# Near braces and $p$-deformed braided groups

Anastasia Doikou  and  Bernard Rybołowicz  (A.Doikou $\&$ B. Rybołowicz)
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, and
Maxwell Institute for Mathematical Sciences, Edinburgh, UK<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Motivated by recent findings on the derivation of parametric non-involutive
solutions of the Yang-Baxter equation we reconstruct the underlying algebraic
structures, called near braces. Using the notion of the near braces we produce
new multi-parametric, non-degenerate, non-involutive solutions of the set-
theoretic Yang-Baxter equation. These solutions are generalisations of the
known ones coming from braces and skew braces. Bijective maps associated to
the inverse solutions are also constructed. Furthermore, we introduce the
generalized notion of $p$-deformed braided groups and $p$-braidings and we
show that every $p$-braiding is a solution of the braid equation. We also show
that certain multi-parametric maps within the near braces provide special
cases of $p$-braidings.

###### Key words and phrases:

Groups; skew braces, braiding, set-theoretic Yang-Baxter equation

###### 2010 Mathematics Subject Classification:

16S70; 16Y99; 08A99

##  1\. Introduction

The aim of the present study is two-fold: on the one hand, motivated by recent
fidings on parametric solutions [17] of the set-theoretic [18, 21] Yang-Baxter
equation (YBE) [3, 38] we derive the underlying algebraic structure associated
to these solutions. On the other hand using the derived algebraic frame we
introduce novel multi-parametric classes of solutions of the YBE.

It is well established now that braces, first introduced by Rump [36],
describe all non-degenerate involutive solutions of the YBE, whereas skew
braces were later introduced to describe non-involutive, non-degenerate
solutions of the YBE [27]. Indeed, based on the ideas of [36] and [27] and on
recent findings regarding parametric solutions of the set-theoretic YBE [17]
we construct the generic algebraic structure, called near brace, that provides
solutions to the set-theoretic braid equation. Moreover, motivated by the
definition of the braided group [34] and the work of [26], we introduce an
extensive definition of a $p$-deformed braided group and $p$-braidings, which
are solutions of the set-theoretic braid equation. All the parametric
solutions derived here are indeed $p$-braidings. It is worth noting that the
study of solutions of the set-theoretic Yang-Baxter equation and the
associated algebraic structures have created a particularly active new field
during the last decade or so (see for instance [1, 2, 10, 9, 12, 11]). The key
observation is that by relaxing more and more conditions on the underlying
algebraic structures one identifies more general classes of solutions (see
e.g. [8, 9, 28, 29, 32, 33, 37], [23]–[25]). It is also worth noting that
interesting links with quantum integrable systems [13, 14] as well the quasi-
triangular quasi-bialgebras [15]-[17] have been recently established, opening
up new intriguing paths of investigations.

We briefly describe what is achieved in this study, and in particular what are
the findings in each section. In the remaining of this section we review some
necessary ideas on non-degenerate set-theoretic solutions of the YBE and the
associated algebraic structures, i.e. braces and skew braces. In Section 2
inspired by the parametric solutions of the YBE introduced in [17] we
reconstruct the generic associated algebraic structure called near brace. In
fact, every near brace can turn to a skew brace by defining a suitably
modified (deformed) addition; this is described in Theorem 2.6. The key idea
is to simultaneously consider $\check{r}$ and its inverse given that we are
exclusively interested in non-degenerate solutions of the braid equation.
Having derived the underlying algebraic structure we move to Subsection 2.1
where we extract multi-parametric bijective maps and hence to identify non-
degenerate, multi-parametric solutions of the YBE as well as their inverses.
In Subsection 2.2 we provide a generalized definition of the braided group and
braidings ($p$-braidings, $p$ stands for parametric) by relaxing some of the
conditions appearing in the definition of [34] (see also relevant findings in
[26].) Furthermore, we show that the generalized $p$-braidings are non-
degenerate solutions of the YBE and the bijective maps coming for the near
braces provide automatically $p$-braidings.

### Preliminaries

Before we start our analysis and present our findings in the subsequent
section we review below basic preliminary notions relevant to our
investigation. Specifically, we recall the problem of solving the set-
theoretic braid equation and some fundamental results. Let $X=\\{x_{1},\ldots
x_{n}\\}$ be a set and ${\check{r}}^{z}:X\times X\rightarrow X\times X,$ where
$z\in X$ is a fixed parameter, first introduced in [17]. We denote

${\check{r}}^{z}(x,y)=(\sigma^{z}_{x}(y),\tau^{z}_{y}(x)).$ (1.1)

We say that $\check{r}^{z}$ is non-degenerate if $\sigma^{z}_{x}$ and
$\tau^{z}_{y}$ are bijective maps, and $(X,{\check{r}})$ is a set-theoretic
solution of the braid equation if

$({\check{r}}^{z}\times\mbox{id})(\mbox{id}\times{\check{r}}^{z})({\check{r}}^{z}\times\mbox{id})=(\mbox{id}\times{\check{r}}^{z})({\check{r}}^{z}\times\mbox{id})(\mbox{id}\times{\check{r}}^{z}).$
(1.2)

The map $\check{r}$ is called involutive if
$\check{r}^{z}\circ\check{r}^{z}=\mbox{id}.$

We also introduce the map $r:X\times X\rightarrow X\times X,$ such that
$r^{z}=\check{r}^{z}\pi,$ where $\pi:X\times X\rightarrow X\times X$ is the
flip map: $\pi(x,y)=(y,x).$ Hence,
$r^{z}(y,x)=(\sigma_{x}^{z}(y),\tau_{y}^{z}(x)),$ and it satisfies the YBE:

$r^{z}_{12}\ r^{z}_{13}\ r^{z}_{23}=r^{z}_{23}\ r^{z}_{13}\ r^{z}_{12},$ (1.3)

where we denote $r^{z}_{12}(y,x,w)=(\sigma_{x}^{z}(y),\tau_{y}^{z}(x),w),$
$r^{z}_{23}(w,y,x)=(w,\sigma_{x}^{z}(y),\tau_{y}^{z}(x))$ and
$r^{z}_{13}(y,w,x)=(\sigma_{x}^{z}(y),w,\tau^{z}_{y}(x)).$

We review below the constraints arising by requiring $(X,\check{r}^{z})$ to be
a solution of the braid equation ([18, 21, 35, 36]). Let,

$({\check{r}}^{z}\times\mbox{id})(\mbox{id}\times{\check{r}}^{z})({\check{r}}^{z}\times\mbox{id})(\eta,x,y)=(L_{1},L_{2},L_{3}),$
$(\mbox{id}\times{\check{r}}^{z})({\check{r}}^{z}\times\mbox{id})(\mbox{id}\times{\check{r}}^{z})(\eta,x,y)=(R_{1},R_{2},R_{3}),$

where, after employing expression (1.1) we identify:

$\displaystyle
L_{1}=\sigma^{z}_{\sigma^{z}_{\eta}(x)}(\sigma^{z}_{\tau^{z}_{x}(\eta)}(y)),\quad
L_{2}=\tau^{z}_{\sigma^{z}_{\tau^{z}_{x}(\eta)}(y)}(\sigma^{z}_{\eta}(x)),\quad
L_{3}=\tau^{z}_{y}(\tau^{z}_{x}(\eta)),$ $\displaystyle
R_{1}=\sigma^{z}_{\eta}(\sigma^{z}_{x}(y)),\quad
R_{1}=\sigma^{z}_{\tau^{z}_{\sigma^{z}_{x}(y)}(\eta)}(\tau^{z}_{y}(x)),\quad
R_{3}=\tau^{z}_{\tau^{z}_{y}(x)}(\tau^{z}_{\sigma^{z}_{x}(y)}(\eta)).$

And by requiring $L_{i}=R_{i},$ $i\in\\{1,2,3\\}$ we obtain the following
fundamental constraints for the associated maps:

$\displaystyle\sigma^{z}_{\eta}(\sigma^{z}_{x}(y))=\sigma^{z}_{\sigma^{z}_{\eta}(x)}(\sigma^{z}_{\tau^{z}_{x}(\eta)}(y)),$
(1.4)
$\displaystyle\tau^{z}_{y}(\tau^{z}_{x}(\eta))=\tau^{z}_{\tau^{z}_{y}(x)}(\tau^{z}_{\sigma^{z}_{x}(y)}(\eta)),$
(1.5)
$\displaystyle\tau^{z}_{\sigma^{z}_{\tau^{z}_{x}(\eta)}(y)}(\sigma^{z}_{\eta}(x))=\sigma^{z}_{\tau^{z}_{\sigma^{z}_{x}(y)}(\eta)}(\tau^{z}_{y}(x)).$
(1.6)

Note that the constraints above are the ones of the set-theoretic solution
(1.1), given that $z$ is a fixed element of the set, i.e. for different
elements $z$ we obtain in principle distinct solutions of the braid equation.

We review now the basic definitions of the algebraic structures that provide
set-theoretic solutions of the braid equation, such as left skew braces and
braces. We also present some key properties associated to these structures
that will be useful when formulating some of the main findings of the present
study, summarized in Section 2.

######  Definition 1.1 ([35, 36, 12]).

A left skew brace is a set $B$ together with two group operations
$+,\circ:B\times B\to B$, the first is called addition and the second is
called multiplication, such that for all $a,b,c\in B$,

$a\circ(b+c)=a\circ b-a+a\circ c.$ (1.7)

If $+$ is an abelian group operation $B$ is called a left brace. Moreover, if
$B$ is a left skew brace and for all $a,b,c\in B$ $(b+c)\circ a=b\circ
a-a+c\circ a$, then $B$ is called a skew brace. Analogously if $+$ is abelian
and $B$ is a skew brace, then $B$ is called a brace.

######  Remark 1.2.

In the literature often left brace is just called a brace and left skew brace
is called a skew brace. In that case various authors call skew braces two-
sided skew braces.

The additive identity of a left skew brace $B$ will be denoted by $0$ and the
multiplicative identity by $1$. In every left skew brace $0=1$. Indeed, this
is easy to show:

$\displaystyle a\circ b=a\circ(b+0)\ \Rightarrow\ a\circ b=a\circ b-a+a\circ
0\ \Rightarrow\ a\circ 0=a\ \Rightarrow\ 0=1.$

The two theorems that follow concern the case wher the parameter $z=1$. Rump
showed the following theorem for involutive set-theoretic solutions.

######  Theorem 1.3.

(Rump’s theorem, [35, 36]). Assume $(B,+,\circ)$ is a left brace. If the map
$\check{r}_{B}:B\times B\to B\times B$ is defined as
${\check{r}}_{B}(x,y)=(\sigma_{x}(y),\tau_{y}(x))$, where
$\sigma_{x}(y)=x\circ y-x$, $\tau_{y}(x)=t\circ x-t$, and $t$ is the inverse
of $\sigma_{x}(y)$ in the circle group $(B,\circ),$ then $(B,\check{r}_{B})$
is an involutive, non-degenerate solution of the braid equation.
Conversely, if $(X,\check{r})$ is an involutive, non-degenerate solution of
the braid equation, then there exists a left brace $(B,+,\circ)$ (called an
underlying brace of the solution $(X,\check{r})$) such that $B$ contains $X,$
$\check{r}_{B}(X\times X)\subseteq X\times X,$ and the map $\check{r}$ is
equal to the restriction of $\check{r}_{B}$ to $X\times X.$ Both the additive
$(B,+)$ and multiplicative $(B,\circ)$ groups of the left brace $(B,+,\circ)$
are generated by $X.$

######  Remark 1.4 (Rump).

Let $(N,+,\cdot)$ be an associative ring. If for $a,b\in N$ we define

$a\circ b=a\cdot b+a+b,$

then $(N,+,\circ)$ is a brace if and only if $(N,+,\cdot)$ is a radical ring.

Guarnieri and Vendramin [27], generalized Rump’s result to left skew braces
and non-degenerate, non-involutive solutions.

######  Theorem 1.5 (Theorem [27]).

Let $B$ be a left skew brace, then the map $\check{r}_{GV}:B\times B\to
B\times B$ given for all $a,b\in B$ by

$\check{r}_{GV}(a,b)=(-a+a\circ b,\ (-a+a\circ b)^{-1}\circ a\circ b)$

is a non-degenerate solution of set-theoretic YBE.

##  2\. Set-theoretic solutions of the YBE and near braces

In this section starting from a generic $z$-parametric set-theoretic solution
of the YBE [17] we reconstruct the underlying algebraic structure, which is
similar to a skew brace. Indeed, we introduce in what follows suitable
algebraic structures that satisfy the fundamental constraints (1.4)-(1.6),
i.e. provide solutions of the braid equation and generalize the findings of
Rump and Guarnieri $\&$ Vendramin. The following generalizations are greatly
inspired by recent results in [17].

For the rest of the subsection we consider $X$ to be a set with an arbitrary
group operation $\circ:X\times X\to X$, with a neutral element $1\in X$ and an
inverse $x^{-1}\in X,$ for all $x\in X.$ There also exists a family of
bijective functions indexed by $X,$ $\sigma^{z}_{x}:X\to X,$ such that
$y\mapsto\sigma^{z}_{x}(y),$ where $z\in X$ is some fixed parameter. We then
define another binary operation $+:X\times X\to X,$ such that

$y+x:=x\circ\sigma^{z}_{x^{-1}}(y\circ z)\circ z^{-1}.$ (2.1)

For convenience we will omit henceforth the fixed $z\in X$ in
$\sigma^{z}_{x}(y)$ and simply write $\sigma_{x}(y).$

######  Remark 2.1.

The operation $+$ is associative if and only if for all $x,y,c\in X$,

$\sigma_{c^{-1}}(y\circ z^{-1}\circ\sigma_{z\circ
y^{-1}}(x))=\sigma_{c^{-1}}(y)\circ
z^{-1}\circ\sigma_{(c\circ\sigma_{c^{-1}}(y)\circ z^{-1})^{-1}}(x).$ (2.2)

From now on we will assume that the operation $+$ is associative, that is
condition (2.2) holds.

Also, we recall that we focus only on non-degenerate, invertible solutions
$\check{r}.$ Given that $\sigma_{x}$ and $\tau_{y}$ are bijections the inverse
maps also exist such that

$\sigma^{-1}_{x}(\sigma_{x}(y))=\sigma_{x}(\sigma^{-1}_{x}(y))=y,\quad\tau^{-1}_{y}(\tau_{y}(x))=\tau_{y}(\tau^{-1}_{y}(x))=x$
(2.3)

Let the inverse $\check{r}^{-1}(x,y)=(\hat{\sigma}_{x}(y),\hat{\tau}_{y}(x))$
exist with $\hat{\sigma}_{x},\ \hat{\tau}_{y}$ being also bijections, that
satisfy:

$\sigma_{\hat{\sigma}_{x}(y)}(\hat{\tau}_{y}(x))=x=\hat{\sigma}_{\sigma_{x}(y)}(\tau_{y}(x)),\quad\tau_{\hat{\tau}_{y}(x)}(\hat{\sigma}_{x}(y))=y=\hat{\tau}_{\tau_{y}(x)}(\sigma_{x}(y)).$
(2.4)

Taking also into consideration (2.3) and (2.4) and that $\sigma_{x},\tau_{y}$
and $\hat{\sigma}_{x},\hat{\tau}_{y}$ are bijections, we deduce:

$\hat{\sigma}^{-1}_{\sigma_{x}(y)}(x)=\tau_{y}(x),\quad\hat{\tau}^{-1}_{\tau_{y}(x)}(y)=\sigma_{x}(y).$
(2.5)

We assume that the map $\hat{\sigma}$ appearing in the inverse matrix
$\check{r}^{-1}$ has the general form

$\hat{\sigma}_{x}(y):=x\circ(x^{-1}\circ z_{2}+y\circ z_{1})\circ\xi,$ (2.6)

where the parameters $z_{1,2},\ \xi$ are to be identified. The derivation of
$\check{r}$ goes hand in hand with the derivations of $\check{r}^{-1}$ (see
details in [17] and later in the text when deriving a generic $\check{r}$ and
its inverse). In the involutive case the two maps coincide and $x+y=y+x.$
However, for any non-degenerate, non-involutive solution both bijective maps
$\sigma_{x},\hat{\sigma}_{x}$ should be considered together with the
fundamental conditions (2.4).

We present below a series of useful Lemmas that will lead to one of our main
theorems.

######  Remark 2.2.

This is just a reminder of a well known fact. We recall that $\sigma_{x}$ is a
bijective function. Recalling also definiton (2.1):

$\sigma_{x}(y_{1})=\sigma_{x}(y_{2})\Leftrightarrow y_{1}\circ
z^{-1}+x^{-1}=y_{2}\circ z^{-1}+x^{-1},$ (2.7)

which implies right cancellation of $+.$ Similarly $\hat{\sigma}_{x}$ is a
bijective function and this leads to left cancellation.

######  Lemma 2.3.

For all $x\in X$, the operations $+x,\ x+:X\to X$ are bijections.

###### Proof.

Let $y_{1},y_{2}\in X$ be such that $y_{1}+x=y_{2}+x$, then

$x\circ\sigma_{x^{-1}}(y_{1}\circ z)\circ
z^{-1}=x\circ\sigma_{x^{-1}}(y_{2}\circ z)\circ
z^{-1}\implies\sigma_{x^{-1}}(y_{1}\circ z)=\sigma_{x^{-1}}(y_{2}\circ z),$

since $\circ$ is a group operation and $\sigma_{x^{-1}}$ is injective, we get
that $y_{1}=y_{2}$ and $+x$ is injective for any $x\in X$. From the
surjectivity, we observe that since $\sigma_{x^{-1}}$ is bijective, we can
consider $d=\sigma^{-1}_{x^{-1}}(x^{-1}\circ c\circ z)\circ z^{-1}$, one can
easily see that $d+x=c$, and since $c$ is arbitrary we get that $+x$ is a
surjection. Thus $+x$ is a bijection. Similarly, from the bijectivity of
$\hat{\sigma}_{x}$ and (2.6) we show that $x+$ is also a bijection. ∎

We now introduce the notion of neutral elements in $(X,+)$

######  Lemma 2.4.

Let $(X,+)$ be a semigroup, then for all $x\in X$ there exists $0_{x}\in X$
such that $0_{x}+x=x.$ Moreover, for all $x,y\in X,$ $0_{x}=0_{y}=0,$ i.e. $0$
is the unique left neutral element. The left neutral element $0$ is also right
neutral element.

###### Proof.

Notice that due to bijectivity of $\sigma_{x}$, we can consider the element

$0_{x}:=\sigma^{-1}_{x^{-1}}(z)\circ z^{-1}\in X,$

recall also the definition of $+$ in (2.1), then simple computation shows:

$0_{x}+x=x\circ\sigma_{x^{-1}}(\sigma^{-1}_{x^{-1}}(z))\circ z^{-1}=x\circ
z\circ z^{-1}=x.$ (2.8)

We have,

$0_{x}+x=x\Rightarrow 0_{x}+x+y=x+y,$

but also

$0_{x+y}+x+y=x+y.$

The last two equations lead to $0_{x}+x+y=0_{x+y}+x+y,$ and due Lemma 2.3
right cancellation holds, so we get that $0_{x}=0_{x+y}$ for all $y\in X$.
Observe that by the Lemma 2.3, $x+$ is a surjection, that is for all $w\in X$
exists $y\in X$ such that $x+y=w$, that is $0:=0_{x}=0_{w}$ for all $w\in X$.

Moreover, $0+y=y\Rightarrow x+0+y=x+y$ and due to associativity and right
cancellativity (Lemma 2.3) we get $x+0=x,$ for all $x\in X$. ∎

######  Lemma 2.5.

Let $0$ be the neutral element in $(X,+)$, then for all $x\in X$ there exists
$-x\in X,$ such that $-x+x=0$ (left inverse). Moreover, $-x\in X$ is a right
inverse, i.e. $x+(-x)=0$ $\forall x\in X.$ That is $(X,+,0)$ is a group.

###### Proof.

Observe that due to bijectivity of $\sigma_{x}$, we consider the element

$-x:=\sigma_{x^{-1}}^{-1}(x^{-1}\circ 0\circ z)\circ z^{-1}\in X.$ (2.9)

Simple computation shows it is a left inverse,

$-x+x=x\circ\sigma_{x^{-1}}(\sigma_{x^{-1}}^{-1}(x^{-1}\circ 0\circ z)\circ
z^{-1}\circ z)\circ z^{-1}=0.$

By associativity we deduce that $x+(-x)+x=0+x$, we get that $x+(-x)=0$, and
$-x$ is the inverse. ∎

To conclude, having only assumed associativity in $+$ (2.1) we deduced that
$(X,+)$ is a group. We may now present our main findings described in the
following central theorem.

######  Theorem 2.6.

Let $(X,\circ)$ be a group and $\check{r}:X\times X\to X\times X$ be such that
$\check{r}(x,y)=(\sigma_{x}(y),\tau_{x}(y))$ is a non-degenerate solution of
the set-theoretic braid equation. Moreover, we assume that:

  1. (A)

The pair $(X,+)$ ($+$ is defined in (2.1)) is a group.

  2. (B)

There exists $\phi:X\to X$ such that for all $a,b,c\in X$ $a\circ(b+c)=a\circ
b+\phi(a)+a\circ c.$

  3. (C)

For $h\in\\{z,\xi\\}\in X$ appearing in $\sigma_{x}(y)$ and
$\hat{\sigma}_{x}(y)$ there exist $\widehat{\phi}:X\to X$ such that for all
$a,b\in X$ $(a+b)\circ h=a\circ h+\widehat{\phi}(h)+b\circ h.$

  4. (D)

The neutral element $0$ of $(X,+)$ has a left and right distributivity.

Then for all $a,b,c\in X$ the following statements hold:

  1. (1)

$\phi(a)=-a\circ 0$ and $~{}\widehat{\phi}(h)=-0\circ h$,

  2. (2)

$\sigma_{a}(b)=(a\circ b\circ z^{-1}-a\circ 0+1)\circ z=a\circ b-a\circ 0\circ
z+z.$

  3. (3)

$~{}a-a\circ 0=1$ and (i) $0\circ 0=-1$ (ii) $1+1=0^{-1}.$

  4. (4)

If $~{}z_{2}\circ\xi=0^{-1},$ $~{}0\circ\xi=z_{1}\circ\xi=z^{-1}\circ 0^{-1},$
then (i) $\hat{\sigma}_{a}(b)\circ\hat{\tau}_{b}(a)=a\circ
b=\sigma_{a}(b)\circ\tau_{b}(a)$ (ii) $-a\circ 0+a=1.$

###### Proof.

  1. (1)

In the following the distributivity rule $a\circ(b+c)=a\circ b+\phi(a)+b\circ
c$ holds, then

$a=a\circ(0+1)=a\circ 0+\phi(a)+a\circ 1\ \Rightarrow\phi(a)=-a\circ 0,$

Also, for those $z\in X$ such that $(a+b)\circ z=a\circ z+\hat{\phi}(z)+b\circ
z$ we have

$z=(0+1)\circ z=0\circ z+\hat{\phi}(z)+z\ \Rightarrow\hat{\phi}(z)=-0\circ z.$

  2. (2)

Using the distributivity rule we obtain

$\sigma_{a}(b)=(a\circ b\circ z^{-1}-a\circ 0+1)\circ z.$ (2.10)

Before we move on with the rest of the proof it is useful to calculate
$(-a)\circ z,$ indeed:

$\displaystyle 0\circ z$ $\displaystyle=$ $\displaystyle(a-a)\circ
z\Rightarrow 0\circ z=a\circ z-0\circ z+(-a)\circ z$ (2.11)
$\displaystyle\Rightarrow$ $\displaystyle(-a)\circ z=0\circ z-a\circ z+0\circ
z.$

The latter then leads to the following convenient identity (see also [17] and
Lemma 2.9 later in the text)

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

and hence (2.10) becomes $~{}\sigma_{a}(b)=a\circ b-a\circ 0\circ z+z.$

  3. (3)

Due to the fact that $\check{r}$ satisfies the braid equation we may employ
(1.4) and the general distributivity rule (see also (2.10)):

$\displaystyle\sigma_{a}(\sigma_{b}(c))$ $\displaystyle=$
$\displaystyle(a\circ\sigma_{b}(c)\circ z^{-1}-a\circ 0+1)\circ z$
$\displaystyle=$ $\displaystyle(a\circ b(c\circ z^{-1}+b^{-1})\circ z\circ
z^{-1}-a\circ 0+1)\circ z$ $\displaystyle=$ $\displaystyle\big{(}a\circ b\circ
c\circ z^{-1}-a\circ b\circ 0+a-a\circ 0+1\big{)}\circ z.$

But due to condition (1.4) and by setting $c=0\circ z,$ we deduce that
$a-a\circ 0=\zeta,$ for all $a\in X$ ($\zeta$ is a fixed element in $X$), but
for $a=1$ we immediately obtain $\zeta=1,$ i.e.

$a-a\circ 0=1.$ (2.12)

(i) By setting $a=0$ in (2.12) we have $0\circ 0=-1.$

(ii) $0\circ(1+1)=0\circ 1-0\circ 0+0\circ 1\Rightarrow
0\circ(1+1)=1\Rightarrow 1+1=0^{-1}.$

  4. (4)

For the following we set $~{}z_{2}\circ\xi=0^{-1},$
$~{}0\circ\xi=z_{1}\circ\xi=z^{-1}\circ 0^{-1}.$

(i) Recall the form of $\hat{\sigma}_{a}(b)$ (2.6), and use the distributivity
rules, then

$\hat{\sigma}_{a}(b)=z_{2}\circ\xi-a\circ 0\circ\xi+a\circ b\circ
z_{1}\circ\xi.$ (2.13)

We consider now the fixed constants: $z_{2}\circ\xi=0^{-1},$
$0\circ\xi=z_{1}\circ\xi=z^{-1}\circ 0^{-1}.$ Note that if $z$ satisfies the
right distributivity then so does $z^{-1}$ (see Proposition 2.3 in [17]) and
also $0\circ z,$ given that $0$ has left and right distributivity. We recall
relations (2.4) for the maps, then

$\displaystyle\sigma_{\hat{\sigma}_{a}(b)}(\hat{\tau}_{b}(a))=a\Rightarrow\hat{\sigma}_{a}(b)\circ\hat{\tau}_{b}(a)-\hat{\sigma}_{a}(b)\circ
0\circ z+z=a\Rightarrow$
$\displaystyle\hat{\sigma}_{a}(b)\circ\hat{\tau}_{b}(a)-(0^{-1}-a\circ
z^{-1}\circ 0^{-1}+a\circ b\circ z^{-1}\circ 0^{-1})\circ 0\circ z+z=a$
$\displaystyle\hat{\sigma}_{a}(b)\circ\hat{\tau}_{b}(a)-a\circ
b+a-z+z=a\Rightarrow$
$\displaystyle\hat{\sigma}_{a}(b)\circ\hat{\tau}_{b}(a)=a\circ b.$

Similarly,
$\hat{\sigma}_{\sigma_{a}(b)}(\tau_{b}(a))=a\Rightarrow\sigma_{a}(b)\circ\tau_{b}(a)=a\circ
b.$

(ii) We consider $z_{2}\circ\xi=0^{-1},$ $0\circ\xi=z_{1}\circ\xi=z^{-1}\circ
0^{-1},$ and consequently, as shown above,
$\sigma_{a}(b)\circ\tau_{b}(a)=a\circ
b=\hat{\sigma}_{a}(b)\circ\hat{\tau}_{b}(a).$ We also recall condition (1.5)
of the braid equation and $a\circ b=\sigma_{a}(b)\circ\tau_{b}(a),$ indeed

$\tau_{c}(\tau_{b}(a))=\sigma_{\tau_{b}(a)}(c)^{-1}\circ\sigma_{a}(b)^{-1}\circ
a\circ b\circ c$

and due to the form of (1.5) we conclude

$\sigma_{a}(b)\circ\sigma_{\tau_{b}(a)}(c)=\sigma_{a}(\sigma_{b}(c))\circ\sigma_{\tau_{\sigma_{b}(c)}(a)}(\tau_{c}(b)).$
(2.14)

We focus on

$\displaystyle\sigma_{a}(b)\circ\sigma_{\tau_{b}^{z}(a)}(c)$ $\displaystyle=$
$\displaystyle\sigma_{a}(b)\circ(\tau_{b}(a)\circ c\circ
z^{-1}-\tau_{b}(a)\circ 0+1)\circ z$ (2.15) $\displaystyle=$
$\displaystyle(a\circ b\circ c\circ z^{-1}-a\circ b\circ 0+\sigma_{a}(b))\circ
z$ $\displaystyle=$ $\displaystyle(a\circ b\circ c\circ z^{-1}-a\circ b\circ
0+a\circ b-a\circ 0\circ z+z)\circ z.$

Taking into consideration the form of (2.14) and (2.15) and the fact that
$b\circ c=\sigma_{c}(c)\circ\tau_{c}(b),$ we conclude that $\forall a\in X,$
$-~{}a\circ 0+a=\hat{\zeta},$ where $\hat{\zeta}\in X$ is a fixed element, and
for $a=1$ we deduce that $\hat{\zeta}=1,$ i.e. $-a\circ 0+a=1.$

∎

######  Remark 2.7.

Due to $a-a\circ 0=-a\circ 0+a=1,$ $\forall a\in B,$ we deduce that $a+1=1+a,$
for all $a\in B.$

We call the algebraic construction deduced in Theorem 2.6 a near brace, in
analogy to near rings, specifically:

######  Definition 2.8.

A near brace is a set $B$ together with two group operations $+,\circ:B\times
B\to B$, the first is called addition and the second is called multiplication,
such that for all $a,b,c\in B$,

$a\circ(b+c)=a\circ b-a\circ 0+a\circ c,.$ (2.16)

We denote by $0$ the neutral element of the $(B,+)$ group and by $1$ the
neutral element of the $(B,\circ)$ group. We say that a near brace $B$ is an
abelian near brace if $+$ is abelian.

We say that a near brace $B$ is a singular near brace if for all $a\in B,$
$~{}a-a\circ 0=-a\circ 0+a=1.$ Near braces will be particularly useful in the
next subsection, where we introduce a method of finding solutions depending on
multiple parameters.

In the special case where $0=1,$ we recover a skew brace. We also show below
some useful properties for near braces.

######  Lemma 2.9.

[17] Let $(B,+,\circ)$ be a near brace, then

  1. (1)

$a\circ(-b)=a\circ 0-a\circ b+a\circ 0.$

  2. (2)

Condition (2.16) is equivalent to the following condition:

$\forall{a,b,c,d\in B}\ \ a\circ(b-c+d)=a\circ b-a\circ c+a\circ d.$

###### Proof.

  1. (1)

$a\circ(b-b)=a\circ 0\Rightarrow a\circ b-a\circ 0+a\circ(-b)=a\circ 0,$ which
leads to $a\circ(-b)=a\circ 0-a\circ b+a\circ 0.$

  2. (2)

Let (2.16) hold then

$\displaystyle a\circ(b-c+d)=a\circ(b-c)-a\circ 0+a\circ d=$ (2.17)
$\displaystyle a\circ b-a\circ 0+a\circ 0-a\circ c+a\circ 0-a\circ 0+a\circ
d=$ (2.18) $\displaystyle a\circ b-a\circ c+a\circ d.$ (2.19)

Conversely, let $a\circ(b-c+d)=a\circ b-a\circ c+a\circ d$ hold, then

$a\circ(b+c)=a\circ(b-0+c)=a\circ b-a\circ 0+a\circ c.$ (2.20)

∎

######  Example 2.10.

Let $(B,\circ)$ be a group with neutral element 1 and define
$a+b:=a\circ\kappa^{-1}\circ b,$ where $1\neq\kappa\in B$ is an element of the
center of $(B,\circ).$ Then $(B,+,\circ)$ is a singular near brace with
neutral element $0=\kappa,$ and we call it the trivial near brace 111We are
indebted to Paola Stefanelli for sharing this example with us. .

######  Example 2.11.

Let us consider the following near-truss introduced in [7, Page 710]:

$\displaystyle\mathrm{Q}(O(i))=\left\\{\frac{m}{2p+1}+\frac{n}{2q+1}i\;|\;p,q\in{\mathbb{Z}},\;\mbox{$m+n$
is an odd integer}\right\\}.$

Then $\mathrm{Q}(O(i))$ together with operations $a+_{i}b=a-i+b$ and $a\circ
b=a\cdot b$ forms a near brace, where $+,\cdot$ are addition and
multiplication of complex numbers, respectively.

######  Definition 2.12.

Let $(B,+,\circ)$ and $(S,+,\circ)$ be near braces. We say that $f:B\to S$ is
a near brace morphism if for all $a,b\in B$,

$f(a+b)=f(a)+f(b)\qquad\&\qquad f(a\circ b)=f(a)\circ f(b).$

######  Lemma 2.13.

Let $f:X\to X$ be a map, such that for all $a,b\in X$ $f(a\circ b-a\circ
z+z)=f(a)\circ f(b)-f(a)\circ 0\circ z+z,$ and there is $e\in X,$ $f(e)=1.$ If
such a map $f$ exists then $0=1.$

###### Proof.

We assume that such map $f:X\to X$ exists. Then for all $a\in X$,

$f(z)=f(a\circ z-a\circ z+z)=f(a)\circ f(z)-f(a)\circ 0\circ z+z,$

and by setting $a=e,$ $f(e)=1$:

$f(z)=f(z)-0\circ z+z\implies 0\circ z=z\implies 0=1,$

where the last implication follows from the fact that $z$ is invertible. ∎

######  Remark 2.14.

Observe that since every bijection is surjective, the preceding Lemma states
that if $\sigma^{{}^{\prime}}_{a}(b):=a\circ b-a\circ 0\circ z+z$ gives a
solution of YBE, it is isomorphic to the solution given by
$\sigma_{a}(b):=a\circ b-a\circ z+z$ if $0=1,$ that is near skew brace is a
left skew brace.

######  Corollary 2.15.

Let $B$ be near brace. Then by the Lemma 2.9 (2), the triple
$\mathrm{T}(B):=(B,[-,-,-],\circ)$, where for all $a,b,c\in B,$
$[a,b,c]:=a-b+c,$ is a near-truss such that $(B,\circ)$ is a group. Thus the
triple $(B,+_{1},\circ)$, where $a+_{1}b:=a-1+b$ for all $a,b\in B,$ is a left
skew brace. That is, the near brace $\mathrm{T}(B)$ and the left skew brace
$(B,+_{1},\circ)$ are isomorphic as near-trusses.

######  Lemma 2.16.

Let $(X,+,\circ)$ be a near brace and $z,w\in X$ satisfy the right
distributivity. Consider also the maps $\sigma,\sigma^{\prime}:X\times X\to X$
such that $\sigma_{a}(b)=a\circ b-a\circ 0\circ z+z$ and
$\sigma^{\prime}_{a}(b)=a\circ b-a\circ 0\circ w+w.$ If
$\sigma_{a}(b)=\sigma^{\prime}_{a}(b)$ then $z^{-1}\circ w-1=w-z.$

###### Proof.

The proof is straightforward by setting $a=z^{-1}\circ 0^{-1}$ in both
$\sigma_{a}(b)$ and $\sigma^{\prime}_{a}(b).$ ∎

###  2.1. Generalized bijective maps $\&$ solutions of the braid equation

Inspired by the findings of the preceding section we introduce below more
general, multi-parametric bijective maps $\sigma^{p}_{a},\tau^{p}_{b}$ ($p$
stands for parametric) that provide solutions of the set-theoretic braid
equation.

######  Proposition 2.17.

Let $(B,+,\circ)$ be a near brace and let us denote $\sigma^{p}_{a}(b):=a\circ
b\circ z_{1}-a\circ\xi+z_{2}$ and
$\tau^{p}_{b}(a):=\sigma^{p}_{a}(b)^{-1}\circ a\circ b$, where $a,b\in B$, and
$h\in\\{\xi,z_{i}\\}\in B,$ $i\in\\{1,2\\}$ are fixed parameters, such that
there exist $c_{1},c_{2}\in B$ such that for all $a,b,c\in B,$ $(a-b+c)\circ
h=a\circ h-b\circ h+c\circ h,$ $~{}a\circ z_{2}\circ z_{1}-a\circ\xi=c_{1}$
and $-a\circ\xi+a\circ z_{1}\circ z_{2}=c_{2}.$

Then for all $a,b,c\in B$ the following properties hold:

  1. (1)

$\sigma^{p}_{a}(b)\circ\tau^{p}_{b}(a)=a\circ b.$

  2. (2)

$\sigma^{p}_{a}(\sigma^{p}_{b}(c))=a\circ b\circ c\circ z_{1}\circ
z_{1}-a\circ b\circ\xi\circ z_{1}+c_{1}+z_{2}$.

  3. (3)

$\sigma^{p}_{a}(b)\circ\sigma^{p}_{\tau^{p}_{b}(a)}(c)=a\circ b\circ c\circ
z_{1}+c_{2}-a\circ\xi\circ z_{2}+z_{2}\circ z_{2}.$

###### Proof.

Let $a,b,c\in B$, then:

  1. (1)

$\sigma^{p}_{a}(b)\circ\tau^{p}_{b}(a)=\sigma^{p}_{a}(b)\circ\sigma^{p}_{a}(b)^{-1}\circ
a\circ b=a\circ b.$

  2. (2)

To show condition (2) we recall that $a\circ z_{2}\circ
z_{1}-a\circ\xi=c_{1}.$ Then,

$\displaystyle\sigma^{p}_{a}(\sigma^{p}_{b}(c))$ $\displaystyle=$
$\displaystyle\sigma^{p}_{a}(b\circ c\circ z_{1}-b\circ\xi+z_{2})$
$\displaystyle=$ $\displaystyle a\circ(b\circ c\circ
z_{1}-b\circ\xi+z_{2})\circ z_{1}-a\circ\xi+z_{2}$ $\displaystyle=$
$\displaystyle a\circ b\circ c\circ z_{1}\circ z_{1}-a\circ b\circ\xi\circ
z_{1}+a\circ z_{2}\circ z_{1}-a\circ\xi+z_{2}$ $\displaystyle=$ $\displaystyle
a\circ b\circ c\circ z_{1}\circ z_{1}-a\circ b\circ\xi\circ
z_{1}+c_{1}+z_{2}.$

  3. (3)

To show condition (3) we use (1) and $-a\circ\xi+a\circ z_{1}\circ
z_{2}=c_{2}.$

$\displaystyle\sigma^{p}_{a}(b)\circ\sigma^{p}_{\tau^{p}_{b}(a)}(c)$
$\displaystyle=\sigma^{p}_{a}(b)\circ(\tau^{p}_{b}(a)\circ c\circ
z_{1}-\tau^{p}_{b}(a)\circ\xi+z_{2})$
$\displaystyle=\sigma^{p}_{a}(b)\circ\tau^{p}_{a}(b)\circ c\circ
z_{1}-\sigma^{p}_{a}(b)\circ\tau^{p}_{a}(b)\circ\xi+\sigma^{p}_{a}(b)\circ
z_{2}$ $\displaystyle=a\circ b\circ c\circ z_{1}-a\circ
b\circ\xi+\sigma^{p}_{a}(b)\circ z_{2}$ $\displaystyle=a\circ b\circ c\circ
z_{1}-a\circ b\circ\xi+(a\circ b\circ z_{1}-a\circ\xi+z_{2})\circ z_{2}$
$\displaystyle=a\circ b\circ c\circ z_{1}+c_{2}-a\circ\xi\circ
z_{2}+z_{2}\circ z_{2}.$

∎

######  Example 2.18.

A simple example of the above generic maps is the case where $z_{1}\circ
z_{2}=\xi\circ 0=z_{2}\circ z_{1},$ then $c_{1}=c_{2}=1.$

Having showed the fundamental properties above we may now proceed in proving
the following theorem.

######  Theorem 2.19.

Let $(B,+,\circ)$ be a near brace and $z\in B$ such that there exist
$c_{1},c_{2}\in B$ such that for all $a,b,c\in B$, $(a-b+c)\circ z_{i}=a\circ
z_{i}-b\circ z_{i}+c\circ z_{i},$ $i\in\\{1,2\\},$ $a\circ z_{2}\circ
z_{1}-a\circ\xi=c_{1}$ and $-a\circ\xi+a\circ z_{1}\circ z_{2}=c_{2}.$ We
define a map $\check{r}:B\times B\to B\times B$ given by

$\check{r}(a,b)=(\sigma^{p}_{a}(b),\tau^{p}_{b}(a)),$

where $\sigma^{p}_{a}(b)=a\circ b\circ z_{1}-a\circ\xi+z_{2},$
$\tau^{p}_{b}(a)=\sigma^{p}_{a}(b)^{-1}\circ a\circ b.$ The pair
$(B,\check{r})$ is a solution of the braid equation.

###### Proof.

To prove this we need to show that the maps $\sigma,\tau$ satisfy the
constraints (1.4)-(1.6). To achieve this we use the properties proven in
Proposition 2.17.

Indeed, from Proposition 2.17, (1) and (2), it follows that (1.4) is
satisfied, i.e.

$\sigma^{p}_{\eta}(\sigma^{p}_{x}(y))=\sigma^{p}_{\sigma^{p}_{\eta}(x)}(\sigma^{p}_{\tau^{p}_{x}(\eta)}(y)).$

We observe that

$\tau^{p}_{b}(\tau^{p}_{a}(\eta))=T^{p}\circ\tau^{p}_{a}(\eta)\circ
b=T^{p}\circ t^{p}\circ\eta\circ a\circ b=T^{p}\circ
t^{p}\circ\eta\circ\sigma^{p}_{a}(b)\circ\tau^{p}_{b}(a),$

where $T^{p}=\sigma^{p}_{\tau^{z}_{a}(\eta)}(b)^{-1}$ and
$t^{p}=\sigma^{p}_{\eta}(a)^{-1}$ (the inverse in the circle group). Due to
(1), (2), (3) of Proposition 2.17 we then conclude that

$\tau^{p}_{b}(\tau^{p}_{a}(\eta))=\tau^{p}_{\tau^{p}_{b}(a)}(\tau^{p}_{\sigma^{p}_{a}(b)}(\eta)),$

so (1.5) is also satisfied.

To prove (1.6), we employ (3), (1) of Proposition 2.17 and use the definition
of $\tau^{p}$,

$\sigma^{p}_{\tau^{p}_{\sigma^{p}_{x}(y)}(\eta)}(\tau^{p}_{y}(x))=\sigma^{p}_{\eta}(\sigma_{x}^{p}(y))^{-1}\circ\sigma^{p}_{\eta}(x)\circ\sigma^{p}_{\tau^{p}_{x}(\eta)}(y)=\tau^{p}_{\sigma^{p}_{\tau^{p}_{x}(\eta)}(y)}(\sigma^{p}_{\eta}(x)).$

Thus, (1.6) is satisfied, and
$\check{r}(a,b)=(\sigma^{p}_{a}(b),\tau^{p}_{b}(a))$ is a solution of the
braid equation. ∎

######  Lemma 2.20.

Let $(B,+,\circ)$ be a near brace and $z,w\in B$ satisfy the right
distributivity. Consider also the multi-parametric maps
$\sigma^{p},\sigma^{{}^{\prime}p}:B\times B\to B$ as defined in Proposition
2.17, such that $\sigma^{p}_{a}(b)=a\circ b\circ z_{1}-a\circ\xi+z_{2}$ and
$\sigma^{{}^{\prime}p}_{a}(b)=a\circ b\circ z_{2}-a\circ\xi+z_{1}.$ If
$\sigma_{a}(b)=\sigma^{\prime}_{a}(b)$ then $0\circ z_{1}^{-1}\circ
z_{2}=z_{2}-z_{1}.$

###### Proof.

The proof is straightforward by setting $a=0\circ\xi^{-1}$ and $b=\xi\circ
z_{1}^{-1}$ in both $\sigma^{p}_{a}(b)$ and $\sigma^{{}^{\prime}p}_{a}(b).$ ∎

######  Remark 2.21.

In the special case where $z_{1}=1$ and $\xi=z_{2}=z$ we recover the
$\sigma^{z}_{x}(y),\ \tau_{y}^{z}(x)$ bijective maps and the $\check{r}_{z}$
solutions of the braid equation introduced in [17].

######  Example 2.22.

We consider the brace $\mathrm{Q}(O(i))$ from Example 2.11. Then, we can have
for instance the following choice of parameters:

  1. (1)

$z_{1}=z_{2}=i,\ c_{1}=c_{2}=i,\ \xi=-1,$ then $\sigma^{p}_{a}(b)=a\circ
b\circ i+_{i}a$

  2. (2)

$z_{1}=i,\ z_{2}=-i,\ c_{1}=c_{2}=i,\ \xi=1,$ then $\sigma^{p}_{a}(b)=a\circ
b\circ i-_{i}a$

  3. (3)

$z_{1}=5,\ z_{2}=3,\ c_{1}=c_{2}=i,\ \xi=15,$ then $\sigma^{p}_{a}(b)=a\circ
b\circ 5-_{i}15\circ a+_{i}3.$

In the following Proposition we provide the explicit expressions of the
inverse $\check{r}$-matrices as well as the corresponding bijective maps.

######  Proposition 2.23.

Let $\check{r}^{*},\check{r}:B\times B\to B\times B$ such that
$\check{r}^{*}:(x,y)\mapsto(\hat{\sigma}^{p}_{x}(y),\hat{\tau}^{p}_{y}(x)),$
$\check{r}:(x,y)\mapsto(\sigma^{p}_{x}(y),\tau^{p}_{y}(x))$ be solutions of
the braid equation. Then the following statements hold.

  1. (1)

$\check{r}^{*}=\check{r}^{-1}$ if and only if

$\displaystyle\hat{\sigma}^{p}_{\sigma_{x}^{p}(y)}(\tau_{y}^{p}(x))=x,\
\hat{\tau}^{p}_{\tau^{p}_{y}(x)}(\sigma^{p}_{x}(y))=y\ \mbox{and}\
\sigma^{p}_{\hat{\sigma}_{x}^{p}(y)}(\hat{\tau}_{y}^{p}(x))=x,\
\tau^{p}_{\hat{\tau}^{p}_{y}(x)}(\hat{\sigma}^{p}_{x}(y))=y.$ (2.21)

  2. (2)

Let $\sigma^{p}_{x}(y)=x\circ y\circ z_{1}-x\circ\xi+z_{2},$ $\
\tau^{p}_{y}(x)=\sigma^{p}_{x}(y)^{-1}\circ x\circ y,$ and $\xi,z_{i}\in B$
$i\in\\{1,2\\}$ are fixed elements, such that there exists $c_{1},c_{2}\in B$
such that for all $a,b,c\in B,$ $(a-b+c)\circ z_{i}=a\circ z_{i}-b\circ
z_{i}+c\circ z_{i},$ $a\circ z_{2}\circ z_{1}-a\circ\xi=c_{1}$ and
$-a\circ\xi+a\circ z_{1}\circ z_{2}=c_{2}.$Then
$\hat{\sigma}^{p}_{x}(y)=\hat{z}_{2}-x\circ\hat{\xi}+x\circ
y\circ\hat{z}_{1},$ $\ \hat{\tau}^{p}_{x}(y)=\hat{\sigma}^{p}_{x}(y)^{-1}\circ
x\circ y,$ where $\hat{\xi}=\xi^{-1},$ $\hat{z}_{1,2}=z_{1,2}\circ\xi^{-1}.$

###### Proof.

We prove the two parts of Proposition 2.23 below:

  1. (1)

If $\check{r}^{*}=\check{r}^{-1},$ then
$\check{r}\check{r}^{*}=\check{r}^{*}\check{r}=\mbox{id}$ and

$\check{r}\check{r}^{*}(x,y)=(\sigma^{p}_{\hat{\sigma}_{x}(y)}(\hat{\tau}^{p}_{y}(x)),\tau^{p}_{\hat{\tau}^{p}_{y}(x)}(\hat{\sigma}^{p}_{x}(y))).$

Thus

$\sigma^{z}_{\hat{\sigma}_{x}^{p}(y)}(\hat{\tau}_{y}^{p}(x))=x,\quad\tau^{p}_{\hat{\tau}^{p}_{y}(x)}(\hat{\sigma}^{z}_{x}(y))=y.$

And vice versa if
$\sigma^{p}_{\hat{\sigma}_{x}^{p}(y)}(\hat{\tau}_{y}^{p}(x))=x,$
$\tau^{p}_{\hat{\tau}^{p}_{y}(x)}(\hat{\sigma}^{p}_{x}(y))=y,$ then it
automatically follows that $\check{r}^{*}=\check{r}^{-1}.$

Similarly, $\check{r}^{*}\check{r}(x,y)=(x,y)$ leads to
$\hat{\sigma}^{p}_{\sigma_{x}^{p}(y)}(\tau_{y}^{p}(x))=x,$
$\hat{\tau}^{p}_{\tau^{p}_{y}(x)}(\sigma^{p}_{x}(y))=y,$ and vice versa.

  2. (2)

For the second part of the Proposition it suffices to show (2.21). Indeed, we
recall that that $\hat{\xi}=\xi^{-1}$ and
$\hat{z}_{1,2}=z_{1,2}\circ\xi^{-1},$ then

$\displaystyle\hat{\sigma}^{p}_{\sigma^{p}_{x}(y)}(\tau^{p}_{y}(x))$
$\displaystyle=$
$\displaystyle\hat{z}_{2}-\sigma^{p}_{x}(y)\circ\hat{\xi}+\sigma^{p}_{x}(y)\circ\tau_{y}^{p}(x)\circ\hat{z}_{1}$
$\displaystyle=$
$\displaystyle\hat{z}_{2}-\sigma^{p}_{x}(y)\circ\hat{\xi}+x\circ
y\circ\hat{z}_{2}$ $\displaystyle=$ $\displaystyle\hat{z}_{2}-(x\circ y\circ
z_{1}-x\circ\xi+z_{2})\circ\hat{\xi}+x\circ y\circ\hat{z}_{1}=x.$

Also,
$\hat{\tau}^{p}_{\tau^{p}_{y}(x)}(\sigma^{p}_{x}(y))=x^{-1}\circ\sigma^{p}_{x}(y)\circ\tau_{y}^{p}(x)=y.$

Similarly, we show

$\displaystyle\sigma^{p}_{\hat{\sigma}_{x}^{p}(y)}(\hat{\tau}_{y}^{p}(x))$
$\displaystyle=$
$\displaystyle\hat{\sigma}^{p}_{x}(y)\circ\hat{\tau}_{y}^{p}(x)\circ
z_{1}-\hat{\sigma}^{p}_{x}(y)\circ\xi+z_{2}$ $\displaystyle=$ $\displaystyle
x\circ y\circ z_{1}-(\hat{z}_{2}-x\circ\hat{\xi}+x\circ
y\circ\hat{z}_{1})\circ\xi+z_{2}=x.$

And as above we immediately deduce that
$\tau^{p}_{\hat{\tau}^{p}_{y}(x)}(\hat{\sigma}^{p}_{x}(y))=y.$ ∎

With this we conclude our analysis on the general bijective maps coming from
near braces and the corresponding solutions of the braid equation.

###  2.2. $p$-deformed braided groups and near braces

Motivated by the definition of braided groups and braidings in [34] as well as
the relevant work presented in [26] we provide a generic definition of the
$p$-deformed braided group and braiding that contain extra fixed parameters,
i.e. for multi-parametric braidings ($p$-braidings).

######  Definition 2.24.

Let $(G,\circ)$ be a group, $m(x,y)=x\circ y$ and $\check{r}$ is an invertible
map $\check{r}:G\times G\to G\times G,$ such that for all $x,y\in G,$
$\check{r}(x,y)=(\sigma^{p}_{x}(y),\tau^{p}_{y}(x)),$ where $\sigma^{p}_{x},\
\tau^{p}_{y}$ are bijective maps in $G$. The map $\check{r}$ is called a
$p$-braiding operator (and the group is called $p$-braided) if

  1. (1)

$x\circ y=\sigma^{p}_{x}(y)\circ\tau^{p}_{y}(x).$

  2. (2)

$(\mbox{id}\times m)\ \check{r}_{12}\ \check{r}_{23}(x,y,w)=(f^{p}_{x\circ
y}(w),\ f^{p}_{x\circ y}(w)^{-1}\circ x\circ y\circ w).$

  3. (3)

$(m\times\mbox{id})\ \check{r}_{23}\ \check{r}_{12}(x,y,w)=(g^{p}_{x}(y\circ
w),\ g^{p}_{x}(y\circ w)^{-1}\circ x\circ y\circ w).$

for some bijections $f_{x}^{p},g_{x}^{p}:G\to G,$ given for all $x\in G.$

######  Proposition 2.25.

Let $(G,\circ)$ be a group, and the invertible map $\check{r}:G\times G\to
G\times G,$ $\check{r}(x,y)=(\sigma^{p}_{x}(y),\tau^{p}_{y}(x)),$ be a
$p$-braiding operator for the group $G.$ Then $\check{r}$ is a non-degenerate
solution of the braid equation.

###### Proof.

We start from the LHS of condition (2) of Proposition 2.25

$\displaystyle(\mbox{id}\times m)\ \check{r}_{12}\ \check{r}_{23}\
(x,y,w)=(\sigma^{p}_{x}(\sigma^{p}_{y}(w)),\
\tau^{p}_{\sigma^{p}_{y}(w)}(x)\circ\tau^{p}_{w}(y)),$

which leads to

$\sigma^{p}_{x}(\sigma^{p}_{y}(w))=f^{p}_{x\circ
y}(w)=f^{p}_{\sigma^{p}_{x}(y)\circ\tau^{p}_{y}(x)}(w)=\sigma^{p}_{\sigma^{p}_{x}(y)}(\sigma^{p}_{\tau^{p}_{y}(x)}(w))$

i.e. the fundamental condition (1.4) is satisfied. Moreover, using condition
(1) we show

$\displaystyle\tau^{p}_{\sigma^{p}_{y}(w)}(x)\circ\tau^{p}_{w}(y)=\sigma^{p}_{x}(\sigma^{p}_{y}(w))^{-1}\circ
x\circ\sigma^{p}_{y}(w)\circ\sigma^{p}_{y}(w)^{-1}\circ y\circ w=f^{p}_{x\circ
y}(w)^{-1}\circ x\circ y\circ w,$

as expected compatible with condition (2) of Proposition 2.25.

Similarly, from the LHS of condition (3)

$\displaystyle(m\times\mbox{id})\ \check{r}_{23}\ \check{r}_{12}\
(x,y,w)=(\sigma_{x}^{p}(y)\circ\sigma^{p}_{\tau^{p}_{y}(x)}(w),\
\tau^{p}_{w}(\tau^{p}_{y}(x))).$

The latter expression leads to

$\sigma_{x}^{p}(y)\circ\sigma^{p}_{\tau^{p}_{y}(x)}(w)=g_{x}^{p}(y\circ w).$
(2.22)

Also, via condition (1)

$\displaystyle\tau^{p}_{w}(\tau^{p}_{y}(x))$ $\displaystyle=$
$\displaystyle\sigma^{p}_{\tau^{p}_{y}(x)}(w)^{-1}\circ\sigma_{x}^{p}(y)^{-1}\circ
x\circ y\circ w=g^{p}_{x}(y\circ w)^{-1}\circ x\circ y\circ w$ (2.23)
$\displaystyle=$
$\displaystyle\tau^{p}_{\tau^{p}_{w}(y)}(\tau^{p}_{\sigma^{p}_{y}(w)}(x))$

compatible with condition (3) of Proposition 2.25, and this shows condition
(1.6).

Having shown properties 1.4 and (1.5) and taking into account that $x\circ
y=\sigma_{x}^{p}(y)\circ\tau_{y}^{p}(x)$ we also show condition (1.6), i.e. we
conclude that $\check{r},$ as defined in Proposition 2.25, is a solution of
the braid equation. ∎

######  Lemma 2.26.

Let $B$ be a near brace, and consider the map $\check{r}:B\times B\to B\times
B,$ $\check{r}(x,y)=(\sigma^{p}_{x}(y),\ \tau^{p}_{y}(x))$ of Proposition
2.19. Then $\check{r}$ is a $p$-braiding.

###### Proof.

The proof is straightfroward via Proposition 2.17. Indeed, all the conditions
of the $p$-braiding Definition 2.24 are satisfied and:
$f^{p}_{a}(b)=a\circ b\circ z_{1}\circ z_{1}-a\circ\xi\circ
z_{1}+c_{1}+z_{2},$ $~{}g^{p}_{a}(b)=a\circ b\circ z_{1}+c_{2}-a\circ\xi\circ
z_{2}+z_{2}\circ z_{2}.$ ∎

With this we conclude our analysis on $p$-braidings and their connection to
the YBE and the notion of the near braces. One of the fundamental open
problems in this frame and a natural next step is the solution of the set-
theoretic reflection equation for this new class of solutions of the set-
theoretic YBE. We hope to address this problem and generalize the notion of
the $p$-braiding to include the reflection equation, in the near future.
Another key question, which we hope to tackle soon, is what the effect of non-
associativity in $(X,+)$ on the construction of the algebraic structures
emerging from solutions of the set-theoretic YBE would be. This is quite a
challenging problem, the analysis of which will yield yet more generalized
classes of solutions of the YBE.

### Acknowledgments

Support from the EPSRC research grant EP/V008129/1 is acknowledged.

## References

  * [1] D. Bachiller, Counterexample to a conjecture about braces, J. Algebra, 453 (2016) 160–176.
  * [2] D. Bachiller, F. Cedó, E. Jespers and J. Okniński, Iterated matched products of finite braces and simplicity; new solutions of the Yang-Baxter equation, Trans. Amer. Math. Soc. 370 (2018) 4881-4907.
  * [3] R.J. Baxter, Exactly solved models in statistical mechanics, Academic Press (1982).
  * [4] T. Brzeziński, _Trusses: Between braces and rings_ , Trans. Amer. Math. Soc. 372 (2019), 4149-4176.
  * [5] T. Brzeziński, _Trusses: Paragons, ideals and modules_ , J. Pure Appl. Algebra 224 (2020), 106258.
  * [6] T. Brzeziński & B. Rybołowicz, _Congruence classes and extensions of rings with an application to braces_ , Comm. Contemp. Math. 23 (2021), 2050010.
  * [7] T. Brzeziński, S. Mereta ans B. Rybołowicz, From pre-trusses to skew braces, Publ. Mat., 66(2) (2022), 683-714.
  * [8] F. Catino, I. Colazzo and P. Stefanelli, Semi-braces and the Yang-Baxter equation, J. Algebra, 483 (2017) 163-187.
  * [9] F. Catino, M. Mazzotta, P. Stefanelli, Inverse semi-braces and the Yang-Baxter equation, J. Algebra, 573 (2021) 576-619.
  * [10] F. Cedó, E. Jespers and C. Verwimp, Structure monoids of set-theoretic solutions of the Yang-Baxter equation. Publicacions Matemàtiques, (2021)
  * [11] F. Cedó, Left braces: solutions of the Yang-Baxter equation, Adv. Group Theory Appl., Vol. 5 (2018) 33-90.
  * [12] F. Cedó, E. Jespers, and J. Okninski, Braces and the Yang-Baxter equation. Comm. Math. Phys., 327(1) (2014) 101–116.
  * [13] A. Doikou and A. Smoktunowicz, From Braces to Hecke algebras $\&$ Quantum Groups, J. Algebra and Applications (2022) 2350179.
  * [14] A. Doikou and A. Smoktunowicz, Set-theoretical Yang-Baxter and reflection equations $\&$ quantum group symmetries, Lett. Math. Phys. 111 (2021) 105.
  * [15] A. Doikou, Set-theoretic Yang–Baxter equation, braces and Drinfeld twists, J. Phys A54 (2021) 415201.
  * [16] A. Doikou, A. Ghionis and B. Vlaar, Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation, Lett. Math. Phys. 112, 78 (2022).
  * [17] A. Doikou and B. Rybołowicz, Novel non-involutive solutions of the Yang-Baxter equation from (skew) braces, arXiv:2204.11580 (2022).
  * [18] V.G. Drinfeld, On some unsolved problems in quantum group theory, Lecture Notes in Math., vol. 1510, Springer-Verlag, Berlin (1992) 1-8.
  * [19] V.G. Drinfeld, Hopf algebras $\&$ the quantum Yang–Baxter equation, Soviet. Math. Dokl. 32 (1985) 254;
A new realization of Yangians and quantized affine algebras, Soviet. Math.
Dokl. 36 (1988) 212.

  * [20] V.G. Drinfeld, Quasi-Hopf algebras, Algebra i Analiz (1989) Volume 1, Issue 6, 114.
V.G. Drinfeld, Quasi-Hopf algebras and Knizhnik-Zamolodchikov equations. In
Problems of modern quantum field theory, Springer, Berlin, Heidelberg (1989)
1-13.

  * [21] P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100 (1999) 169-209.
  * [22] P. Etingof, Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, Comm. Algebra 31 (2003) 1961.
  * [23] T. Gateva-Ivanova, A combinatorial approach to the set-theoretic solutions of the Yang–Baxter equation, J. Math. Phys. 45, 3828 (2004).
  * [24] T. Gateva-Ivanova, Binomial skew polynomial rings, Artin–Schelter regularity, and binomial solutions of the Yang-Baxter equation, Serdica Mathematical Journal Volume: 30, Issue: 2-3 (2004 )431-470.
  * [25] T. Gateva-Ivanova, Set-theoretic solutions of the Yang–Baxter equation, braces and symmetric groups, Adv. Math., 388(7) (2018) 649-701.
  * [26] T. Gateva-Ivanova and S. Majid, Matched pairs approach to set theoretic solutions of the Yang–Baxter equation, J. Algebra 319 (2008) 1462–1529.
  * [27] L. Guarnieri and L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comp. 86(307) (2017) 2519–2534.
  * [28] E. Jespers, E. Kubat and A. Van Antwerpen, The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang-Baxter equation, Trans. Amer. Math. Soc. 372 (2019) 7191–7223.
  * [29] E. Jespers, E. Kubat, A. Van Antwerpen and L. Vendramin, Factorizations of skew braces, Math. Ann. 375 (2019) no. 3-4, 1649–1663.
  * [30] E. Jespers and J. Okniński, Monoids and groups of I-type, Alg. Represent. Theory 8 (2005) 709–729.
  * [31] A. Koch and P. J. Truman. Opposite skew left braces and applications, J. Algebra 546 (2020) 218–235.
  * [32] A. Konovalov, A. Smoktunowicz and L. Vendramin, On skew braces and their ideals, Experimental Mathematics, Volume 30 (2021) 95–104.
  * [33] V. Lebed and L. Vendramin, On structure groups of set-theoretical solutions to the Yang-Baxter equation, Proc. Edinburgh Math. Soc., Volume 62, Issue 3 (2019) 683 -717.
  * [34] J.H. Lu, M. Yan, and Y.C. Zhu, On the set-theoretical Yang-Baxter equation, Duke Math. Journal, 104 (2000) 1-18.
  * [35] W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation. Adv. Math., 193(1) (2005) 40–55.
  * [36] W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307(1) (2007) 153–170.
  * [37] A. Smoktunowicz and L. Vendramin, On Skew Braces (with an appendix by N. Byott and L. Vendramin), Journal of Combinatorial Algebra Volume 2, Issue 1 (2018) 47-86.
  * [38] C.N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Rev. Lett. 19 (1967) 1312.


* Lunnan et al. (2012) Lunnan, R., Vogelsberger, M., Frebel, A., et al. 2012, ApJ, 746, 109, doi: 10.1088/0004-637X/746/1/109
  * Macias & Ramirez-Ruiz (2019) Macias, P., & Ramirez-Ruiz, E. 2019, ApJ, 877, L24, doi: 10.3847/2041-8213/ab2049
  * Magain (1984) Magain, P. 1984, A&A, 134, 189
  * Magg et al. (2022) Magg, M., Schauer, A. T. P., Klessen, R. S., et al. 2022, ApJ, 929, 119, doi: 10.3847/1538-4357/ac5aac
  * Magg et al. (2020) Magg, M., Nordlander, T., Glover, S. C. O., et al. 2020, MNRAS, 498, 3703, doi: 10.1093/mnras/staa2624
  * Marino et al. (2008) Marino, A. F., Villanova, S., Piotto, G., et al. 2008, A&A, 490, 625, doi: 10.1051/0004-6361:200810389
  * Marshall et al. (2018) Marshall, J. L., Hansen, T., Simon, J. D., et al. 2018, arXiv e-prints, arXiv:1812.01022.  https://arxiv.org/abs/1812.01022
  * Martini et al. (2006) Martini, P., Kelson, D. D., Kim, E., Mulchaey, J. S., & Athey, A. A. 2006, ApJ, 644, 116, doi: 10.1086/503521
  * Mashonkina et al. (2014) Mashonkina, L., Christlieb, N., & Eriksson, K. 2014, A&A, 569, A43, doi: 10.1051/0004-6361/201424017
  * Mashonkina et al. (2007) Mashonkina, L., Korn, A. J., & Przybilla, N. 2007, A&A, 461, 261, doi: 10.1051/0004-6361:20065999
  * Mashonkina et al. (2017) Mashonkina, L., Sitnova, T., & Belyaev, A. K. 2017, A&A, 605, A53, doi: 10.1051/0004-6361/201731236
  * Mashonkina et al. (2016) Mashonkina, L. I., Sitnova, T. N., & Pakhomov, Y. V. 2016, Astronomy Letters, 42, 606, doi: 10.1134/S1063773716080028
  * Mateo et al. (2012) Mateo, M., Bailey, J. I., Crane, J., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy IV, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 84464Y, doi: 10.1117/12.926448
  * McConnachie & Venn (2020) McConnachie, A. W., & Venn, K. A. 2020, Research Notes of the American Astronomical Society, 4, 229, doi: 10.3847/2515-5172/abd18b
  * Mckinney (2010) Mckinney, W. 2010
  * McQuinn et al. (2015) McQuinn, K. B. W., Skillman, E. D., Dolphin, A., et al. 2015, ApJ, 815, L17, doi: 10.1088/2041-8205/815/2/L17
  * McWilliam et al. (2013) McWilliam, A., Wallerstein, G., & Mottini, M. 2013, ApJ, 778, 149, doi: 10.1088/0004-637X/778/2/149
  * Miller et al. (2020) Miller, J. M., Sprouse, T. M., Fryer, C. L., et al. 2020, ApJ, 902, 66, doi: 10.3847/1538-4357/abb4e3
  * Minor et al. (2019) Minor, Q. E., Pace, A. B., Marshall, J. L., & Strigari, L. E. 2019, MNRAS, 487, 2961, doi: 10.1093/mnras/stz1468
  * Molero et al. (2021) Molero, M., Romano, D., Reichert, M., et al. 2021, MNRAS, 505, 2913, doi: 10.1093/mnras/stab1429
  * Mösta et al. (2018) Mösta, P., Roberts, L. F., Halevi, G., et al. 2018, ApJ, 864, 171, doi: 10.3847/1538-4357/aad6ec
  * Muñoz et al. (2018) Muñoz, R. R., Côté, P., Santana, F. A., et al. 2018, ApJ, 860, 66, doi: 10.3847/1538-4357/aac16b
  * Mutlu-Pakdil et al. (2018) Mutlu-Pakdil, B., Sand, D. J., Carlin, J. L., et al. 2018, ApJ, 863, 25, doi: 10.3847/1538-4357/aacd0e
  * O’Brian et al. (1991) O’Brian, T. R., Wickliffe, M. E., Lawler, J. E., Whaling, W., & Brault, J. W. 1991, Journal of the Optical Society of America B Optical Physics, 8, 1185, doi: 10.1364/JOSAB.8.001185
  * Ojima et al. (2018) Ojima, T., Ishimaru, Y., Wanajo, S., Prantzos, N., & François, P. 2018, ApJ, 865, 87, doi: 10.3847/1538-4357/aada11
  * Okamoto et al. (2008) Okamoto, T., Gao, L., & Theuns, T. 2008, MNRAS, 390, 920, doi: 10.1111/j.1365-2966.2008.13830.x
  * Pan et al. (2013) Pan, L., Scannapieco, E., & Scalo, J. 2013, ApJ, 775, 111, doi: 10.1088/0004-637X/775/2/111
  * Pasquini et al. (2002) Pasquini, L., Avila, G., Blecha, A., et al. 2002, The Messenger, 110, 1
  * Patel et al. (2020) Patel, E., Kallivayalil, N., Garavito-Camargo, N., et al. 2020, ApJ, 893, 121, doi: 10.3847/1538-4357/ab7b75
  * Placco et al. (2021) Placco, V. M., Sneden, C., Roederer, I. U., et al. 2021, Research Notes of the American Astronomical Society, 5, 92, doi: 10.3847/2515-5172/abf651
  * Price-Whelan et al. (2018) Price-Whelan, A. M., Sipőcz, B. M., Günther, H. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Radice et al. (2018) Radice, D., Perego, A., Hotokezaka, K., et al. 2018, ApJ, 869, 130, doi: 10.3847/1538-4357/aaf054
  * Revaz et al. (2016) Revaz, Y., Arnaudon, A., Nichols, M., Bonvin, V., & Jablonka, P. 2016, A&A, 588, A21, doi: 10.1051/0004-6361/201526438
  * Ritter et al. (2015) Ritter, J. S., Sluder, A., Safranek-Shrader, C., Milosavljević, M., & Bromm, V. 2015, MNRAS, 451, 1190, doi: 10.1093/mnras/stv982
  * Robertson et al. (2005) Robertson, B., Bullock, J. S., Font, A. S., Johnston, K. V., & Hernquist, L. 2005, ApJ, 632, 872, doi: 10.1086/452619
  * Robin et al. (2003) Robin, A. C., Reylé, C., Derrière, S., & Picaud, S. 2003, A&A, 409, 523, doi: 10.1051/0004-6361:20031117
  * Rodriguez Wimberly et al. (2019) Rodriguez Wimberly, M. K., Cooper, M. C., Fillingham, S. P., et al. 2019, MNRAS, 483, 4031, doi: 10.1093/mnras/sty3357
  * Roederer (2017) Roederer, I. U. 2017, ApJ, 835, 23, doi: 10.3847/1538-4357/835/1/23
  * Roederer et al. (2016a) Roederer, I. U., Mateo, M., Bailey, J. I., et al. 2016a, MNRAS, 455, 2417, doi: 10.1093/mnras/stv2462
  * Roederer et al. (2014) Roederer, I. U., Preston, G. W., Thompson, I. B., et al. 2014, AJ, 147, 136, doi: 10.1088/0004-6256/147/6/136
  * Roederer et al. (2016b) Roederer, I. U., Mateo, M., Bailey, III, J. I., et al. 2016b, AJ, 151, 82, doi: 10.3847/0004-6256/151/3/82
  * Ryabchikova et al. (2015) Ryabchikova, T., Piskunov, N., Kurucz, R. L., et al. 2015, Phys. Scr, 90, 054005, doi: 10.1088/0031-8949/90/5/054005
  * Ryan et al. (1996) Ryan, S. G., Norris, J. E., & Beers, T. C. 1996, ApJ, 471, 254, doi: 10.1086/177967
  * Sacchi et al. (2021) Sacchi, E., Richstein, H., Kallivayalil, N., et al. 2021, ApJ, 920, L19, doi: 10.3847/2041-8213/ac2aa3
  * Safarzadeh et al. (2019a) Safarzadeh, M., Ramirez-Ruiz, E., Andrews, J. J., et al. 2019a, ApJ, 872, 105, doi: 10.3847/1538-4357/aafe0e
  * Safarzadeh et al. (2019b) Safarzadeh, M., Sarmento, R., & Scannapieco, E. 2019b, ApJ, 876, 28, doi: 10.3847/1538-4357/ab1341
  * Safarzadeh & Scannapieco (2017) Safarzadeh, M., & Scannapieco, E. 2017, MNRAS, 471, 2088, doi: 10.1093/mnras/stx1706
  * Safranek-Shrader et al. (2012) Safranek-Shrader, C., Agarwal, M., Federrath, C., et al. 2012, MNRAS, 426, 1159, doi: 10.1111/j.1365-2966.2012.21852.x
  * Salvadori & Ferrara (2009) Salvadori, S., & Ferrara, A. 2009, MNRAS, 395, L6, doi: 10.1111/j.1745-3933.2009.00627.x
  * Santana et al. (2013) Santana, F. A., Muñoz, R. R., Geha, M., et al. 2013, ApJ, 774, 106, doi: 10.1088/0004-637X/774/2/106
  * Schönrich & Weinberg (2019) Schönrich, R. A., & Weinberg, D. H. 2019, MNRAS, 487, 580, doi: 10.1093/mnras/stz1126
  * Siegel et al. (2019) Siegel, D. M., Barnes, J., & Metzger, B. D. 2019, Nature, 569, 241, doi: 10.1038/s41586-019-1136-0
  * Simon (2019) Simon, J. D. 2019, ARA&A, 57, 375, doi: 10.1146/annurev-astro-091918-104453
  * Simon et al. (2010) Simon, J. D., Frebel, A., McWilliam, A., Kirby, E. N., & Thompson, I. B. 2010, ApJ, 716, 446, doi: 10.1088/0004-637X/716/1/446
  * Simon et al. (2015) Simon, J. D., Drlica-Wagner, A., Li, T. S., et al. 2015, ApJ, 808, 95, doi: 10.1088/0004-637X/808/1/95
  * Simon et al. (2017) Simon, J. D., Li, T. S., Drlica-Wagner, A., et al. 2017, ApJ, 838, 11, doi: 10.3847/1538-4357/aa5be7
  * Simon et al. (2020) Simon, J. D., Li, T. S., Erkal, D., et al. 2020, ApJ, 892, 137, doi: 10.3847/1538-4357/ab7ccb
  * Simon et al. (2022) Simon, J. D., Brown, T. M., Mutlu-Pakdil, B., et al. 2022, ApJ, in press, arXiv:2212.00810.  https://arxiv.org/abs/2212.00810
  * Smith & Raggett (1981) Smith, G., & Raggett, D. S. J. 1981, Journal of Physics B Atomic Molecular Physics, 14, 4015, doi: 10.1088/0022-3700/14/21/016
  * Sneden et al. (2008) Sneden, C., Cowan, J. J., & Gallino, R. 2008, ARA&A, 46, 241, doi: 10.1146/annurev.astro.46.060407.145207
  * Sneden (1973) Sneden, C. A. 1973, PhD thesis, The University of Texas at Austin.
  * Sobeck et al. (2011) Sobeck, J. S., Kraft, R. P., Sneden, C., et al. 2011, AJ, 141, 175, doi: 10.1088/0004-6256/141/6/175
  * Speagle (2020) Speagle, J. S. 2020, MNRAS, 493, 3132, doi: 10.1093/mnras/staa278
  * Surman et al. (2006) Surman, R., McLaughlin, G. C., & Hix, W. R. 2006, ApJ, 643, 1057, doi: 10.1086/501116
  * Tanaka et al. (2017) Tanaka, M., Utsumi, Y., Mazzali, P. A., et al. 2017, PASJ, 69, 102, doi: 10.1093/pasj/psx121
  * Tanvir et al. (2017) Tanvir, N. R., Levan, A. J., González-Fernández, C., et al. 2017, ApJ, 848, L27, doi: 10.3847/2041-8213/aa90b6
  * Tarumi et al. (2021) Tarumi, Y., Suda, T., van de Voort, F., et al. 2021, MNRAS, 505, 3755, doi: 10.1093/mnras/stab1487
  * Tarumi et al. (2020) Tarumi, Y., Yoshida, N., & Inoue, S. 2020, MNRAS, 494, 120, doi: 10.1093/mnras/staa720
  * Theler et al. (2020) Theler, R., Jablonka, P., Lucchesi, R., et al. 2020, A&A, 642, A176, doi: 10.1051/0004-6361/201937146
  * Tody (1986) Tody, D. 1986, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 627, Instrumentation in astronomy VI, ed. D. L. Crawford, 733, doi: 10.1117/12.968154
  * Tody (1993) Tody, D. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 52, Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch, R. J. V. Brissenden, & J. Barnes, 173
  * Tolstoy et al. (2009) Tolstoy, E., Hill, V., & Tosi, M. 2009, ARA&A, 47, 371, doi: 10.1146/annurev-astro-082708-101650
  * Tsujimoto et al. (2017) Tsujimoto, T., Matsuno, T., Aoki, W., Ishigaki, M. N., & Shigeyama, T. 2017, ApJ, 850, L12, doi: 10.3847/2041-8213/aa9886
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science & Engineering, 13, 22, doi: http://dx.doi.org/10.1109/MCSE.2011.37
  * Vargas et al. (2013) Vargas, L. C., Geha, M., Kirby, E. N., & Simon, J. D. 2013, ApJ, 767, 134, doi: 10.1088/0004-637X/767/2/134
  * Venn et al. (2004) Venn, K. A., Irwin, M., Shetrone, M. D., et al. 2004, AJ, 128, 1177, doi: 10.1086/422734
  * Venn et al. (2017) Venn, K. A., Starkenburg, E., Malo, L., Martin, N., & Laevens, B. P. M. 2017, MNRAS, 466, 3741, doi: 10.1093/mnras/stw3198
  * Walker et al. (2015) Walker, M. G., Mateo, M., Olszewski, E. W., et al. 2015, ApJ, 808, 108, doi: 10.1088/0004-637X/808/2/108
  * Waskom et al. (2016) Waskom, M., Botvinnik, O., O’Kane, D., et al. 2016, seaborn: v0.7.0 (January 2016), doi: 10.5281/zenodo.45133
  * Webster et al. (2015) Webster, D., Bland-Hawthorn, J., & Sutherland, R. 2015, ApJ, 799, L21, doi: 10.1088/2041-8205/799/2/L21
  * Webster et al. (2016) Webster, D., Frebel, A., & Bland-Hawthorn, J. 2016, ApJ, 818, 80, doi: 10.3847/0004-637X/818/1/80
  * Webster et al. (2014) Webster, D., Sutherland, R., & Bland-Hawthorn, J. 2014, ApJ, 796, 11, doi: 10.1088/0004-637X/796/1/11
  * Weidner et al. (2013) Weidner, C., Kroupa, P., Pflamm-Altenburg, J., & Vazdekis, A. 2013, MNRAS, 436, 3309, doi: 10.1093/mnras/stt1806
  * Weisz et al. (2014a) Weisz, D. R., Dolphin, A. E., Skillman, E. D., et al. 2014a, ApJ, 789, 147, doi: 10.1088/0004-637X/789/2/147
  * Weisz et al. (2014b) —. 2014b, ApJ, 789, 148, doi: 10.1088/0004-637X/789/2/148
  * Wenger et al. (2000) Wenger, M., Ochsenbein, F., Egret, D., et al. 2000, A&AS, 143, 9, doi: 10.1051/aas:2000332
  * Wheeler et al. (2019) Wheeler, C., Hopkins, P. F., Pace, A. B., et al. 2019, MNRAS, 490, 4447, doi: 10.1093/mnras/stz2887
  * Willman & Strader (2012) Willman, B., & Strader, J. 2012, AJ, 144, 76, doi: 10.1088/0004-6256/144/3/76
  * Wise & Abel (2007) Wise, J. H., & Abel, T. 2007, ApJ, 665, 899, doi: 10.1086/520036
  * Wolf et al. (2010) Wolf, J., Martinez, G. D., Bullock, J. S., et al. 2010, MNRAS, 406, 1220, doi: 10.1111/j.1365-2966.2010.16753.x
  * Wu et al. (2016) Wu, M.-R., Fernández, R., Martínez-Pinedo, G., & Metzger, B. D. 2016, MNRAS, 463, 2323, doi: 10.1093/mnras/stw2156
  * Zechmeister et al. (2014) Zechmeister, M., Anglada-Escudé, G., & Reiners, A. 2014, A&A, 561, A59, doi: 10.1051/0004-6361/201322746

##  Appendix A Radial Velocities and Binarity

As a relatively nearby UFD of great scientific interest, Ret II has obtained
many different epochs of radial velocities. We have collected all currently
available literature velocities in Table 7. The literature velocities are
mostly derived from coadding spectra taken across 1-4 adjacent nights, so the
MJD reported here is only accurate to 2 days of precision. These velocities
are not homogeneous and may suffer from systematic effects.

Table 7: Literature Radial Velocities ID | $v_{\rm hel}$ | $\sigma_{\rm vhel}$ | $v_{\rm S15}$ | $\sigma_{\rm S15}$ | $v_{\rm K15}$ | $\sigma_{\rm K15}$ | $v_{\rm J16}$ | $\sigma_{\rm J16}$ | $v_{\rm R16}$ | $\sigma_{\rm R16}$ | $v_{\rm VLT}$ | $\sigma_{\rm VLT}$ | $v_{\rm HR}$ | $\sigma_{\rm HR}$ | $v_{\rm W15}$ | $\sigma_{\rm W15}$ | $v_{\rm MR}$ | $\sigma_{\rm MR}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  | MJD=57072 | MJD=57090 | MJD=57298 | MJD=57341 | MJD=58052 | MJD=58073 | MJD=57072 | MJD=57639
1 | $+65.3$ | $0.2$ |  $\cdots$ |  $\cdots$ | $+66.3$ | $0.2$ | $+66.8$ | $1.0$ | $+65.5$ | $1.0$ | $+67.6$ | $0.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+56.3$ | $2.4$
2 | $+61.1$ | $3.2$ | $+59.1$ | $0.9$ | $+61.4$ | $0.4$ | $+62.7$ | $1.0$ | $+62.0$ | $1.0$ | $+65.5$ | $0.7$ | $+64.9$ | $0.7$ | $+61.8$ | $0.4$ | $+55.1$ | $2.4$
3 | $+62.0$ | $0.4$ | $+62.3$ | $1.0$ |  $\cdots$ |  $\cdots$ | $+62.0$ | $1.0$ | $+62.2$ | $1.0$ | $+63.5$ | $0.7$ | $+66.8$ | $0.8$ | $+63.2$ | $0.5$ | $+52.4$ | $2.4$
4 | $+58.5$ | $0.3$ | $+57.7$ | $1.0$ | $+59.6$ | $0.5$ | $+60.9$ | $1.0$ | $+59.7$ | $1.0$ | $+61.2$ | $0.7$ | $+60.4$ | $0.8$ | $+60.4$ | $0.7$ | $+47.3$ | $2.4$
5 | $+61.7$ | $0.3$ |  $\cdots$ |  $\cdots$ | $+63.5$ | $0.5$ | $+61.9$ | $1.0$ |  $\cdots$ |  $\cdots$ | $+63.8$ | $0.7$ | $+63.2$ | $0.8$ |  $\cdots$ |  $\cdots$ | $+54.0$ | $2.4$
6 | $+64.3$ | $0.4$ | $+64.4$ | $1.1$ | $+65.6$ | $0.9$ | $+63.5$ | $1.0$ |  $\cdots$ |  $\cdots$ | $+67.4$ | $0.7$ | $+67.0$ | $0.8$ | $+62.9$ | $1.2$ | $+59.7$ | $2.4$
7 | $+63.2$ | $0.4$ | $+65.2$ | $1.2$ | $+65.9$ | $1.2$ | $+62.7$ | $1.0$ |  $\cdots$ |  $\cdots$ | $+65.7$ | $0.7$ | $+65.0$ | $0.9$ | $+63.9$ | $2.3$ | $+43.6$ | $2.4$
8 | $+60.2$ | $0.4$ | $+59.8$ | $1.2$ | $+61.9$ | $0.8$ | $+61.9$ | $1.0$ |  $\cdots$ |  $\cdots$ | $+62.5$ | $0.7$ | $+61.7$ | $0.8$ | $+61.8$ | $1.4$ | $+51.6$ | $2.4$
9 | $+69.2$ | $0.4$ | $+69.7$ | $1.4$ | $+70.8$ | $1.1$ | $+71.6$ | $1.0$ |  $\cdots$ |  $\cdots$ | $+71.2$ | $0.7$ | $+70.8$ | $0.8$ | $+70.0$ | $1.7$ | $+59.6$ | $2.4$
10 | $+62.1$ | $3.9$ | $+62.3$ | $1.1$ | $+69.1$ | $1.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+64.0$ | $0.7$ | $+61.4$ | $0.8$ | $+65.6$ | $1.1$ |  $\cdots$ |  $\cdots$
11 | $+67.0$ | $0.9$ | $+67.9$ | $1.1$ | $+65.4$ | $1.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+70.2$ | $3.2$ | $+67.9$ | $1.3$ | $+61.3$ | $2.4$
12 | $+64.6$ | $0.5$ | $+65.7$ | $1.1$ | $+65.0$ | $1.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+67.2$ | $0.7$ | $+66.7$ | $0.8$ | $+69.1$ | $1.5$ | $+53.2$ | $2.4$
13 | $+63.6$ | $2.6$ | $+65.6$ | $1.3$ | $+68.2$ | $1.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+67.8$ | $0.7$ | $+63.1$ | $0.8$ | $+70.4$ | $1.9$ |  $\cdots$ |  $\cdots$
14 | $+62.3$ | $0.7$ | $+59.3$ | $1.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+65.3$ | $0.7$ | $+69.0$ | $3.3$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
15 | $+62.6$ | $0.6$ | $+63.2$ | $1.4$ | $+63.4$ | $1.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+65.3$ | $0.7$ | $+65.1$ | $3.3$ | $+62.5$ | $1.9$ |  $\cdots$ |  $\cdots$
16 | $+64.6$ | $0.9$ | $+59.1$ | $8.2$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+67.4$ | $0.9$ | $+0.0$ | $0.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
17 | $+60.2$ | $0.7$ | $+57.4$ | $2.4$ | $+60.0$ | $2.1$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+63.3$ | $0.7$ | $+64.9$ | $3.3$ | $+60.1$ | $2.1$ | $+40.7$ | $2.4$
18 | $+59.3$ | $7.1$ | $+66.3$ | $1.4$ | $+62.9$ | $3.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+59.4$ | $0.8$ | $+73.6$ | $3.3$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
19 | $+60.7$ | $10.5$ | $+67.9$ | $1.4$ | $+78.9$ | $1.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+57.9$ | $0.8$ | $+64.4$ | $3.3$ | $+70.2$ | $3.3$ | $+26.5$ | $2.4$
20 | $+63.6$ | $0.7$ | $+63.5$ | $1.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+66.3$ | $0.7$ | $+70.9$ | $3.4$ | $+66.4$ | $2.9$ |  $\cdots$ |  $\cdots$
21 | $+61.0$ | $0.8$ | $+56.7$ | $1.9$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+64.7$ | $0.9$ | $+64.2$ | $3.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
22 | $+65.6$ | $0.7$ | $+64.7$ | $1.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+68.6$ | $0.8$ | $+68.4$ | $3.5$ | $+66.7$ | $2.0$ |  $\cdots$ |  $\cdots$
23 | $+61.6$ | $0.7$ | $+59.8$ | $1.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+64.7$ | $0.8$ | $+65.2$ | $3.5$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
24 | $+63.6$ | $0.8$ | $+68.0$ | $3.5$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+65.9$ | $0.8$ | $+71.6$ | $3.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
25 | $+62.0$ | $0.7$ | $+61.9$ | $2.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+64.9$ | $0.8$ | $+63.0$ | $3.4$ | $+65.0$ | $2.9$ |  $\cdots$ |  $\cdots$
26 | $+61.5$ | $0.8$ | $+61.7$ | $4.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+64.0$ | $0.9$ | $+69.1$ | $4.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
97 | $+67.5$ | $0.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+70.3$ | $0.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
99 | $+67.2$ | $0.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+69.9$ | $0.8$ | $+70.9$ | $3.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
100 | $+61.0$ | $0.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+63.4$ | $0.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
102 | $+66.8$ | $0.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+69.1$ | $0.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
134 | $+62.2$ | $1.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+65.0$ | $1.0$ | $+0.0$ | $0.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
142 | $+61.0$ | $0.7$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+64.0$ | $0.9$ | $+63.0$ | $1.1$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
143 | $+58.9$ | $0.9$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+61.7$ | $0.9$ | $+0.0$ | $0.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
144 | $+65.0$ | $1.1$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+67.6$ | $1.2$ | $+68.5$ | $2.6$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
151 | $+68.0$ | $1.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+70.8$ | $1.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
154 | $+61.9$ | $1.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+64.6$ | $1.0$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
157 | $+67.9$ | $0.9$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+70.5$ | $0.9$ | $+73.1$ | $3.4$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
188 | $+68.4$ | $1.2$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+71.2$ | $1.2$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
192 | $+69.1$ | $0.9$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+72.2$ | $0.9$ | $+66.9$ | $3.5$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
195 | $+68.1$ | $0.8$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $+71.1$ | $0.8$ | $+65.2$ | $4.2$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$

In a first attempt to calibrate the systematic effects, we adopt the S15
velocities as a reference velocity scale, as they have the most stars with
common velocities compared to other literature sources. For matched stars in
each sample, we calculate a weighted mean velocity offset. After removing this
offset, stars with velocity variations inconsistent with a chi-squared test
with $p<0.01$ are identified as likely binaries (e.g., Chiti et al., 2022). We
iterate this process until convergence, resulting in five binary stars: Star
2, 13, 18, 19, and 21. The final mean velocity offsets relative to S15 are
1.01 $\mathrm{\,km\,s^{-1}}$ for K15, 0.63 $\mathrm{\,km\,s^{-1}}$ for J16,
1.02 $\mathrm{\,km\,s^{-1}}$ for Roederer et al. (2016b), 2.79
$\mathrm{\,km\,s^{-1}}$ for our VLT spectra, and 2.37 $\mathrm{\,km\,s^{-1}}$
for our HighRes M2FS spectra. Our MedRes M2FS data have a very large offset of
$-8.9\mathrm{\,km\,s^{-1}}$, and we thus decided to exclude it from any
velocity studies.

##  Appendix B Effect of Microturbulence on Barium Abundances

Microturbulence ($\nu_{t}$) is a parameter introduced to 1D stellar
atmospheres to account for unmodeled 3D atmospheric effects. It affects lines
at the saturated part of the curve of growth, where a higher microturbulence
effectively desaturates strong lines by adding some extra doppler broadening
(e.g., Gray, 2008). When lots of Fe I lines are available, microturbulence is
usually found by balancing Fe I abundance as a function of line strength.
Empirical measurements of microturbulence show that red giants with lower
surface gravities (and temperatures) tend to have higher microturbulence
values (e.g., Barklem et al., 2005).

In our VLT spectra, the Ba 6496Å line is at or near saturation when detected.
For our coolest giants, increasing $\nu_{t}$ by 0.2 $\mathrm{\,km\,s^{-1}}$
reduces [Ba/H] by 0.15 dex. Since we aim to resolve abundance scatter on the
order of 0.20 dex, this systematic effect is often the dominant uncertainty,
especially for the brightest giants where the equivalent width is well-
measured. There are not enough Fe lines to self-consistently measure $\nu_{t}$
in our stars, so we must use existing correlations between $\log g$ and
$\nu_{t}$. Here, we investigate five different datasets that measured
$\nu_{t}$ using high-resolution spectroscopy of metal-poor red giant stars,
examining systematic differences in $\nu_{t}-\log g$ relations, as well as the
scatter around those relations. The effect of these choices on our barium
abundance results is investigated in Appendix C.

Figure 9 shows the result of our investigation. The left column in Figure 9
plots $\log g$ vs $\nu_{t}$ for data from Barklem et al. (2005) (B05), Marino
et al. (2008) (M08), Cohen et al. (2013) (C13), Roederer et al. (2014) (R14),
and Jacobson et al. (2015) (J15), while the other columns show the measured
[Ba/H]LTE compared to $T_{\rm eff}$, [Fe/H], and the signal-to-noise ratio
(SNR). We separate Roederer et al. (2014) into a giants-only sample as well
(R14g). When available, we use the stellar parameters as tabulated in JINAbase
(Abohalima & Frebel, 2018). Horizontal branch stars have higher
microturbulence than RGB stars, so they are removed with a cut $\log
g>0.00286\,T_{\rm eff}-12.7$. To each dataset, we fit a linear and quadratic
polynomial for $\nu_{t}$ as a function of $\log g$, using a robust fitter
based on the routine robust_poly_fit in the AstroIDL library. The scatter
around the fit is measured with the biweight scale of the residuals (a robust
standard deviation). The coefficients and scatter around each relation are
given in Table 8.

Note that the right column of Figure 9 shows that our stars with SNR $<25$
have a much larger scatter than stars above that threshold. Visual examination
of the spectra (Figure 3) suggests that these stars are adversely affected by
inaccurate sky subtraction that is blended with the Ba line. We thus have
decided to exclude the low-SNR stars from most analyses. Also note that we
used MOOG LTE/ATLAS [Ba/H] abundances for this figure, though the conclusions
are robust if using NLTE instead.

###  B.1 Mean Relations

There are clear differences in the average $\nu_{t}-\log g$ relation across
different literature samples: at low $\log g$, the first three rows (B05, C13,
J15) have systematically higher $\nu_{t}$ than the last three rows (M08, R14,
R14g). The origin of this difference is not clear. One possibility is the
spectra for M08 and R14 have SNR $\sim 100$, substantially higher than B05 and
J15 which typically have SNR $\sim 30$. The low SNR of weak iron lines could
bias microturbulence too high (as described by Magain 1984). However, C13 also
have SNR $\sim 100$ and obtain higher microturbulence values. Another
possibility is that NLTE effects bias the microturbulence due to underlying
correlations between excitation potential, line strength, and typical NLTE
correction size (Bergemann et al., 2012). Further investigation of these
differences would be valuable but is beyond the scope of this paper appendix.

In this paper, we have decided to pick a $\nu_{t}-\log g$ relation that leaves
no trend in the [Ba/H] and $T_{\rm eff}$ for our stars. In Figure 9, the 2nd
column shows [Ba/H] vs $T_{\rm eff}$, where the first three rows of Figure 9
have a strong systematic trend such that cooler stars have lower [Ba/H]. The
bottom three rows do not have a significant trend. The coolest stars have the
highest SNR and lowest statistical uncertainty, so the different mean trend
makes a significant difference on our final inferred Ba scatter. Figure 10
shows this specifically for the high SNR and clear member stars for the R14g
and B05 relations. The R14g $\nu_{t}-\log g$ relation clearly has less trend
with $T_{\rm eff}$ than B05. Note that the trends are primarily driven by the
two coolest and brightest stars ($T_{\rm eff}\sim 4500\,$K). Because these two
stars have low statistical uncertainty on their [Ba/H] abundances, the
systematic effect of microturbulence can make a large difference in the
inferred [Ba/H] scatter.

We have decided to adopt the quadratic $\nu_{t}-\log g$ relation from the
giant stars in Roederer et al. (2014) (R14g) as our fiducial results. R14g
have the highest SNR spectra of metal-poor giants with the largest wavelength
coverage out of all these data samples. The results of this paper would not
change if we used the very similar M08 or R14 relations instead. However, this
$\nu_{t}-\log g$ relation is different from most previous studies of dwarf
galaxy stellar abundances (Ji et al., 2016b; Ji & Frebel, 2018; Roederer et
al., 2016b). Thus in Appendix C, we give all results using the $\nu_{t}-\log
g$ relation from B05 as well, which matches those previous abundance studies.
We note that adding NLTE effects for Ba exacerbates the trend for [Ba/H] vs
$T_{\rm eff}$ when using the B05 $\nu_{t}$-$\log g$ relation, because the NLTE
corrections are larger (more negative) when microturbulence is higher. It is
also worth noting that past studies of Ret II had clear [Ba/H] trends with
temperature that can be explained by microturbulence.

###  B.2 Microturbulence Scatter

Typically, a systematic uncertainty of ${\approx}$0.2 $\mathrm{\,km\,s^{-1}}$
is adopted for microturbulence, which accounts for the systematic mean
differences described above. However, because we are interested in the
abundance scatter within Ret II, another crucial value is the _intrinsic
scatter_ in microturbulence around a “true” $\nu_{t}-\log g$ relation, i.e.,
changes in the atmospheric structure that are unmodeled by $\log g$ alone.
This error is likely _smaller_ than the observational scatter, because the
microturbulence measurements themselves are noisy.

Examining our five datasets, two have a scatter of ${\sim}0.2$
$\mathrm{\,km\,s^{-1}}$ (C13, J15), while three have a scatter of ${\sim}0.1$
$\mathrm{\,km\,s^{-1}}$ (B05, M08, R14). Using a smaller intrinsic $\nu_{t}$
scatter would increase the significance of our scatter detections, while a
larger intrinsic $\nu_{t}$ scatter reduces the significance. Since we have
adopted the mean $\nu_{t}-\log g$ relation from R14g, we also decide to adopt
the intrinsic scatter of $0.13\mathrm{\,km\,s^{-1}}$ from that data sample. In
our systematic investigations using the B05 sample, we adopt the corresponding
intrinsic scatter of $0.12\mathrm{\,km\,s^{-1}}$.

Figure 9: Impact of different $\nu_{t}-\log g$ relations. Left column: $\log g$ vs $\nu_{t}$ for each sample of stars. The thin red lines at the bottom indicate the $\log g$ of Ret II stars. The best quadratic fit is plotted as a thick red line. The grey points show stars from all other rows for context. Left-middle column: [Ba/H] vs $T_{\rm eff}$. Right-middle column: [Ba/H] vs [Fe/H]. Right column: [Ba/H] vs SNR. Stars with SNR $<25$ display substantially larger [Ba/H] scatter, likely due to residuals from sky subtraction. Overall, the first three rows (B05, C13, J15) have similar trends, showing an upturn in $\nu_{t}$ at low $\log g$ but a very noticable trend in [Ba/H] vs $T_{\rm eff}$. The last three rows have smaller $\nu_{t}$ and little trend in [Ba/H] vs $T_{\rm eff}$ (excluding stars with SNR $<25$). Figure 10: Comparison of [Ba/H] vs $T_{\rm eff}$ slopes for different microturbulence relations. In contrast to Fig 9, here we only include the high SNR clear member stars. The left and right panels show the NLTE and LTE [Ba/H] abundances, respectively. The red circles and blue squares indicate [Ba/H] derived using the R14g and B05 $\nu_{t}-\log g$ relations, respectively. The red and blue solid lines indicate a fit to all stars using the R14g and B05 relations, respectively; while the dashed lines indicate the fit removing the coolest star (ID 1). It is clear that the trend with $T_{\rm eff}$ is flatter using the R14g relation, regardless of whether the coolest star is included. For the NLTE panel, the slopes of the lines (in units of dex per 100K) are 0.03 and 0.08 for R14g and B05, respectively; or 0.01 and 0.05 when removing the cold star for R14g and B05, respectively.  Table 8: Fit parameters for $\nu_{t}$ Sample | Quadratic Fit | $\sigma$ | Linear Fit | $\sigma$
---|---|---|---|---
B05 | $0.1001\log g^{2}+-0.7394\log g+2.847$ | 0.12 | $-0.2527\log g+2.316$ | 0.15
C13 | $0.1048\log g^{2}+-0.7744\log g+2.965$ | 0.18 | $-0.2189\log g+2.300$ | 0.20
J15 | $0.1307\log g^{2}+-0.9812\log g+3.322$ | 0.21 | $-0.5217\log g+2.973$ | 0.22
M08 | $0.0175\log g^{2}+-0.3242\log g+2.009$ | 0.08 | $-0.2545\log g+1.944$ | 0.08
R14 | $0.0471\log g^{2}+-0.3474\log g+1.969$ | 0.18 | $-0.1201\log g+1.764$ | 0.18
R14 giants | $0.0386\log g^{2}+-0.3313\log g+1.960$ | 0.13 | $-0.2247\log g+1.897$ | 0.10

##  Appendix C Systematic Effects on Barium Scatter

Here we explore the effect of different data subsets and microturbulence
relations on the main result of this paper, the [Ba/H] mean and scatter. For
the data samples, we consider permutations of membership (clear members only
vs including candidate members) and MULTI NLTE/MARCS vs MOOG LTE/ATLAS. For
the microturbulence relations, we use the fiducial R14 giants (R14g) relation,
as well as the B05 relation that has a higher microturbulence for the
coolest/lowest gravity giants. For each of these permutations, we fit the two-
component Ba scatter model described in Section 4.4. Note that while our
fiducial model is run with a very large number of steps, for the other models
we only sampled to reach ${\gtrsim}100$ effective samples, and thus the
uncertainties and limits on the parameters will be less accurate.

Table 9 gives the results of the model fits. The first row is our fiducial
value, while the other rows show various data permutations. $\mu_{1}$ and
$\sigma_{1}$ are the most important values, indicating the mean and intrinsic
spread on the detected [Ba/H] abundances. $\mu_{2}$ and $\sigma_{2}$ are the
mean and scatter of the undetected [Ba/H] component, which is not well-
constrained given that no low [Ba/H] abundances were detected. $p_{2}$ is the
fraction of stars in the undetected [Ba/H] component, i.e., $1-p_{2}$ is the
fraction of $r$-enhanced stars. The uncertainties are $1\sigma$, and the limit
on $\sigma_{1}$ is a 95% limit.

We point out three main conclusions of Table 9. First, the MULTI NLTE/MARCS
mean [Ba/H] abundances ($\mu_{1}$) are typically lower than the MOOG LTE/ATLAS
abundances by about 0.3 dex. Figure 11 shows that the typical [Ba/H]
correction going from MOOG to MULTI is $-0.27\pm 0.04$ dex in a way that is
fairly close to a constant offset. This is a result both of the different
model atmospheres as well as the effect of NLTE. Second, when considering just
the clear member stars, none of the models detect a significant [Ba/H]
dispersion $\sigma_{1}$. The constraint is stronger when using NLTE, but
weaker when using the B05 microturbulence relation instead of the R14g
microturbulence relation. This is driven primarily by the coolest Ret II star
(ID 1), which is most affected by the different microturbulence relations
(Figure 9). Third, when including the candidate member stars, all the upper
limits on $\sigma_{1}$ get looser, and actually in one case (B05 MOOG with
candidates) the intrinsic dispersion is resolved at $2\sigma$. This is
predominantly because of the outlier star 14, which has a weak but detected Ba
line. If this star is actually part of Ret II, then our two-component model
for [Ba/H] is likely insufficient to describe the data because star 14 is well
outside of the main peak of [Ba/H] detections, but well above the more
stringent [Ba/H] upper limits.

After examining Table 9 and Figure 9, we decided that using the R14g
MULTI/NLTE results with only clear members is the most reliable measurement.
It is clear that using NLTE and definite members will result in a better
measurement, and eliminating the trend with stellar parameters discussed in
Appendix B justifies using R14g instead of B05. However, for completeness, we
show several permutations of best-fit [Ba/H] distributions in Figure 12. The
top-left panel shows the R14g and NLTE abundances used in the main paper, but
plotting all low-SNR detections as small data points and low-SNR upper limits
as grey arrows. As also seen in the right column of Figure 9, the low-SNR data
are skewed towards higher [Ba/H] abundances primarily due to bad sky
subtraction (Figure 3). The top-right panel shows three alternate fits to
different permutations of data used (members and candidates; high- and low-SNR
data). The bottom left panel shows the effect of changing the radiative
transfer, and the bottom-right panel shows the effect of changing the
microturbulence relation. These differences make a relatively small change to
the Ba dispersion (which is not resolved) but a fairly large change to the
mean abundance.

Figure 11: Differences between NLTE (MULTI/MARCS) and LTE (MOOG/ATLAS) abundances for stars with SNR $>25$. Left: differences as a function of [Ba/H] (LTE). Right: differences as a function of $T_{\rm eff}$. The NLTE correction for saturated lines clusters closely around $\Delta$[Ba/H]$=-0.27\pm 0.04$ (the outlier is the candidate member star 14 with a relatively low [Ba/H] abundance). Figure 12: Exploration of different best-fit models by permuting the data sample (top-right panel), radiative transfer (bottom-left panel), and $\log g-\nu_{t}$ relation (bottom-right panel). Table 9:  [Ba/H] distribution fits (stars with SNR $>25$) Data | $\mu_{1}$ | $\sigma_{1}$ |  $\sigma_{1}$ limit | $\mu_{2}$ | $\sigma_{2}$ | $p_{2}$
---|---|---|---|---|---|---
R14g MULTI NLTE members | $-1.68^{+0.07}_{-0.07}$ | $0.05^{+0.08}_{-0.03}$ | $<0.20$ | $-4.32^{+0.49}_{-0.46}$ | $0.08^{+0.32}_{-0.06}$ | $0.28^{+0.12}_{-0.10}$
R14g MOOG LTE members | $-1.38^{+0.06}_{-0.06}$ | $0.06^{+0.09}_{-0.04}$ | $<0.22$ | $-4.26^{+0.53}_{-0.50}$ | $0.09^{+0.32}_{-0.07}$ | $0.28^{+0.12}_{-0.10}$
B05 MULTI NLTE members | $-1.91^{+0.07}_{-0.06}$ | $0.11^{+0.10}_{-0.08}$ | $<0.28$ | $-4.31^{+0.49}_{-0.47}$ | $0.09^{+0.36}_{-0.07}$ | $0.26^{+0.12}_{-0.10}$
B05 MOOG LTE members | $-1.60^{+0.07}_{-0.07}$ | $0.14^{+0.09}_{-0.09}$ | $<0.30$ | $-4.22^{+0.48}_{-0.51}$ | $0.11^{+0.37}_{-0.09}$ | $0.23^{+0.14}_{-0.13}$
R14g MULTI NLTE with candidates | $-1.73^{+0.07}_{-0.08}$ | $0.12^{+0.11}_{-0.09}$ | $<0.31$ | $-4.12^{+0.57}_{-0.56}$ | $0.23^{+0.55}_{-0.20}$ | $0.26^{+0.12}_{-0.11}$
R14g MOOG LTE with candidates | $-1.43^{+0.07}_{-0.08}$ | $0.13^{+0.13}_{-0.09}$ | $<0.36$ | $-3.77^{+0.62}_{-0.69}$ | $0.62^{+0.28}_{-0.50}$ | $0.29^{+0.14}_{-0.11}$
B05 MULTI NLTE with candidates | $-1.95^{+0.07}_{-0.04}$ | $0.19^{+0.10}_{-0.10}$ | $<0.37$ | $-4.25^{+0.52}_{-0.51}$ | $0.10^{+0.46}_{-0.08}$ | $0.22^{+0.12}_{-0.09}$
B05 MOOG LTE with candidates | $-1.69^{+0.11}_{-0.10}$ | $0.24^{+0.11}_{-0.10}$ | $<0.46$ | $-3.83^{+0.74}_{-0.61}$ | $0.40^{+0.44}_{-0.37}$ | $0.25^{+0.15}_{-0.13}$

##  Appendix D Additional Chemical Abundances

We provide a table of member stars with sufficiently high S/N in the M2FS
HiRes Mg b data to measure detailed chemical abundances. This illustrates the
usefulness of the M2FS Mg Wide configuration for measuring detailed chemical
abundances in metal-poor stars.

Table 10: M2FS Mg b Abundances ID | SNR | [Mg/H] | $\sigma_{\rm Mg}$ | [Ca/H] | $\sigma_{\rm Ca}$ | [Ti/H] | $\sigma_{\rm Ti}$ | [Cr/H] | $\sigma_{\rm Cr}$ | [Fe/H] | $\sigma_{\rm Fe}$ | [Nd/H] | $\sigma_{\rm Nd}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
2 | 39.3 | $-2.48$ | $0.12$ | $-2.21$ | $0.07$ | $-2.29$ | $0.10$ | $-2.83$ | $0.13$ | $-2.63$ | $0.11$ | $-1.20$ | $0.09$
3 | 34.2 | $-2.59$ | $0.13$ | $-2.38$ | $0.07$ | $-2.57$ | $0.08$ | $-3.35$ | $0.11$ | $-2.78$ | $0.13$ | $-1.35$ | $0.08$
4 | 37.3 | $-2.55$ | $0.11$ | $-2.47$ | $0.08$ | $-2.71$ | $0.08$ | $-3.38$ | $0.09$ | $-3.14$ | $0.08$ |  $\cdots$ |  $\cdots$
5 | 19.4 | $-2.52$ | $0.13$ | $-1.86$ | $0.13$ | $-1.65$ | $0.19$ | $-2.10$ | $0.20$ | $-1.93$ | $0.17$ | $-0.89$ | $0.15$
6 | 23.5 | $-2.75$ | $0.13$ | $-2.24$ | $0.15$ | $-2.55$ | $0.10$ |  $\cdots$ |  $\cdots$ | $-2.81$ | $0.12$ |  $\cdots$ |  $\cdots$
7 | 13.8 | $-2.67$ | $0.22$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
8 | 15.2 | $-2.03$ | $0.16$ | $-1.52$ | $0.17$ | $-1.76$ | $0.14$ | $-2.42$ | $0.23$ | $-2.17$ | $0.12$ | $-0.90$ | $0.12$
9 | 17.0 | $-2.72$ | $0.13$ |  $\cdots$ |  $\cdots$ | $-2.55$ | $0.12$ | $-3.32$ | $0.14$ | $-2.67$ | $0.11$ |  $\cdots$ |  $\cdots$
10 | 15.3 | $-2.09$ | $0.18$ |  $\cdots$ |  $\cdots$ | $-2.44$ | $0.17$ | $-3.49$ | $0.16$ | $-2.86$ | $0.16$ |  $\cdots$ |  $\cdots$
12 | 15.2 | $-2.84$ | $0.15$ | $-1.94$ | $0.16$ | $-2.40$ | $0.13$ | $-3.07$ | $0.22$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$
13 | 9.9 | $-2.60$ | $0.16$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$ | $-2.46$ | $0.27$ |  $\cdots$ |  $\cdots$ |  $\cdots$ |  $\cdots$