-1}}(x)=z\lambda_{z^{-1}}(x)$ $\displaystyle\Rightarrow\;$
$\displaystyle x\rho_{x}(y^{-1})^{-1}=x\rho_{x}(z^{-1})^{-1}$
$\displaystyle\Rightarrow\;$ $\displaystyle\rho_{x}(y^{-1})=\rho_{x}(z^{-1})$
$\displaystyle\Rightarrow\;$ $\displaystyle y=z$ ( since $\rho_{x}$ is
injective. )

Thus $(G,+)$ is left cancellative.

Finally we show that the left semibrace relation is satisfied. For $x,y,z\in
G$ we have

$\displaystyle xy+x(x^{-1}+z)=\;$ $\displaystyle
xy+xz\lambda_{z^{-1}}(x^{-1})$ $\displaystyle=\;$ $\displaystyle
xz\lambda_{z^{-1}}(x^{-1})\lambda_{xz\lambda_{z^{-1}}(x^{-1})}(xy)$
$\displaystyle=\;$ $\displaystyle
xz\lambda_{z^{-1}}(x^{-1})\lambda_{\rho_{x^{-1}}(z^{-1})}(xy)$
$\displaystyle=\;$ $\displaystyle xz\lambda_{z^{-1}}(x^{-1}xy)$ (by (6))
$\displaystyle=\;$ $\displaystyle xz\lambda_{z^{-1}}(y)$ $\displaystyle=\;$
$\displaystyle x(y+z).$

Thus $(G,+,\cdot)$ is indeed a left semibrace.

For the remaining claims, we note that for $x\in G$ we have
$x+e=e\lambda_{e^{-1}}(x)=\lambda_{e}(x)$; thus $G+e=\lambda_{e}(G)=H$.
Finally, we have

$\displaystyle x+x=x$ $\displaystyle\Leftrightarrow\;$ $\displaystyle
x\lambda_{x^{-1}}(x)=x$ $\displaystyle\Leftrightarrow\;$
$\displaystyle\lambda_{x^{-1}}(x)=e$ $\displaystyle\Leftrightarrow\;$
$\displaystyle\lambda_{x^{-1}}(x)\odot e=e$ (note $\lambda_{x^{-1}}(x)\in H$)
$\displaystyle\Leftrightarrow\;$ $\displaystyle\gamma_{x^{-1}}(x\odot e)=e$
(relationship between $\lambda$ and $\gamma$) $\displaystyle\Leftrightarrow\;$
$\displaystyle x\odot e=e$ ($\gamma_{x}\in\operatorname{\mbox{Aut}}(H,\star)$)
$\displaystyle\Leftrightarrow\;$ $\displaystyle x\in S.$

Thus $E=S$.

Conversely, suppose that $+$ is a binary operation on $G$ such that
$(G,+,\cdot)$ is a left semibrace in which $G+e=H$ and $E=S$. For $h,k\in H$
define

$h\star k=k+h;$

then $(H,\star)$ is the opposite group to $(H,+)$. In addition, for $x\in G$
and $h\in H$ define $x\odot h=xh+e$. We claim that $(G,\cdot,H,\star,\odot)$
is a bracoid containing a brace in which $\operatorname{\mbox{Stab}}(e)=S$. It
is straightforward to verify that $\odot$ is a transitive left action of
$(G,\cdot)$ on $H$; it remains to show that the bracoid relation (8) is
satisfied. By the definition of the binary operation $\star$, it suffices to
show that

$x\odot(h+k)=(x\odot h)-(x\odot e)+(x\odot k)$ (20)

for all $x\in G$ and $h,k\in H$. Beginning with the right hand side of (20) we
have:

$\displaystyle(x\odot h)-(x\odot e)+(x\odot k)$ $\displaystyle=$
$\displaystyle(x\odot h)-(x\odot e)+(xk+e)$ $\displaystyle=$
$\displaystyle(x\odot h)-(x\odot e)+(x(e+k)+e)$ $\displaystyle=$
$\displaystyle(x\odot h)-(x\odot e)+(xe)+\mathcal{L}_{x}(k)+e$
$\displaystyle=$ $\displaystyle(x\odot h)-(x\odot
e)+(xe+e)+\mathcal{L}_{x}(k)+e$ $\displaystyle=$ $\displaystyle
xh+\mathcal{L}_{x}(k)+e$ $\displaystyle=$ $\displaystyle x(h+k)+e$
$\displaystyle=$ $\displaystyle x\odot(h+k).$

Hence (20) holds, and so $(G,\cdot,H,\star,\odot)$ is indeed a left bracoid.

For the remaining claims: first we note that $x\in S$ if and only if $x+e=e$,
which occurs if and only if $x\in E$; second, for $h\in H$ we have $h\odot
e=h+e=h$, so the action of $H$ on itself is transitive, and so $(H,+,\cdot)$
is a brace.

The constructions described above are mutually inverse: if
$(G,\cdot,H,\star,\odot)$ is a bracoid as in the statement of the theorem and
$(G,+,\cdot)$ is the corresponding semibrace, then the bracoid corresponding
to $(G,+,\cdot)$ is $(G,\cdot,H,\hat{\star},\hat{\odot})$, where for all
$h,k\in H$ and $x\in G$ we have $h\;\hat{\star}\;k=k+h$ and
$x\;\hat{\odot}\;h=xh+e$. Since $k+h=h\star k$, we quickly see that
$\hat{\star}$ conincides with $\star$. Since $\hat{\odot}$ and $\odot$ are
both transitive actions of $(G,\cdot)$ on $H$, they coincide if and only if
$x\;\hat{\odot}\;e=x\odot e$ for all $x\in G$. Writing $x=hs$, with $h\in H$
and $s\in S$, we have:

$\displaystyle x\;\hat{\odot}\;e=\;$ $\displaystyle hs\;\hat{\odot}\;e$
$\displaystyle=\;$ $\displaystyle h\;\hat{\odot}\;e$ ( since $s$ is idempotent
in $(G,+)$) $\displaystyle=\;$ $\displaystyle h+e$ $\displaystyle=\;$
$\displaystyle h$ $\displaystyle=\;$ $\displaystyle h\odot e$
$\displaystyle=\;$ $\displaystyle hs\odot e$ $\displaystyle=\;$ $\displaystyle
x\odot e.$

Thus the bracoid corresponding to $(G,+,\cdot)$ is the original bracoid
$(G,\cdot,H,\star,\odot)$.

Conversely, if $(G,+,\cdot)$ is a semibrace as in the statement of the theorem
and $(G,\cdot,H,\star,\odot)$ is the corresponding bracoid, then the semibrace
corresponding to $(G,\cdot,H,\star,\odot)$ is $(G,\hat{+},\cdot)$, where
$x\;\hat{+}\;y=y\gamma_{y^{-1}}(x)$. By the bracoid relation (7) we may write

$\gamma_{y^{-1}}(x)=y^{-1}\odot((y\odot e)\star(x\odot e)),$

and so

$\displaystyle x\;\hat{+}\;y=\;$ $\displaystyle y\lambda_{y^{-1}}(x)$
$\displaystyle=\;$ $\displaystyle y(y^{-1}\odot((y\odot e)\star(x\odot e)))$
$\displaystyle=\;$ $\displaystyle y(y^{-1}\odot((y+e)\star(x+e)))$
$\displaystyle=\;$ $\displaystyle y(y^{-1}\odot(x+e+y+e))$ $\displaystyle=\;$
$\displaystyle y(y^{-1}(x+y+e)+e)$ $\displaystyle=\;$ $\displaystyle
x+y+e+\mathcal{L}_{y}(e)$ $\displaystyle=\;$ $\displaystyle x+y$

Thus the semibrace corresponding to $(G,\cdot,H,\star,\odot)$ is the original
semibrace $(G,+,\cdot)$.

Hence we obtain the bijection in the statement of the theorem. ∎

######  Proposition 5.2.

Suppose that the left bracoid $(G,\cdot,H,\star,\odot)$ and the left semibrace
$(G,+,\cdot)$ correspond as in Theorem 5.1. Let $r$ be the right nondegenerate
solution arising from $(G,\cdot,H,\star,\odot)$, let $s$ be the left
nondegenerate solution arising from $(G,+,\cdot)$, and define
$\iota,\tau:G\times G\rightarrow G\times G$ by

$\iota(x,y)=(x^{-1},y^{-1})\mbox{ and }\tau(x,y)=(y,x).$

Then

$s(x,y)=\iota\tau r\tau\iota(x,y).$

###### Proof.

Let $x,y\in G$ and recall that $r(x,y)=(\lambda_{x}(y),\rho_{y}(x))$ and
$s(x,y)=(\mathcal{L}_{x}(y),\mathcal{R}_{y}(x))$. From the first of these we
obtain quickly that

$\iota\tau
r\tau\iota(x,y)=(\rho_{x^{-1}}(y^{-1})^{-1},\lambda_{y^{-1}}(x^{-1})^{-1}).$

The relationship between $+$ and $\lambda$ implies that
$x^{-1}+y=y\lambda_{y^{-1}}(x^{-1})$, so

$\lambda_{y^{-1}}(x^{-1})^{-1}=(x^{-1}+y)^{-1}y=\mathcal{R}_{y}(x).$

Now we recall that

$\lambda_{y^{-1}}(x^{-1})\rho_{x^{-1}}(y^{-1})=y^{-1}x^{-1}=(xy)^{-1},$

so

$\rho_{x^{-1}}(y^{-1})^{-1}\lambda_{y^{-1}}(x^{-1})^{-1}=xy=\mathcal{L}_{x}(y)\mathcal{R}_{y}(x).$

We have already shown that $\lambda_{y^{-1}}(x^{-1})^{-1}=\mathcal{R}_{y}(x)$,
so we must also have $\rho_{x^{-1}}(y^{-1})^{-1}=\mathcal{L}_{x}(y)$. This
completes the proof. ∎

######  Remark 5.3.

If $G=H$ then bracoids and the semibraces considered in Theorem 5.1 are
braces, and the corresponding solutions are bijective and nondegenerate. In
this case our constructions pair a brace with its opposite, and a solution
with its inverse. If we adjust our method of constructing a semibrace from a
bracoid by defining $\hat{\lambda}_{x}(y)=x\odot(y\odot e)\star(x^{-1}\odot
e)$ (this is the $\gamma$-function of the opposite bracoid) and
$x+y=y\hat{\lambda}_{y^{-1}}(x)$ then we obtain a correspondence such that in
the case $G=H$ the braces (and solutions) coincide. This may be more
satisfying from an algebraic perspective, but the connection between the left
and right degenerate solutions appears to be much less transparent under this
formulation.

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