text:
\section{Introduction}

The Yang--Baxter equation (YBE) is a fundamental object in pure mathematics and physics. It arose from the work of the Nobel prize-winning
physicist Yang \cite{Ya} on statistical mechanics and independently
in the work of Baxter \cite{Ba} concerning the 8-vertex model. A number of exciting developments in physics and mathematics have led to the conclusion that the YBE is a fundamental mathematical structure, with connections to various topics such as knot theory,
braid theory, operator theory, Hopf algebras, quantum groups, 3-manifolds and the monodromy of differential equations.

A solution to the YBE is a pair $(V,R)$, where
$V$ is a vector space and $R\colon V\otimes V\to V\otimes V$ is an
linear map such that
\[
(R\otimes\mathrm{id})(\mathrm{id}\otimes R)(R\otimes\mathrm{id})=(\mathrm{id}\otimes R)(R\otimes\mathrm{id})(\mathrm{id}\otimes R).
\]
An absolutely fascinating family of solutions consist of those maps $R$
induced by linear extensions of a bijective map
$r\colon X\times X\to X\times X$,
where $X$ is a basis of $V$.
As this idea captures all the combinatorics behind the YBE,
one says that the pair $(X,r)$ is a combinatorial solution, also called set-theoretic solution by Drinfel'd \cite{MR1183474}.

The fundamental problem of constructing combinatorial solutions
with given properties is nowadays based on the use of certain (associative and non-associative) algebraic structures acting on the solution. Examples of such structures are
groups of I-type \cite{MR1637256}, bijective 1-cocycles \cite{MR1769723,MR1769723,MR1809284}, skew braces \cite{MR3177933,MR2278047,MR3647970} and cycle sets \cite{MR2132760}.
Properties of solutions can be described in terms of such structures. For that reason, the algebra behind
the YBE is now the focus of intensive research. One particular algebraic structure stands out: skew braces, whose theory has its origins in Jacobson radical rings, and now appears in several
different areas of mathematics \cite{MR3291816,MR3763907}. These algebraic structures cover the class of bijective
non-degenerate combinatorial solutions, i.e. bijective solutions $(X,r)$ such that if one writes
\[
r(x,y)=(\sigma_x(y),\tau_y(x))
\]
for any $x,y \in X$, then all the maps  $\sigma_x$ and $\tau_y$ are bijective.
For simplicity's sake we call bijective non-degenerate solutions just solutions throughout the paper.

A first source of inspiration for this paper is the work
of Diaconis about properties of random algebraic objects \cite{MR1159288}.
Our motivation is the following
question: Which properties can be found in a randomly chosen solution to the YBE?

A solution $(X,r)$ is said to be \emph{multipermutation} if $(X,r)$
can be retracted into the trivial solution
over the singleton after finitely many steps of identifying elements $x$ and $y$ with $\sigma_x =\sigma_y$ and $\tau_x =\tau_y$.
Our first goal is to \emph{provide a rigorous framework}
for the following experimental observation: \emph{almost all}
finite solutions are multipermutation solutions.
In fact, among all involutive
solutions with at most ten elements, more than 95\% are
multipermutation solutions.

The importance of multipermutation solutions was first recognized
in the seminal work of Etingof, Schedler and Soloviev \cite{MR1722951} for involutive solutions. This was
later confirmed by several existing connections
between multipermutation solutions
and other topics such as left-ordered groups and right nilpotency skew braces, see for example
\cite{MR3177933,MR2652212,MR3861714,MR2885602,MR3814340}.

At the moment, we are far from being able to
understand properties of randomly chosen solutions to the YBE. However,
to provide a rigorous framework
that proves that almost every solution is a multipermutation solution,
it is somewhat clear that this will be achieved
studying nilpotency of skew braces.

There are two skew braces that control the structure of a non-degenerate bijective solution $(X,r)$.
The first one is the structure skew brace $G(X,r)$. The multiplicative
group of this brace is isomorphic to the group with generators
in $X$ and relations $xy=\sigma_x(y)\tau_y(x)$ for $x,y\in X$.
The additive
group of $G(X,r)$ is isomorphic to the group
with generators $X$ and relations $y^{-1}xy=\sigma_y \tau_{\sigma^{-1}_x (y) }(x)$.

To study multipermutation solutions right nilpotency
was introduced by Rump  \cite{MR2278047} for skew braces braces of abelian type (that is, the additive group of the respective  skew brace is abelian) and later
extended to arbitrary skew braces in \cite{MR3763907,MR3957824}.
A combination of results of Ced\'{o}, Smoktunowicz and Vendramin \cite[Theorem 2.20]{MR3957824}
and Ced\'{o}, Jespers, Kubat, Van Antwerpen and Verwimp \cite[Theorem 4.13]{cedo2020various}
proved that the following equivalence  for solutions:
\begin{align*}
(X,r)\text{ \it is multipermutation}
&\Longleftrightarrow
G(X,r)\text{ \it is right nilpotent of nilpotent type},
\end{align*}
where being of nilpotent type means that the additive structure of the respective skew brace is nilpotent. This was shown by Gateva--Ivanova \cite{MR3861714} for involutive solutions.
The second skew brace associated to a non-degenerate bijective solution is the permutation skew brace $\mathcal{G}(X,r)$, i.e.
the subgroup of $\mathbb{S} (X) \times \mathbb{S} (X)$  generated by the elements $(\sigma_x,\tau_x^{-1})$. This skew brace is a quotient of $G(X,r)$ by an ideal contained in
its socle $\operatorname{Soc}(G(X,r))$.
Moreover, using Proposition 2.19 in \cite{MR3957824}, one obtains the characterization $$(X,r)\text{ \it is multipermutation} \Longleftrightarrow \mathcal{G}(X,r)\text{ \it is right nilpotent of nilpotent type}.$$ In particular, if $(X,r)$ is finite, this characterization is realized by the  finite skew brace $\mathcal{G}(X,r)$.

By analogy, Rump \cite{MR2278047}, and Ced\'{o}, Smoktunowicz and Vendramin \cite{MR3957824}
also defined left nilpotency of a skew brace, where the latter authors showed that left nilpotency of skew braces of nilpotent type is equivalent to the multiplicative group being nilpotent.
Left and right $p$-nilpotency of finite
skew braces were studied in \cite{MR4062375,MR3935814}.
Recently, motivated by universal algebra,
Bonatto and Jedli{\v c}ka \cite{BonJed2021} introduced
the notion of central nilpotency, which yields a subclass of left and right nilpotent skew braces. As this class of skew braces is defined through iterative use of the annihilator, we call this annihilator nilpotent.

Computer experiments show that among the 36116 involutive multipermutation solutions of size at most eight 34770 of them admit a left nilpotent $\mathcal{G}(X,r)$ permutation brace. This implies that at least for solutions of size at most $8$ almost all of their permutation braces are annihilator nilpotent.

There are several reasons to consider annihilator nilpotent skew braces. On the one hand, the previous observation suggests that the study of
annihilator nilpotent skew braces is the first step towards
the problem of finding a precise framework that allows us to prove
that almost all solutions are multipermutation solutions. On the other hand,
this structural property is extremely important for the general algebraic
theory of skew braces, as annihilator nilpotent skew braces are to skew
braces what nilpotent groups are for groups.

\medskip
The paper is organized as follows. In Section \ref{preliminaries}
we discuss definitions and basic properties of skew braces. We introduce
annihilator nilpotent skew braces and present several examples.
This section also includes several
characterizations of anihilator nilpotency and results regarding
connections between this
and similar nilpotency notions. In Section \ref{noetherian} we study
annihilator nilpotent skew braces with ascending chain condition (ACC) on sub skew braces.
It turns out that  an annihilator nilpotent skew brace
has ACC on sub skew braces
if and only if it is finitely generated as a skew brace.
In Section \ref{torsion}
we study the relationship between additive and multiplicative torsion
of annihilator nilpotent skew braces. Finally, in Section \ref{generating}
we prove a brace-theoretic analog of a result of
Hirsch about non-generating elements (Theorem \ref{thm:HirschBraceNilp})
and study its consequences.

\section{Definitions and preliminaries}
\label{preliminaries}
In this section we fix notation and introduce some crucial definitions and lemmas.
A \emph{skew brace} is a triple $(B,+,\circ)$, where $(B,+)$ and $(B,\circ)$ are
groups (called the additive and multiplicative group respectively) and
\[
a\circ (b+c)=a\circ b-a+a\circ c
\]
holds for all $a,b,c\in B$. If $B$
is a skew brace and $a\in B$, then $\overline{a}$ denotes the inverse of
$a$ with respect to the multiplicative operation.
For  a skew brace $B$ the map $\lambda\colon (B,\circ)\to\operatorname{Aut}(B,+)$, $a\mapsto \lambda_a$, with
$\lambda_a(b)=-a+a\circ b$, is a group homomorphism.

A skew brace is said to be \emph{trivial} if both operations coincide.

We regularly make use of the following lemma, see for example \cite{MR4023387}.
Recall that for elements $a$ and $b$ of a
skew left brace $B$ one  puts
\[
a * b = \lambda_a(b)-b=-a+a\circ b-b.
\]
This operation $*$ measures the ``difference'' between the additive
and multiplicative operations. The center of a group $(G,\cdot)$ is denoted $Z(G)$ or $Z(G,\cdot)$ to emphasize the operation.

\begin{lem}
\label{lem:id*}
 Let $B$ be a skew brace. For any $x,y,z \in B$  the following properties hold:
 \begin{enumerate}
 \item  $x*(y+z)=x*y +y +x*z-y$,
 \item $(x+ y)*z=x*(\lambda_x^{-1}(y) * z)+\lambda_x^{-1}(y)*z +x*z$,
 \item $(x \circ y) *z = x*(y*z) + y*z + x*z$.
 \end{enumerate}
\end{lem}

\begin{defn}
Let $B$ be a skew brace. Recall that the socle  of $B$, denoted $\operatorname{Soc}(B)$, is defined as $$ \operatorname{Soc}(B) = \Ker \lambda \cap Z(B,+).$$ Moreover, the annihilator $\operatorname{Ann}(B)$ of $B$ is defined as $$ \operatorname{Ann}(B) = \operatorname{Soc}(B) \cap \mathrm{Fix}(B),$$
where $\mathrm{Fix}(B)=\{x\in B:\lambda_a(x)=x\text{ for all $a\in B$}\}$.
\end{defn}

Note that
\[
\operatorname{Ann}(B)=\{ x\in B : x\circ a =a\circ x =x+a =a+x, \mbox{ for all } x\in B\}.
\]
Similarly to the upper central series in nilpotent groups, one can define the $n$-th annihilator of a skew brace. If $\operatorname{Ann}_n(B)$ is defined, then denote $\pi:B \longrightarrow B/\operatorname{Ann}_n(B)$ the canonical morphism. One denotes
\[
\operatorname{Ann}_{n+1}(B) = \pi^{-1}(\operatorname{Ann}(B/\operatorname{Ann}_n(B))).
\]
By the isomorphism theorems, these are all ideals in $B$.

In \cite{BonJed2021}, Bonatto and Jedli{\v c}ka introduced the following notion of nilpotency. Note that they call this centrally nilpotent, as the annihilator of a skew brace corresponds to a notion of center coming from universal algebra. However, as this would further mystify the terminology, we opt to call this notion annihilator nilpotent.

\begin{defn}
Let $B$ be a skew brace. An ascending series
\[
0=I_0 \subseteq I_1 \subseteq ... \subseteq I_n = B
\]
of ideals of $B$ is called an annihilator series of $B$, if $I_{j+1}/I_j \subseteq \operatorname{Ann}(B/I_{j})$ for $0\leq j <n$. If $B$ has an annihilator series, then $B$ is called annihilator nilpotent. Clearly, $B$ is annihilator nilpotent if and only if $\operatorname{Ann}_n(B)=B$ for some positive integer $n$.
\end{defn}

Obviously, sub skew braces and epimorphic images of annihilator nilpotent skew braces also are annihilator skew braces. Note that if $B$
is an annihilator nilpotent skew brace,
then both $(B,+)$ and $(B,\circ)$ are nilpotent groups.

\begin{exa}
Let $(G,\circ)$ be a group and denote by $\circ^{opp}$ the opposite operation of $\circ$. Then $(G,\circ,\circ^{opp})$ is annihilator nilpotent if and only if the group
$(G,\circ)$ is nilpotent. In particular, every result on annihilator nilpotent skew braces recovers a similar result on nilpotent groups.
\end{exa}

Recall that a skew brace $B$ is said to be \emph{two-sided} if
\[
(a+b)\circ c=a\circ c-c+b\circ c
\]
holds for all $a,b,c\in B$.

\begin{exa}
It was shown in \cite{MR2278047} that a Jacobson radical ring
$(B,+,*)$ corresponds to a two-sided brace $B$, where
$a\circ b = a+ a*b +b$. Moreover, ring theoretic ideals of $B$
coincide with brace theoretic ideals of $B$.
Hence nilpotent Jacobson radical rings (for example finite Jacobson radical rings) are annihilator nilpotent as braces.
To prove this one makes use of the following fact:  if the $*$-product
of $n$ factors of $B$ is  zero, then the $*$-product of $n-1$ factors
of $B$ is contained  in $\operatorname{Ann}(B)$ as brace.
\end{exa}

\begin{exa}
The operation $x\circ y=x+y+2xy$, using the ring structure on $\mathbb{Z}/4$, induces an annihilator nilpotent skew brace $(\mathbb{Z}/4,+,\circ)$ with multiplicative group
isomorphic to $\mathbb{Z}/2\times\mathbb{Z}/2$ and additive group $\mathbb{Z}/4$.
\end{exa}

\begin{exa}
Of the $47$ skew braces of order $8$, there exist only $2$ skew braces that are not annihilator nilpotent, which even turn out to be of abelian type. This indicates that they dominate skew $p$-braces. Note that only $3$ of these are not two-sided. Of order $16$ only $40$ of $1605$ skew braces are not annihilator nilpotent. Of order $27$ only $4$ of the $101$ are not annihilator nilpotent of which $27$ are not two-sided.
\end{exa}

We now link the notion of annihilator nilpotent skew braces to the known notions of nilpotency. For this we first recall these. The following notions of left and right nilpotent were defined by Rump \cite{MR2278047}. It was shown \cite{MR3814340,MR3957824} that left nilpotency coincides for finite skew braces of nilpotent type
(i.e. the additive group is nilpotent) to the nilpotency of the multiplicative group. Moreover, it was shown \cite{MR3957824} that such a finite skew brace is the direct product of braces of prime power order.

Let $B$ be a skew brace and let $X$ and $Y$ be subsets of $B$. Then $X*Y$ is, by definition,
the additive subgroup of $B$ generated by $\{x*y:x\in X,\,y\in Y\}$.

\begin{defn}
Let $B$ be a skew brace. Denote $B^{n+1} = B*B^n$ with $B^1=B$ for any positive integer $n$.
Then the skew brace $B$ is said to be left nilpotent if $B^n=0$ for some positive integer $n$.
\end{defn}

On the other hand, as mentioned in the introduction, right nilpotency is related to
multipermutation solutions. In fact, a solution $(X,r)$ is multipermutation if and only if $G(X,r)$ is a right nilpotent skew brace of nilpotent type.

\begin{defn}
Let $B$ be a skew brace. Denote $B^{(n+1)}=B^{(n)}*B$ with $B^{(1)}=B$.
Then the skew brace $B$ is said to be right nilpotent if $B^{(n)}=0$ for some positive integer $n$.
\end{defn}

Smoktunowicz introduced \cite{MR3814340} strongly nilpotent skew braces.

\begin{defn}
Let $B$ be a skew brace. Denote
\[
B^{\left[n+1\right]}=\sum_{i=1}^n B^{[i]}*B^{[n-i]}
\]
with $B^{[1]}=B$. Then the skew brace $B$ is said to be strongly nilpotent, if $B^{[n]}=0$ for some positive integer $n$.
\end{defn}

It was shown by Ced\'{o}, Smoktunowicz and Vendramin \cite{MR3957824} (and earlier for left braces by Smoktunowicz) that strongly nilpotent skew braces coincide with skew braces that are both left and right nilpotent.
Hence we have the following link between the different notions of nilpotency of skew braces.

\[\begin{tikzcd}
 & \text{Annihilator nilpotent} \arrow{d} &
 \\
 & \text{Strongly nilpotent} \arrow{dl} \arrow{dr}&
 \\
 \text{Left nilpotent} & & \text{Right nilpotent}
\end{tikzcd}\]

\begin{exa}
    Let $X=\{1,\dots,5\}$ and $(X,r)$ be the solution
    given by $\sigma_1=\sigma_2=\sigma_3=\mathrm{id}$, $\sigma_4=(23)(45)$ and
    $\sigma_5=(12)(45)$. Then $(X,r)$ is a multipermutation
    solution. Moreover, $\mathcal{G}(X,r)$ is a skew brace
    of abelian type with additive group isomorphic to $\mathbb{Z}/6$ and
    multiplicative group isomorphic to $\mathbb{S}_3$. The skew brace
    $\mathcal{G}(X,r)$ is right nilpotent and not left nilpotent.
\end{exa}

Let $A$ be a skew brace. An \emph{s-series} of $A$ is a sequence
\[
A=I_0\supseteq I_1\supseteq I_2\supseteq\cdots\supseteq I_n=0
\]
of ideals of $A$ such that $I_{j-1}/I_j\subseteq\operatorname{Soc}(A/I_j)$ for each $j$.

In \cite[Lemma 2.14]{MR3957824} it is proved that
if $A=I_0\supseteq I_1\supseteq I_2\supseteq\cdots\supseteq I_n=0$ is an $s$-series for $A$, then $A^{(i+1)}\subseteq I_i$
for all i.

\begin{pro}\label{pro:leftandrightgivesmult}
Let $B$ be a skew brace of nilpotent type. If $B$ is left and right nilpotent, then $(B,\circ)$ is a nilpotent group.
\end{pro}
\begin{proof}
Let $B$ be left and right nilpotent of nilpotent type. Then $B$ possesses an s-series by \cite[Lemma 2.16]{MR3957824}.
We proceed by induction on the length of this series. If $B = \operatorname{Soc}(B)$, the theorem clearly holds.
Suppose we have shown that the theorem holds for skew braces with a socle series of length at most $k$.

Let $B$ be a left and right nilpotent skew brace of nilpotent type with a socle series of length $k+1$.
Denote $\gamma_n(B,\circ)$ the $n$-th derived group of $(B,\circ)$. Then, applying the induction
hypothesis on $B/\operatorname{Soc}(B)$, one obtains that $\gamma_m(B,\circ) \subseteq \operatorname{Soc}(B)$ for some positive
integer $m$.
Consider $x \in \operatorname{Soc}(B)$ and $b \in B$, then $$ \left\lbrack b,x \right\rbrack_{\circ} =
b \circ x \circ \overline{b} \circ \overline{x} =
 b + \lambda_{b}(x) - \lambda_{b\circ x \circ \overline{b}}(b)
 - \lambda_{\left\lbrack b,x \right\rbrack_{\circ}}(x).$$
As $\lambda_x = \mathrm{id}_B$  and because $\lambda_b(x) \in Z(B,+)$
this becomes $$\left\lbrack b,x \right\rbrack_{\circ} = b + \lambda_b(x) -b-x= b*x.$$
As $\operatorname{Soc}(B)$ is a normal subgroup of $(B,\circ)$ and $\gamma_m(B,\circ) \subseteq \operatorname{Soc}(B)$, it follows that for all positive integers
$l$ it holds that $\gamma_{n+l}(B) \subseteq \operatorname{Soc}(B)$. Hence, applying the formula above, one
obtains that $\gamma_{n+l}(B) \subseteq B^l$. As $B$ is left nilpotent, this shows that
the derived series of $(B,\circ)$ ends in $0$ and thus $(B,\circ)$ is a nilpotent group.
\end{proof}

Inspired by the lower central series of group theory,
Bonatto and Jedli{\v c}ka \cite{BonJed2021} introduced the following series of ideals.

Let $B$ be a skew brace and $X$ and $Y$ be subsets of $B$. We write $[X,Y]_+$ (resp. $[X,Y]_\circ$) to denote
the additive (resp. multiplicative) subgroup generated by commutators
\[
[x,y]_+=-x-y+x+y\qquad\text{(resp. $[x,y]_\circ=\overline{x}\circ\overline{y}\circ x\circ y$)},
\]
$x\in X$ and $y\in Y$.

\begin{defn}
Let $B$ be a skew brace and $I$ an ideal of $B$. Denote $\Gamma_0(I) = I$ and $\Gamma_{n+1}(I) =\left< \Gamma_n(I)*B, B*\Gamma_n(I), [B,\Gamma_n(I)]_+\right>_+$ for $n\geq0$.
\end{defn}

Note that if $B$ is a skew brace and $I$ is an ideal of $B$, then each $\Gamma_n(I)$ is an ideal of $B$.
It was shown by Bonatto and Jedli{\v c}ka \cite{BonJed2021} that the series
  $$\Gamma_0 (B) \supseteq \Gamma_1 (B) \supseteq \Gamma_2 (B) \supseteq \cdots$$
 plays the role of a lower central series for annihilator nilpotency, which we state for further reference.

\begin{thm}[Bonatto--Jedli{\v c}ka]
\label{thm:bonatto-nilpotent}
Let $B$ be a skew brace. Then $B$ is annihilator nilpotent if and only if $\Gamma_c(B) = 0$ for some positive integer $c$.
\end{thm}

\begin{proof}
    See \cite[Theorem 2.7]{BonJed2021}.
\end{proof}

\begin{corollary}
Let $B$ be a skew brace of nilpotent type. Then the following properties  are equivalent:
\begin{enumerate}
    \item $B$ is annihilator nilpotent.
    \item $B$ is left and right nilpotent.
    \item $B$ is strongly nilpotent.
\end{enumerate}
\end{corollary}

\begin{proof}
By \cite[Theorem 2.3]{MR3957824}, statements $(2)$ and $(3)$ are equivalent.

That $(1)$ implies $(2)$ follows immediately from Theorem \ref{thm:bonatto-nilpotent}.

To prove $(2)$ implies $(1)$,
assume  $B$ is left and right nilpotent. We will proceed by induction on the length of the s-series of $B$, which exists
because of \cite[Lemma 2.16]{MR3957824}.
By Proposition~\ref{pro:leftandrightgivesmult}, $(B,\circ)$ is a nilpotent group of class $c$. As $\operatorname{Soc}(B)$ is a normal subgroup of $(B,\circ)$, it follows that
\[
0 \neq \operatorname{Soc}(B)\cap Z(B,\circ) =\operatorname{Ann}(B).
\]
Inductively, one sees that $\operatorname{Soc}(B) \cap Z^k(B,\circ) \subseteq\operatorname{Ann}_k(B)$ for all $k$.
Hence $\operatorname{Soc}(B) \subseteq \operatorname{Ann}_c(B)$. Since, by the induction hypothesis, $B/\operatorname{Soc}(B)$ has a finite annihilator series,
it follows that $B$ has a finite annihilator series and thus $B$, i.e. $B$ is annihilator nilpotent.
\end{proof}

Inspired by the notion of strong nilpotency and a question of Bonatto and Jedli{\v c}ka, we introduce the following chain of ideals.
\begin{defn}
Let $B$ be a skew brace. Denote $$\Gamma_{[n]}(B)=\left< \Gamma_{[i]}(B)*\Gamma_{[n-i]}(B), \left\lbrack\Gamma_{[i]}(B),\Gamma_{[n-i]}(B)\right\rbrack_+: 1\leq i\leq n-1\right>_+,$$ with $\Gamma_1(B)=B$.
\end{defn}

\begin{lem}
Let $B$ be a skew brace. Then each $ \Gamma_{\lbrack n\rbrack}(B)$ is an ideal and $\Gamma_{\lbrack n+1 \rbrack}(B) \subseteq \Gamma_{\lbrack n \rbrack}(B)$ for all positive integers $n$.
\end{lem}

\begin{proof}
We will show the result by induction on the index $n$. Clearly, $\Gamma_{[1]}(B)$ is an ideal. Moreover, as $\Gamma_{[2]}(B)=B*B+\left\lbrack B,B\right\rbrack_+$, it is easy to see that $\Gamma_{[2]}(B)/B*B$ is an ideal of $B/B*B$, which shows that $\Gamma_{[2]}(B)$ is an ideal and, clearly, $\Gamma_{[2]}(B)\subseteq \Gamma_{[1]}(B)$.  Suppose that for $k<n$ we have that $\Gamma_{[k]}(B) \subseteq \Gamma_{[k-1]}(B)$ and $\Gamma_{[k]}(B)$ is an ideal. We show the result for $n$. Note that if $k>1$ then, by the induction hypothesis, $$\Gamma_{[k]}(B) * \Gamma_{[n-k]}(B) \subseteq \Gamma_{[k-1]}(B)*\Gamma_{[n-k]}(B) \subseteq \Gamma_{[n-1]}(B).$$ Analogously, one proves that $\left\lbrack \Gamma_{[k]}(B),\Gamma_{n-k]}(B)\right\rbrack \subseteq \Gamma_{[n-1]}(B)$, which shows that $\Gamma_{[n]}(B) \subseteq \Gamma_{[n-1]}(B)$. In particular, this entails, by definition of $\Gamma_{[n]}(B)$, that $$ B*\Gamma_{[n]}(B) \subseteq B*\Gamma_{[n-1]}(B) \subseteq \Gamma_{[n]}(B).$$ Analogously, one shows that
$\Gamma_{[n]}(B)*B\subseteq \Gamma_{[n]}(B)$ and
\[
\left\lbrack B, \Gamma_{[n]}(B)\right\rbrack_+ \subseteq \Gamma_{[n]}(B).
\]
As $\Gamma_{[n]}(B)$ is an additive subgroup by definition, the result follows.
\end{proof}

The following proposition further substantiates the problem proposed by Bonatto and Jedli{\v c}ka whether all commutators,
coming from a universal algebra setting, are nested binary commutators and whether the latter are immediately satisfied by annihilator nilpotent skew braces.
Furthermore, it allows us to use the series $\Gamma_{[n]}(B)$ in the rest of the paper.

\begin{thm}\label{thm:gamma}
Let $B$ be a skew brace. Then $B$ is annihilator nilpotent if and only if there exists a positive integer $c$ such that $\Gamma_{[c]}(B) = 0$.
\end{thm}
\begin{proof}
The sufficiency is obvious as $\Gamma_n(B) \subseteq \Gamma_{[n]}(B)$ for all $n$.

We prove necessity by induction. If $\Gamma_{2}(B)=0$, then $B$ is trivial brace. Hence $\Gamma_{[2]}(B) = 0$, which proves the base case.
Suppose that we have shown that $\Gamma_{\left[2^k\right]}(B)=0$ that for all skew braces $B$ with $\Gamma_{k}(B)=0$ for $k\leq n$.

Consider a skew brace $B$ such that $\Gamma_{n+1}(B)=0$. Let $a \in \Gamma_{\left[2^{n+1}\right]}(B)$.
Then $a$ is a sum of elements of the form $a_i*b_i$
and $\left\lbrack a_i, b_i \right\rbrack$ with $a_i \in \Gamma_{[q_i]}$ and $b_i\in \Gamma_{\left[2^{n+1}-q_i\right\rbrack}$. Consider the cases where $q_i \geq 2^{n}$. Then applying the induction hypothesis on $B/\Gamma_{n}(B)$, we obtain that $a_i\in \Gamma_n(B)$.
Hence both $a_i*b_i \in \Gamma_{n+1}(B)=0$ and $\left\lbrack a_i, b_i \right\rbrack\in \Gamma_{n+1}(B)=0$,
showing that these terms are all zero. Remain the cases where $q_i<2^n$.
However, applying the same reasoning on $b_i \in \Gamma_{[2^{n+1}-q_i]}(B)\subseteq \Gamma_{\left\lbrack 2^{n}\right\rbrack}(B)$
we obtain that also all these terms are zero. Hence $a=0$.
\end{proof}

\section{Annihilator nilpotent skew braces with ACC on sub skew braces}
\label{noetherian}

A skew brace is said to have  the ascending chain condition (ACC) on sub skew braces if it satisfies the ascending chain condition on sub skew braces.
In this section we show that for annihilator nilpotent skew braces this condition is equivalent with being finitely generated as a skew braces, or equivalent with its additive (respectively multiplicative) group being finitely generated.
The necessity of these conditions is shown in the following lemma.

\begin{lem}\label{lem:noetherianannnilp}
Let $B$ be a skew brace. If $B$ has ACC on sub skew braces and and is annihilator nilpotent, then the following properties are equivalent:
\begin{enumerate}
    \item The skew brace $B$ is finitely generated.
    \item The group $(B,+)$ is finitely generated.
    \item The group $(B,\circ)$ is finitely generated.
\end{enumerate}
\end{lem}
\begin{proof}
Let $B$ be finitely generated. Consider the annihilator series $$ 0 \subseteq \operatorname{Ann}(B) \subseteq ... \subseteq \operatorname{Ann}_k(B) = B.$$ As $B$ has ACC on sub skew braces, the ideal $\operatorname{Ann}(B)$ is finitely generated as a skew brace. As the latter is a trivial brace, this implies that it is both additively and multiplicatively finitely generated. Similarly,
\[
\operatorname{Ann}_{i+1}(B)/\operatorname{Ann}_{i}(B)
\]
is a finitely generated trivial brace. Hence, every factor of the annihilator series is finitely generated both as additive and multiplicative group, which implies that $B$ is finitely generated as additive and multiplicative group.

Clearly, if $(B,+)$ or $(B,\circ)$ are finitely generated, then $B$ is finitely generated, showing the result.
\end{proof}

In the following lemma, we often use, without mention, the following identity (stated in Lemma~\ref{lem:id*}.(3)) that holds in any skew brace
 $$ (a\circ b) * c = a*(b*c) +b*c +a*c.$$

\begin{lem}\label{lem:gammaminus2}
Let $B$ be an annihilator nilpotent skew brace with $\Gamma_{[c]}(B)=0$. If $a \in \Gamma_{\left[c-k\right]}(B)$ and $q,w \in \Gamma_{[k-1]}$, then the following statements hold:
\begin{enumerate}
\item $a*(q + w) = a*q + a*w$,
\item $a*(q\circ w) = a*q + a*w$,
\item $(q\circ w) *a = q*a + w*a$,
\item $(q+w)*a = q*a + w*a$,
\item $\left\lbrack a,q+w\right\rbrack_+ = \left\lbrack a,q\right\rbrack_{+} + \left\lbrack a, w \right\rbrack_{+}$,
\item $\left\lbrack a,q\circ w\right\rbrack_+ = \left\lbrack a,q\right\rbrack_{+} + \left\lbrack a, w \right\rbrack_{+}$.
\end{enumerate}
\end{lem}
\begin{proof}
$(1)$ Note that $$a*(q+w) = a*q + q + a*w -q = a*q + \left\lbrack q, a*w \right\rbrack_{+} +a*w.$$ As $a \in \Gamma_{[c-k]}(B)$ and $a*w \in \Gamma_{[c-1]}(B)$, we have that $ \left\lbrack q, a*w \right\rbrack_{+}\in \Gamma_{[c]}=0$
and thus  one obtains that  $$a*(q + w) = a*q + a*w,$$ showing the first identity.

$(2)$ It easily is verified that $$a*(q \circ w) = a* (q + q*w + w).$$ Hence, by (1),
$$ a*(q\circ w)= a*q + a*(q*w) + a*w.$$ As $a*(q*w) \in \Gamma_{[c-k]}(B)*\Gamma_{[k]}(B)=0$, it follows that  $a*(q\circ w)=a*q+a*w$.

$(3)$ Note that $$(q\circ w)*a = q*(w*a)+w*a+q*a.$$
 As $q*(w*a) \in \Gamma_{[c]}(B)$ and $w*a \in Z(B,+)$, it follows that this is equal to $q*a+w*a$.

$(4)$ Because  $a\circ b = a+\lambda_{b}$, we obtain that $$ (q+w)*a = (q \circ \lambda_{\overline{q}}(w))*a.$$
Using identity $(3)$, we get that  $(q+w)*a =q*a + \lambda_{\overline{q}}(w)*a$. Applying this again on the last term, we obtain $$\lambda_{\overline{q}}(w)*a = (\overline{q}*w+w)*a = (\bar{q}*w)*a + \lambda_{\overline{q}*w}(w)*a.$$ As $(\bar{q}*w)*a \in \Gamma_{[k]}*\Gamma_{[c-k]}=0$, we obtain that $\lambda_{\overline{q}}(w)*a=\lambda_{\overline{q}*w}(w)*a$. Denoting $q_0=\overline{q}$ and $q_{n+1}=q_n*w$, we obtain inductively that $\lambda_{q_n}(w)*a = \lambda_{q_{n+1}}(w)*a$. As $q_n \in B^n$, it follows that $q_c=0$. Hence, we have shown that $$(q+w)*a = q*a + \lambda_{q_c}(w)*a=q*a+w*a.$$

$(5)$ This follows easily from the fact that $[a,q]_+\in Z(B,+)$.

$(6)$ This follows from  $(5)$ and the fact that
\[
[a,q*w]_+ \in [\Gamma_{[c-k]}(B),\Gamma_{k}(B)]_+=0.
\]
This completes the proof.
\end{proof}

\begin{thm}\label{thm:noetherian}
Let $B$ be an annihilator nilpotent skew brace. If $B$ is finitely generated (as a skew brace), then $B$ has ACC on sub skew braces.
\end{thm}

\begin{proof}
By Theorem \ref{thm:gamma}, we can proceed by induction on the positive integer $c$ such that $\Gamma_{[c]}=0$. If this integer is $1$, then $B$ is a trivial skew brace of abelian type. In particular, the result becomes a well-known theorem on finitely generated abelian groups.

Suppose we have shown that every annihilator nilpotent skew brace $B$ with $\Gamma_{[n]}(B)=0$ has ACC on sub skew braces.
Consider $B$ an annihilator nilpotent skew brace such that $\Gamma_{[n+1]}(B)=0$.
Thus, $\Gamma_{[n]}(B) \subseteq \operatorname{Ann}(B)$. Let $H$ be a sub skew brace of $B$.
Then $H/ (H \cap \Gamma_{[n]}(B))$ is finitely generated by applying the
induction hypothesis on $B/\Gamma_{[n]}(B)$. Rests to show that
$H \cap \Gamma_{[n]}(B)$ is finitely generated as a skew brace. As $\Gamma_{[n]}(B) \subseteq \operatorname{Ann}(B)$, it is sufficient to show that $\Gamma_{[n]}(B)$ is finitely generated as a skew brace.

Let $1 \leq k \leq n-1$ and consider $\Gamma_{[k]}(B) * \Gamma_{[n-k]}(B)$. Then, by the induction hypothesis, both $\Gamma_{[k]}(B)$ and $\Gamma_{[n-k]}(B)$ are finitely generated and have ACC on sub skew braces modulo $\Gamma_{[n]}(B)$. By Lemma \ref{lem:noetherianannnilp}, the multiplicative group of the factor $\Gamma_{[k]}(B)/\Gamma_{[n]}(B)$ is finitely generated, say by the natural image of the set $A=\left\lbrace a_1,...,a_r\right\rbrace$ and the additive group of the factor $\Gamma_{[n-k]}(B)/\Gamma_{[n]}(B)$ is finitely generated, say by the natural image of the set $Y=\left\lbrace y_1,...,y_s\right\rbrace$. Without loss of generality, we may assume that the set $Y$ (resp. $A$) is closed under taking additive (resp. multiplicative) inverses. Let $\alpha \in \Gamma_{[k]}(B)$ and $\beta \in \Gamma_{[n-k]}(B)$. Then there exist $s_1,s_2 \in \Gamma_{[n]}(B)$ such that $\alpha = s_1 \circ \left( \Pi_{i=1}^f a_{j_i}\right)$ and $\beta = s_2 + \left(\sum_{i=1}^{g}b_{l_i}\right)$. Note that $$\alpha * \beta= \left(s_1 \circ \left( \Pi_{i=1}^f a_{j_i}\right)\right) * \left(s_2 + \left(\sum_{i=1}^{g}b_{l_i}\right)\right) = \left( \Pi_{i=1}^f a_{j_i}\right) * \left(\sum_{i=1}^{g}b_{l_i}\right),$$ where the last equality holds as $s_1,s_2 \in \Gamma_{[n]}(B) \subseteq \operatorname{Ann}(B)$. Applying parts (1) and (3) of Lemma \ref{lem:gammaminus2}, this is further equal to a sum of elements of the form $a_{j_i}*b_{l_j}$. This shows that the additive group $\Gamma_{[k]}(B)*\Gamma_{[n-k]}(B)$ is finitely generated.

Completely analogously, applying $(5)$ and $(6)$ of Lemma \ref{lem:gammaminus2}, one sees that the additive group $\left\lbrack\Gamma_{[k]}(B), \Gamma_{[n-k]}(B)\right\rbrack_+$ is finitely generated. As, by definition, $\Gamma_{[n]}(B)$ is the additive subgroup generated by $\Gamma_{[k]}(B)*\Gamma_{[n-k]}(B)$ and $\left\lbrack \Gamma_{[k]}(B),\Gamma_{[n-k]}(B)\right\rbrack_+$ for
all $k\in\{1,\dots,n-1\}$, it follows that $\Gamma_{[n]}(B)$ is finitely generated.
\end{proof}

\begin{corollary}
Let $B$ be an annihilator nilpotent skew brace. Then the following statements are equivalent:
\begin{enumerate}
    \item The skew brace $B$ is finitely generated (as a skew brace).
    \item The group $(B,+)$ is finitely generated.
    \item The group $(B,\circ)$ is finitely generated.
\end{enumerate}
\end{corollary}

\begin{proof}
    We prove $(1)\implies(2)$ and $(1)\implies(3)$, as the converses are trivial.
    We proceed by induction on the length of the annihilator series of $B$.
    If the length is one, then the skew brace is trivial and the claim follows. Let us assume the result
    holds for length $k\geq1$. Let $B$ be a skew brace with an annihilator series of length $k+1$.
    By Theorem \ref{thm:noetherian}, $B$ has ACC on sub skew braces. In particular,
    $\operatorname{Ann}(B)$ is finitely generated (as a skew brace). As $\operatorname{Ann}(B)$ is a trivial skew brace, the group
    $(\operatorname{Ann}(B),+)=(\operatorname{Ann}(B),\circ)$ is finitely generated. The quotient $B/\operatorname{Ann}(B)$ is annihilator nilpotent
    of length $k$ and finitely generated (as a skew brace). Thus the claim follows from the
    inductive hypothesis.
\end{proof}

\begin{cor}
\label{cor:Hopfian}
Let $B$ be a finitely generated skew brace that is annihilator nilpotent. Then $B$ is Hopfian, i.e. every surjective skew brace endomorphism of $B$ is an isomorphism.
\end{cor}
\begin{proof}
Let $f: B \longrightarrow B$ be a surjective morphism of braces. Then the induced morphism $\tilde{f}:B/B^2 \longrightarrow B/B^2$ is also surjective. As $B$ is a finitely generated skew brace, it follows that $B/B^2$ is a finitely generated nilpotent group. As $B/B^2$ is Hopfian, it follows that $\tilde{f}$ is an isomorphism. Hence, $\ker f$ is contained in $B^2$. Furthermore, as $f$ is surjective, it can be restricted to a surjective skew brace morphism $f_1: B^2 \longrightarrow B^2$. As $B^2$ is finitely generated by Theorem \ref{thm:noetherian}, the result follows by induction.
\end{proof}

Note that Corollary \ref{cor:Hopfian} can also be proven as a direct consequence of the ACC property, similar to standard proofs in group and module theory.

\section{Periodic elements in annihilator nilpotent skew braces}
\label{torsion}

In this section we link the torsion of the additive group with the torsion of the multiplicative group for annihilator nilpotent skew braces. Futhermore, we show that finitely generated annihilator nilpotent skew braces are residually finite. Recall that in a locally nilpotent group (i.e. finitely generated subgroups are nilpotent) the periodic elements form a normal locally finite subgroup. The periodic elements of a group $G$ will be denoted by $T(G)$. For a skew brace $A=(A,+,\circ)$ we denote by $T_{+}(A)$ the torsion elements of the additive group $(A,+)$, and by $T_{\circ}(A)$ the periodic elements of the multiplicative group $(A,\circ)$.

\begin{lem}\label{lem:torsioncentral}
Let $A$ be an annihilator nilpotent skew brace. If
$T_{+}(\operatorname{Ann}(A)) =\{ 0\}$, then
$T_{+}(\operatorname{Ann}_{n+1}(A)/\operatorname{Ann}_{n}(A))=\{0\}$ for all $n$.
\end{lem}

\begin{proof}
We prove this by induction on the nilpotence class $m$.
If $m=1$ then the result is obvious.
So suppose the result holds for all annihilator nilpotent skew braces of class less than $m$. In particular it holds for $A/\operatorname{Ann}(A)$.
If $T_{+}(\operatorname{Ann}_{n+1}(A)/\operatorname{Ann}_{n}(A))\neq \{ 0\}$ for some $n$, then by the assumption $n>0$ and, thus
by the induction hypothesis $\operatorname{Ann}_{2}(A)$ contains an element $a+\operatorname{Ann}(A)$ that
is nonzero and periodic in $\operatorname{Ann}_{2}/\operatorname{Ann}(A)$. In particular $A*a\subseteq\operatorname{Ann}(A)$
and $ka\in\operatorname{Ann} (A)$ for some positive integer $k$.
By Lemma~\ref{lem:id*} we know that for any $x,y,x\in A$ we have $x*(y+z)  =
x*y +y +x*z -y$.  In particular,
for any $b\in A$ and positive integer $n$,
\begin{align*}
b*(na) &=b* ((n-1)a +a)=
b*(n-1)a +(n-1)a +b*a -(n-1)a.
\end{align*}
As $b*a\in\operatorname{Ann}(A)$, it follows that $b*na = b*(n-1)a + b*a$. Hence, by induction, $b*na =n(b*a)$.
As $ka\in\operatorname{Ann}(A)$ we obtain that $k(b*a)=0$.

Suppose $A*a\neq \{ 0\}$. Then let  $b\in A$ with $b*a\neq 0$. So, by the previous,  $b*a$ is a non-trivial elements in $T_{+}(A)$, a contradiction.

Hence, $A*a=\{ 0\}$.
Then  $na = a^n$ for all positive integers $n$ and thus also $\lambda_a (a^n) =a^n$.
By Lemma~\ref{lem:id*} we know that
\[
(x+y)*z =x*(\lambda_{x}^{-1} *z) +\lambda_{x}^{-1}*z +x*z
\]
for all $x,y,z\in A$.
Hence, for any $b\in A$,
$$(na)*b=(a+a^{n-1})*b = a*(a^{n-1}*b)  +(a^{n-1})*b +a*b.$$
Since $a^{k-1}*b \in \operatorname{Ann}(A)$ this yields
 $$(na)*b=  ((n-1)a)*b + a*b.$$
So, by induction,
  $$(na)*b=n(a*b),$$
for all positive integers $n$.
Because $ka\in\operatorname{Ann}(A)$ we get that $k(a*b)=0$.
Consequently, if  $a*b\neq 0$, for some $b\in A$,
then $T_+(A) \neq \{ 0\}$, a contradiction.

Therefore,  $A*a=a*A=0$. In $(A,+)$ we have
\[
[x+y,z]_+=-y+[x,z]_++y+[y,z]_+
\]
for all $x,y,x\in A$.
Then
\[
[ka,b]_+=[a+(k-1)a,b]_+= -(k-1)a+[a,b]_++(k-1)a+[(k-1)a,b]_+
\]
for all $b\in A$.
As $\left\lbrack a, b\right\rbrack \in \operatorname{Ann}(B) \subseteq Z(A,+)$, we thus get that
 $$[ka,b]_+
= [a,b]_+ + [(k-1)a,b]_+ =...=k[a,b]_+.$$
Now, if $[a,b]_+ \neq 0$, then $T_+(A)$ is non-trivial.
Hence, $A*a=a*A=0$ and $a \in Z(A,+)$, which implies that $a \in \operatorname{Ann}(A)$, again a contradiction.
\end{proof}

\begin{pro}\label{pro:torsionpart}
Let $A$ be an annihilator nilpotent skew brace. Then
\begin{enumerate}
    \item
$T_{+}(A)= T_{\circ}(A)$,
    \item $T_{+}(A)$ is an ideal of $A$,
    \item  $T_{+}(A/T_{+}(A))=\{ 0\}$,
    \item  $T_{+}(A)$ is finite if $A$ is finitely generated as a skew brace.
\end{enumerate}
\end{pro}
\begin{proof}
The annihilator nilpotency yields that $(A,+)$ and $(A,\circ)$ are nilpotent groups. Hence $T_{+}(A)$ (respectively $T_{\circ}(A)$) is a normal subgroup of $(A,+)$ (respectively $(A,\circ)$).
Since each $\lambda_a$ is an additive homomorphism of $(A,+)$ we have that $T_{+}(A)$ is a left ideal of $A$. In particular it is a skew subbrace of $A$. In case $A$ is finitely generated as a skew brace, we get from Theorem~\ref{thm:noetherian} that $T_{+}(A)$ is a skew brace with ACC on sub skew braces.

Assume that $A$ is finitely generated as a skew brace. We claim that $T_{+}(A)$ is finite.  By Theorem~\ref{thm:noetherian}, each $\operatorname{Ann}_n(A)$ is finitely generated as a skew brace. Hence,  in particular $T_{+}(\operatorname{Ann} (A))$ is a finite abelian group and it is an ideal of $A$. By induction on the nilpotence class, $T_{+}(A/\operatorname{Ann} (A))$ is finite. Since $T_{+}(A)/T_{+}(\operatorname{Ann} (A))\subseteq T_{+}(A/\operatorname{Ann} (A))$ it follows that $T_{+}(A)$ is finite.
A similar argument shows that $T_{+}(A)$ is an ideal of $A$ and that $T_{+}(A)\subseteq T_{\circ}(A)$.

We claim that $T_{+}(A)=T_{0}(A)$. To do so, without loss of generality, we may assume that $A$ is finitely generated as a skew brace. Because $T_{+}(A)$ is an ideal of $A$, $T_{+}(A)\subseteq T_{0}(A)$ and $T_{+}(A/T_{+}(A))=\{ 0\}$, we also may assume that $T_{+}(A)=\{ 0\}$. Hence, by Lemma~\ref{lem:torsioncentral}, each $\operatorname{Ann}_{n+1}(A)/\operatorname{Ann}_{n}(A)$ is a trivial brace and a torsion-free abelian (multiplicative) group. It follows that $(A,\circ)$ is torsion-free as well, and the claim follows.
\end{proof}

As a corollary, one immediately obtains the following result:

\begin{corollary}
Let $A$ be an annihilator nilpotent skew brace such that $T_{+}(A)=\{ 0\}$.
For any $a,b\in A$, $a^n=b^n$ or $na=nb$ implies $a=b$.
\end{corollary}

\begin{lem}\label{lem:finiteanngivesfinite}
Let $B$ be an annihilator nilpotent skew brace. If $\operatorname{Ann}(B)$ is finite and $B$ is finitely generated, then $B$ is finite.
\end{lem}
\begin{proof}
Let $B$ be a counterexample with annihilator chain of minimal length. Either $\operatorname{Ann}_2(B)$ is finite or $\operatorname{Ann}_2(B)$ contains an element that is of infinite order modulo $\operatorname{Ann}(B)$. In the first case, $B/\operatorname{Ann}(B)$ is a counterexample with shorter annihilator chain, so this does not occur. Hence, $\operatorname{Ann}_2(B)$ contains an element of infinite order modulo $\operatorname{Ann}(B)$. We will show that this can not happen. Let $x \in \operatorname{Ann}_2(B)$ of infinite order modulo $\operatorname{Ann}(B)$. Denote by $e$ the exponent $(\operatorname{Ann}(B),\circ)$. Consider $g \in B$. Then it holds that $[x,g]_{\circ}$ is contained in $\operatorname{Ann}(B)$ and $$ [x^e,g]_{\circ} = [x,g]_{\circ}^e = 0.$$ Hence, $(\operatorname{Ann}_{2}(B),\circ)/\operatorname{Ann}(B)$ is a finitely generated periodic abelian group, showing that $\operatorname{Ann}_2(B)$ is finite.
\end{proof}

\begin{thm}
Let $B$ be an annihilator nilpotent skew brace. If $B$ is finitely generated, then $B$ is residually finite.
\end{thm}
\begin{proof}
Let $T$ denote the torsion of $B$. Let $g \in T$. As $\operatorname{Ann}(B)$ is a finitely generated abelian group, it follows that this is equal to $T' \times \mathbb{Z}^l$ for a positive integer $l$ and finite abelian group $T'$. Denote $N_1= \mathbb{Z}^l$ in $\operatorname{Ann}(B)$. As $N_1 \subseteq \operatorname{Ann}(B)$, it holds that $N_1$ is an ideal of $B$. Consider $B/N_1$. Clearly, $g \not \in N_1$, which implies that $g+N_1 \neq N_1$. Suppose we have inductively defined ideals $N_k$, which are torsion-free with $N_{k-1}\subseteq N_k$ and $N_k/N_{k-1} \subseteq \operatorname{Ann}(B/N_{k-1})$. By Lemma \ref{lem:finiteanngivesfinite}, $\operatorname{Ann}(B/N_{k-1})$ is either finite, such that we end the chain, or $\operatorname{Ann}(B/N_{k-1})$ is infinite. Hence, $\operatorname{Ann}(B/N_{k-1})=T_k\times \mathbb{Z}^{l_k}$. Denote $N_k = \mathbb{Z}^{l_k}+N_{k-1}$. The chain $N_k$ ends due to $B$ having ACC on sub skew braces. Hence, there exists $N_k$ such that $\operatorname{Ann}(B/N_k)$ is finite. This shows, by Lemma \ref{lem:finiteanngivesfinite}, that $B/N_k$ is finite. As $g \not\in N_k$, this is a suitable finite image. Hence, we have shown that every torsion element survives in a finite homomorphic image of $B$.

Let $g \in B\setminus T$. Then consider $B'=B/T$. Either $g \in \operatorname{Ann}(B')$ or $g \not\in \operatorname{Ann}(B')$. In the latter case, one can continue by induction on the length of the annihilator chain. In the former case, consider for a positive integer $k>1$ the ideal $G=\langle g^k\rangle\subseteq \operatorname{Ann}(B')$ of $B'$. Consider $B'/G$. Clearly, $g+G \neq G$, but $g+G$ is a torsion element of $B'/G$. Hence, by the first case $g+G$ survives in a finite homomorphic image of $B'/G$, which shows the result.
\end{proof}

\section{Generating sets of annhilator nilpotent skew braces}
\label{generating}

A well-known result of Hirsch states that the commutator subgroup of a nilpotent group $G$ is a set of
non-generating elements of $G$. We prove an analogue of this result in the context of skew braces that are annihilator nilpotent.

\begin{thm}\label{thm:HirschBrace}
Let $B$ be a skew brace of right nilpotency class $c+1$, i.e. $B^{(c+1)}=0$ and $B^{(c)}\neq 0$. If $A$ is a sub skew brace of $B$ such that $A+B^2=B$ and $A*B\subseteq A$, then $A=B$.
\end{thm}
\begin{proof}
The proof is by induction on $c$. For $c=1$ it is clear. Consider the ideal series $$ 0 \subseteq B^{(c)} \subseteq ... \subseteq B^{(2)} \subseteq B.$$ Then the induction hypothesis can be applied on $B/B^{(c)}$. This proves that $B/B^{(c)} = A/(A \cap B^{(c)})$. Hence, $B = A + B^{(c)}$.

We now prove that $B*B\subseteq A$; and thus the result follows. To do so, consider arbitrary elements
$a+b,x+y\in B$ with $a,x\in A$ and $b,y\in B^{(c)}$.
Then, by Lemma~\ref{lem:id*},
\begin{align*}
 (a+b)*(x+y) &= (a+b)*x +x + (a*y)-x\\
   &=a*(\lambda_a^{-1}(b) *x)+ \lambda_a^{-1}(b)*x +a*x +x +(a*y)-x
\end{align*}
Since $B*B$ is an ideal of $B$ and $B^{(c)}*B=0$,
\begin{gather*}
a*(\lambda_a^{-1}(b) *x)+ \lambda_a^{-1}(b)*x=0
\shortintertext{and hence}
(a+b)*(x+y)=x +(a*y)-x.
\end{gather*}
The assumption $A*B\subseteq A$ implies that indeed $B*B\subseteq A$.
\end{proof}

\begin{thm}\label{thm:HirschBraceNilp}
Let $B$ be a annihilator nilpotent skew brace. If $A$ is a sub skew brace of $B$ such that $A+B^2=B$, then $A=B$.
\end{thm}
\begin{proof}
The proof is by induction on the length of the annihilator series. Consider the series $$ 0 \subseteq \operatorname{Ann}(B) \subseteq ... \subseteq \operatorname{Ann}_n(B) = B.$$

Consider the skew brace $B/\operatorname{Ann}(B)$, on which the induction hypothesis applies. This shows that $B = A + \operatorname{Ann}(B)$. In particular, this implies that $$ B*B \subseteq A*A + \operatorname{Ann}(B)*A +A *\operatorname{Ann}(B) + \operatorname{Ann}(B)*\operatorname{Ann}(B) \subseteq A*A.$$
Hence, we obtain that $B=A+B*B \subseteq A$, showing the result.
\end{proof}

\begin{cor}\label{cor:surjectivebrace}
Let $B$ be an annihilator nilpotent skew brace.
Let $f:A\rightarrow B$ be a homomorphism of skew braces and let
$\overline{f}:A/A^2 \rightarrow B/B^2$ be the natural induced homomorphism of skew braces.
Then $f$ is surjective if and only if  $\overline{f}$ is surjective.
\end{cor}

\begin{proof}
Assume $\overline{f}$ is surjective. Let $b\in B$. Then there exists $a\in A$ so that
 $$ \overline{f} (a+A^2) =f(a)+ B^2 = b+B^2.$$
Hence, $b\in f(A)+B^2$. Since this holds for arbitrary $b\in B$, it follows that $B=f(A)+B^2$. So, by Theorem~\ref{thm:HirschBraceNilp}, $B=f(A)$. This proves the sufficiency of the statement. The necessity is obvious.
\end{proof}

\begin{cor} Let $B$ be an annihilator  nilpotent skew brace.
A subset
$S\subseteq B$ generates the skew brace  $B$ if and only if $\overline{S}$ (the natural image of $S$ in $B/B^2$) generates the skew brace  $B/B^2$ (and thus it generates $B/B^2$ as an additive group).
In particular, if $T$ is a subset of $B$ that contains a representative from each $\lambda$-orbit then $B$ is generated (as a skew brace) by the set $T$.
\end{cor}
\begin{proof}
Let $A$ be the skew brace of  $B$ generated by $S$.
If $\overline{S}$ generates $B/B^2$ then $B=A +B^2$, where $A$ is the skew brace of $B$ generated by $S$. Hence, by Theorem~\ref{thm:HirschBraceNilp}, $B=A$. The converse is obvious.
To prove the final claim it is sufficient to  note that $\lambda_b (a)-a =b*a\in B^2$.
\end{proof}

\begin{defn}
Let $B$ be a skew brace and $H$ a sub skew brace. We call the largest sub skew brace $N$ of $B$ such that $H$ is an ideal in $N$, the idealizer of $H$ in $B$. We denote this by $I(H)$.
\end{defn}
Note that $I(H)$ can also be characterized as the largest sub skew brace $A$ of $B$ such that $A*H \subseteq H$, $H*A \subseteq H$ and $\left[H,A\right]_+ \subseteq H$. These conditions remind of the notions of idealizer within ring theory and the normalizer within group theory.
Using the length of the annihilator chain, the proof of the following proposition  is straightforward, hence it is omitted
\begin{pro}
Let $B$ be a skew brace.
\begin{enumerate}
\item If $B$ is annihilator nilpotent, then every sub skew brace $A$ has a subideal chain, i.e. there exists a chain of sub skew braces $A=H_1 \subseteq H_2 \subseteq... \subseteq H_n=B $ with $H_i$ an ideal of $H_{i+1}$.
\item If every sub skew brace has a subideal chain, then every proper sub skew brace is strictly contained in its idealizer. In particular, in this case every maximal sub skew brace is an ideal.
\end{enumerate}
\end{pro}
Recall from \cite{MR4256133} the following result: the intersection of all maximal
ideals of a skew brace $B$ equals the set of non-generating elements $b$
of $B$, i.e.
if  the ideal  $(S \cup \{b\}) = B$ then  the ideal  $(S) = B$, for each subset
$S \subseteq B$.
This set is denoted by $\Rad(B)$ and is called the radical of $B$.
If there do not exist maximal ideals. By definition,  $\Rad(B)=B$.
\begin{cor}
Let $B$ be an annihilator nilpotent skew brace. Then every maximal sub brace skew brace is an ideal and  $\Rad(B)$ is the intersection of all maximal sub skew braces.
\end{cor}
Note that, with a similar proof as in the equivalent characterizations of the Frattini subgroup, one can see that the intersection of all maximal sub skew braces coincides with the set of non-generating elements as skew brace. Here, a non-generating element as skew brace is an element $x$ of a skew brace $B$ such that for any subset $S$ of $B$ one has that the sub skew brace generated by $S \cup \left\lbrace x \right\rbrace$ is $B$ implies that the sub skew brace generated by $S$ is $B$.

\subsection*{Acknowledgments}

The third author is supported in part by OZR3762
of Vrije Universiteit Brussel. The second author is supported
by Fonds voor Wetenschappelijk Onderzoek (Flanders).

\bibliographystyle{abbrv}

redpajama_set_name: RedPajamaArXiv


text:
Wonder Falls project sinks in Niagara Falls
State officials have a new vision for the former Rainbow Centre, and it's not centered around tourism
by: Daniel Telvock
First pitched by Governor Andrew Cuomo in 2014 as a project that would transform tourism in the Cataract City, the much-touted Wonder Falls water park hotel resort is now dead.
Within the same minute that the developer of the project, Uniland, announced that the resort was officially dead, Empire State Development, the state's economic agency, revealed that it already has a new vision for some of the former Rainbow Centre: a high-density housing and commercial development.
News 4 Investigates reported in April that state officials and Uniland were still working through finances and federal tax changes that now require state and local incentives, grants and loans to be marked as income for the recipients.
As a result, sources told News 4 that the federal tax changes would have a significant negative impact on the developer's profit on the project, considering that the state offered to assist with funds out of Gov. Cuomo's Buffalo Billion economic initiative.
On Friday, the company cited the tax changes as a big reason for the decision.
"The project team and I are disappointed that the years we spent developing and championing this project will not result in Wonder Falls coming to life," said Uniland Vice President Michael Montante in a prepared statement
"We have consistently maintained that we are responsible stewards of public funds. In this case, the economics do not make sense and the prudent and proper decision was to end the process. As forces such as political policies and economic conditions shifted, so did the viability of the concept."
Uniland said it spent "countless resources" on the proposed $150 million project that would have provided a year-round attraction in a city that desperately needs one to compliment the Niagara Falls and the more than eight million people who visit the park each year. The project included a 300-room hotel, water park, restaurants and other family-focused entertainment.
Providing the city with a new tourist attraction no longer appears to be within the state's new vision.
Empire State Development, through its subsidiary USA Niagara Development Corporation (USAN), said it will pursue plans to transform the Rainbow Centre in downtown by releasing a request for proposals on Monday, Dec. 9. This process will invite developers to make their pitches for what the state is calling a "civic improvement project" that will "establish greater public access and walkability, create commercial opportunity and also improve parking to support new and existing business at and near the site."
The state wants "a high-density, mixed-use development with restored pedestrian access through the center of the building and active ground-level uses."
"Reimagining the former Rainbow Centre mall through adaptive reuse will reconnect the downtown street grid, creating a more pedestrian-friendly area that will attract shoppers and visitors and also support local businesses," saidEmpire State Development Acting Commissioner and President & CEO-designate Eric Gertler in a prepared statement.
ESD did not address the federal tax changes and how it might impact other state economic development projects that get local or state incentives, grants and loans and did not respond to inquiries from News 4.
The former mall has about 200,000-square-feet of available space to include for redevelopment. Part of the former mall houses the Niagara Falls Culinary Institute.
Niagara Falls Mayor-Elect Robert Restaino said the former mall is a critical piece of property that provides the city with a tremendous opportunity.
"As a city-owned property, I will work with USAN to ensure that we improve the site to include high-density, mixed-use redevelopment with an active ground level that will foster future development on adjoining parcels," he said.
LEWISTON, N.Y. (WIVB) -- Summer concert announcements are rolling in and Barenaked Ladies, Gin Blossoms and Toad the Wet Sprocket will be at Artpark on July 7.
Tickets go on sale at 10 a.m. on Friday, Jan. 24. Admission is $37 for both front of stage and reserved seating, and $15 for general admission.
KANSAS CITY, Mo. (via WDAF) -- One of the Kansas City Chiefs' biggest fans is from Niagara Falls, Ontario, and fellow fans are rallying to get him to the AFC Championship game on Sunday.
Dave Chandler is known to his friends as "KDave Canada," and he's been a Chiefs fan for more than 20 years. He's never lived in Kansas City. Chandler said he became a fan when he would go watch the Buffalo Bills play the Chiefs and saw Derek Thomas.
by Andrew Forgotch / Jan 17, 2020
LANCASTER, Pa. (WHTM) - We're still waiting for numbers, but it appears the first year of growing hemp in Pennsylvania may not have been as lucrative as some farmers had hoped.
"I tell people the prices were too high. Now they're back to reality," said Steve Groff, a hemp farmer and industry expert.

redpajama_set_name: RedPajamaCommonCrawl


text:
The 2nd Connecticut Light Artillery Battery, was recruited from Connecticut and served in the Union Army between September 10, 1862 and August 9, 1865 during the American Civil War.

Service
The 2nd Light Battery, under the command of Captains John W. Sterling and Walter S. Hotchkiss was organized in Bridgeport in August 1862 and mustered into the U.S. Service for three years, or the war on September 10. On October 15, it left for Washington via New York City where it remained encamped until December 12.

The Battery then moved to Fairfax Courthouse, Virginia and was assigned to the 2nd Brigade, Casey's Division, under the command of General Stoughton. Near the end of January 1863, it moved to Wolf Run Shoals, forming part of the defences of Washington until June 25, when it reported to General Tyler, commander of the artillery reserve, Army of the Potomac. During the Battle of Gettysburg the Battery was positioned to the left of the center for fifty-six hours, and was fortunate having only three men slightly wounded.

After the battle it moved to Frederick City, then to Washington, arriving on August 20. On August 22, it embarked for New York and served there while the draft was taking place. It then returned to Washington on January 24, 1864, then proceeded to Baltimore, where it embarked for New Orleans for service in the Department of the Gulf. For a time, the Battery was stationed at Brashear City, with one section at Thibodeaux; then proceeded to Algiers, and on July 30, embarked on transport for Dauphin Island, Mobile Harbor. There it was engaged in assisting the fleet under Farragut during the reduction of Fort Gaines and Fort Morgan after which it returned to Algiers then went into winter quarters in New Orleans.

During the winter it was stationed at the mouth of the White River, Kennerville and Greenville, Louisiana, at Fort Morgan, Alabama, and from March 11 to 20, at Barrancas and Pensacola, Florida. Following Pensacola it marched through the Black swamp to Fort Blakely, Alabama, and on April 9, 1865, witnessed the fall of that stronghold. It continued to serve in the Department of the Gulf, returning home in July, where it was mustered out at New Haven, on August 9, 1865.

The Battery served almost three years during which it had marched and travelled about  used up 205 horses.

Casualties
Its losses were one killed, three wounded, eighteen died of disease and accidents, eight were discharged for disability – a total of thirty.

See also

List of Connecticut Civil War units

Notes

References

Bibliography
 Unknown. (2004). Civil War Regiments from Connecticut. eBooksOnDisk.com .

Artillery units and formations of the American Civil War
Military units and formations established in 1862
Military units and formations disestablished in 1865

redpajama_set_name: RedPajamaWikipedia


With shrinking budgets all around, many HR professionals have to do more with less. At times like these, phone interviews can be a great tool to help weed out any unqualified candidates before holding a first round of face-to-face meetings. They can be quick, easy to schedule, and conducted from anywhere.
Perhaps partly due to the convenience, candidates may underestimate the gravity of a phone interview. After all, it's just a chat, right?
Wrong. No matter how casual your HR contact or recruiter sounded when they emailed you to set up a "quick phone call", your phone screening is likely to be the first round of the official process.
To make sure you don't ruin your chances before you've even set foot in the company's office, we've rounded up some tips on how to ace your next phone interview.
"The less you talk, the more you listen and the better you'll do," Bailo told Business Insider. Since you're getting no visual clues from your interviewer, it's harder to gauge if it's time to stop talking or if they're fascinated to hear more.
It's safer to assume the first and keep things concise. If they want to hear more, they can always ask.
While this may sound a little odd, Paul Bailo, author of "The Essential Phone Interview Handbook", told Business Insider that having a photo of your interviewer in front of you during the call will help you connect to the person and make you calmer.
If you can't find their photo, Bailo says you can even use a photo of someone you take seriously, like a well-known authority figure.
You'll almost certainly be asked about some of the information on these documents, says Peter Vogt, Monster senior contributing writer. As such, you may as well take advantage of the fact that your interviewer can't see you, and have your documents ready as a backup.
However, do make sure you know what they contain. You can look at them for support if needed, but there won't be time to read through your whole letter to find out what you wrote down as your differentiating quality.
Unless your interviewer specifically asks, there's no need to mention you'd normally be playing basketball on a Wednesday evening after work. "The point of a phone interview is to focus on getting to know a candidate's professional experience and goals," Mckenzie Roark, campus talent specialist at Lithko Contracting, tells Glassdoor.
Not everyone agrees on this one, but some argue putting on a formal outfit will make you feel more confident and more inclined to take the call seriously. Additionally, it's better to be overdressed for a phone call, than to be wearing your pyamas when it turns out they'd like to conduct the call via Skype.
"Data provides the backbone for making more strategic decisions" – Ross Sparkman..
Would you let staff redistribute their salary?
What if several team members approached you with a request to redistribute some of their pay to a colleague in financial trouble?..


From the Yorkshire town of Ilkley, all the way to Bowness in the Lake District, the Dales Way covers 84 miles (135 km) through the North York Moors on quality running trails.
There are several testing hills as you cross two national parks, taking in classic moorland and the much-loved lumps of the Lake District. Enjoy classic Yorkshire countryside, singing rivers, dreamy dales and old stonewalls.
Start with a long rise and fall through the moors, with smaller hills to tackle towards the end. Muddy your running shoes in two of England's finest national parks.
We also offer the Dales Way as a walking holiday, for those looking to enjoy this trail at a gentler pace. Click here to find out more.
Meadows near the Wharfe, taken by Deborah Jackson.
Wharfedale Countryside, taken by Andrew Flowerday.
Arten Gill Viaduct on Settle-Carlisle line, taken by Andy Flowerday.
The most convenient international airports for the start of the trail are Leeds-Bradford and Manchester.
There is a railway station at the start of the trail (Ilkley) and close to the end of the trail (Bowness-on-Windermere) at Windermere.
From London - There are train routes from London Kings Cross to the start of the trail (Ilkley). There are train routes from Windermere (near the end of the trail) to London Euston.
The most convenient way to return to the start of the trail (Ilkley) is via train from Windermere railway station to Ilkley.
Taking an extra night is the perfect way to extend your trail running holiday and give yourself extra time to relax and explore the Yorkshire Dales, as there is plenty to discover and enjoy at each location along the trail.
Grassington is a delightful village with traditional cottages, pubs and shops surrounding a cobbled market square, making it a very popular choice for a rest day. The village?s longest-serving outdoor clothing and equipment shop, The Mountaineer, is perfect if you need to replenish your running kit, otherwise feel free to explore Grassington?s many little alleyways and hidden gems while you relax and prepare for your next stint out on the trail.
Alternatively, Buckden is a village and civil parish worth exploring on a rest day; an attractive jumble of stone cottages perched on the hillside with fantastic views over the River Wharfe. In the centre of the village sits Buckden Palace, a former residence of the bishops of Lincoln between the 12th and 19th centuries and host to a number of Kings of England who have stayed there, making this just one of many interesting local landmarks to visit. Take a gentle recovery jog around the local area and see what you can discover.
Dent is also worth considering for a rest day. This lovely village complements the surrounding dales wonderfully, with its narrow cobbled streets, quaint shops and ancient church to explore. As well as this, Dent is well-known for its annual music and beer festival at the end of June, and nearby in Cowgill is the independent Dent microbrewery, meaning this area is great for enjoying a locally-produced drink in one of its traditional inns.
At the end of the trail, Bowness-on-Windermere provides a perfect base from which to explore the southern Lake District after completing your running holiday. By spending an extra day in this sprawling tourist town, you will have plenty of opportunity to enjoy the various sights and attractions on offer. In particular, Bowness-on-Windermere is one of the main centres of outdoor activities in the UK and boasts a wide variety of activity options, including sailing, kayaking and other water sports out on the lake. With such a variety of things to do, a stay in Bowness-on-Windermere will ensure your holiday finishes on a high note.


text:
Tecnosistemi
Schiavoni Group
Tecnosistemi Tecnosistemi Schiavoni Group
With a turnover of over 119 million of Euros and about 750 employees, Schiavoni Group is one of the largest business realities in Marche region and ranks among the major international players.
Started about fifty years ago, the history of the group founded by Sergio Schiavoni is full of successes linked first to the commercialization of electrical equipment only and after as an industry group of more developed highly innovative sectors.
At the base of the Schiavoni Group success, there is an entrepreneurial vision careful to changes in society and market developments, vision driven by a strong spirit of initiative and able to look with creativity to the most advanced sectors of the economy.
These characteristics led to a decisive intuition in the 60's: the realization of ground stakes in steel and copper.
This intuition was the beginning of the rapid growth of Schiavoni Group in the commercial sector both for international installation, multidisciplinary engineering and renewable energy.
Today the group is involved in important challenges along the major axes identified for the core business growth: the process of internationalization of its business, which now sees a 60% of turnover generated abroad, close attention to technology and to what these opportunities allow you to grasp.
As part of the Italian economic system, it is undoubtedly a great example of entrepreneurship enhancement and an irreplaceable factor of industrial growth. Thank you, once again, to the clarity of its strategic choices, the excellence of its companies, a solid financial position, and above all the great qualities of those who work in the Group and which have always been his main strength.
Tecnosistemi together with the mother company IMESA S.p.A., has decades of experience in supply Medium and Low Voltage Switchboards and Containerized Electrical Substations. We supplied our equipment to main clients worldwide such as ENI, Petrofac, Weatherford, DNO, Fincantieri,
Tecnosistemi started his activity in the United Arab Emirates, under formally the Italtel name, in the late 1994. The first project was the design, supply, installation, testing and commissioning of a sophisticate Access Control and Alarm Monitoring system for Etisalat in the emirates of Abu Dhabi and Dubai.
Tecnosistemi FZ LLC
Office 119, Building 5, IBM Building
Dubai Internet City, Dubai – U.A.E.
Mail: info@tecnosistemi.ae


Is this bug a big deal? Let's discuss.
Twitter on Thursday advised all 330 million of its users to change their passwords after a software bug caused the passwords to be stored in an unencrypted way for an unspecified period of time.
So should I change my Twitter password?
Probably! But if you're wondering if Twitter's problem is a huge deal, it is not so egregious as, say, the Equifax data breach that exposed extensive amounts of people's personal financial information. For one thing, Twitter said that no one has inappropriately accessed the user passwords.
And even Twitter is framing changing your password as something you can decide to do rather than something you MUST do immediately. In a tweet, the company referred to changing your password as a "precaution" rather than an imperative. Company executives also called it a "decision" as opposed to an obligation.
"We are sharing this information to help people make an informed decision about their account security," Twitter's chief technology officer Parag Agrawal said. "We didn't have to, but believe it's the right thing to do."
"This situation is somewhere between a low-to-medium level security issue. It raises way more questions than it actually answers, but a password leak that only happens internally and not out on the entire internet is a much more ideal situation to be in," said Jessy Irwin, head of security at Tendermint.
Twitter also allows you to skip changing your password when it notifies you in your browser or the Twitter app.
So what actually happened to the passwords?
This whole thing is about how tech companies store passwords, which involves a lot of very hard math, but is not a complicated concept.
Say my Twitter password is Password123 (1, it isn't; and 2, this shouldn't be your password for anything!!). Even though I enter it as Password123, Twitter's systems and employees see what I wrote as a jumbled string of numbers and letters like 64eyb95exmp. That change is a process called hashing, and the jumbled version is called a hashed password. When I enter my password, it goes to Twitter through a bunch of code and appears as the scrambled version rather than what I've actually written, which allows me to log in without someone on the other end of the internet being able to steal my password.
According to Twitter, the company stored people's passwords in an "unmasked" way in an internal log, which means they were stored as I would see them when I'm logging in. So instead of 64eyb95exmp, people looking at one specific database at Twitter would see my password as Password123 (again, do not make this your new password).
That means if someone broke into Twitter's databases, they would easily be able to steal your account by just copying and pasting your password. Twitter said no one outside or inside the company did that, which is good!
We don't know how many people at Twitter had the ability to see the unencrypted passwords in the first place or what in the hashing process went wrong. Twitter's internal security policies would dictate which employees, likely engineers, would have access to the logs. But again, Twitter said no one saw them who wasn't supposed to.
"The info was in an internal log in an obscure field, so unlikely anyone would have seen," said Twitter spokesperson Liz Kelley.
Kelley said an internal investigation into the bug is ongoing, but wouldn't say how long the passwords had been exposed.
What's your new Twitter password? Sound off in the comments!


Photographer and artist, Rinne Allen lives and works in Athens, Georgia (though she also travels the world taking pictures. Follow her on Instagram for a glimpse.) She is deeply committed to her community, having co-founded a children's school for creative arts and working with the University of Georgia on special arts programming. It's an understatement to say that she embraces collaboration. Her local work includes a book and blog with Rebecca Wood and Kristen Bach about Athens called Beauty Everyday along with collaborations with chefs, designers, and makers from Athens and beyond.
Taking yet another step into her community, Rinne photographed the Rinne's Dress Collection on those whom she works with closely.
While we're getting ready for a new collaboration with chef Ashley Christensen, we wanted to share one our favorites again—before we phase out this beautiful collection in two weeks. The basic version of the Rinne's Dress will continue to be available for purchase from the Alabama Chanin Collection.
This dress looks fantastic on the models above. Thank you for showing what the dress looks like in the real world. They really are beautiful!!
I adore this dress pattern! It reminds me of someone like Ray Eames' uniform, that this is a dress you could wear every day for all things. Do you foresee that this pattern might ever be released for purchase?
Thank you for your comment, Tori. The Rinne Dress is a Collection pattern, and as such is not available for release. However, I will let the Design Team know your kind words about the dress, and they may be able to plan something relatable for The School of Making in the future. On that note, one of this year's Build a Wardrobe Subscription patterns is the Jumpsuit – it will have a dress variation when it launches. Keep an eye out for it, as it may be similar to what you are looking for.


Join Us – Neem India Pvt. Ltd.
We invite Agri background dedicated Dealers with extensive local market knowledge who are willing to do join our hands in making organic farming a new tomorrow and are willing to do ground work for it with us.
To become a member of our distributor team all you need is to download and fill in the Distributors' Appointment Form and send it along with the documents mentioned below, to the registered Office address of the company.
Duly filled Distributors' Appointment Form.
Legal Status of Firm: Proprietorship / Partnership / Pvt. Ltd. Company.
a) Provide affidavit, if the firm is Proprietary.
c) Limited Company : Memorandum and Articles of Association.
11,000/- Dealership Deposit by cheque in favour of Neem India Products Pvt. Ltd.
2 blank letterheads with stamp & signature of the firm.
For any query regarding above, please feel free to contact us on 0240- 2354912.


text:
Search Ends for New NBII Search Engine …Implementation Begins
Access Newsletter, Spring 2009
Search Ends for New NBII Search Engine
…Implementation Begins
In the fall 2008 issue of Access we said a new and improved search engine was coming to the NBII. Now – drum roll – we're most pleased to say, "It's here!" The contract to implement the NBII's new search engine was awarded on February 19, 2009 to Vivísimo, a Pittsburgh-based search software company founded in 2000 that is emerging as an industry leader. An example of a Vivisimo implementation is available on FirstGov.gov, the official Web portal of the U.S. government. "It's safe to say we can now offer our users one of the most advanced search engines available on any biodiversity Web site," said Jim Erwin, coordinator of the NBII Search Engine Evaluation team. The new search engine will offer many advantages over the NBII's existing search function. Here are a few examples of new capabilities that will be available:
• The search interface is easy to use. What's more, the search results arrive preorganized in a variety of groups created on the fly (clustered), which allows users to focus on particular categories or browse through related fields, thus avoiding a common "overload" problem of sorting through too many results.
• The search is set up to cast a wide net. When a user executes a search, the search engine has been customized to crawl – simultaneously – 35 key repositories of biodiversity data and information (this number will grow). Results arrive from
all of those sources in one search. The repositories – which include Web sites, databases, and federated resources – have been painstakingly selected from those offered by governments (federal, state, and local), nonprofits, the private sector, educational institutions, and more.
• Users can do a basic search by putting keywords in the search box (located in the NBII home page banner) or an advanced search with metadata fields. They can also specify how many results they want to come back.
• The search engine will also be integrated with Google Maps. For instance, if you're looking for species records that are geospatially referenced, you'll soon be able to display that information on the appropriate map.
• Users can search for images. The images come back as thumbnails. Just click on a thumbnail to open it up and examine it further. You can also search information associated with an image. And the list of benefits goes on and on. "Choosing and customizing this search engine has been a long, exhaustive process that we expect
to conclude by July 15," said Erwin. "But we really feel like all the effort is
worth it – and we're sure our users will agree."
Posted by NBII at 7:02 PM
The NBII Homepage
Search NBII-Info
NBII-Info
The National Biological Information Infrastructure (NBII) is a broad, collaborative program to provide increased access to data and information on the nation's biological resources. The NBII links diverse, high-quality biological databases, information products, and analytical tools maintained by NBII partners and other contributors in government agencies, academic institutions, non-government organizations, and private industry. NBII partners and collaborators also work on new standards, tools, and technologies that make it easier to find, integrate, and apply biological resources information. Resource managers, scientists, educators, and the general public use the NBII to answer a wide range of questions related to the management, use, or conservation of this nation's biological resources.
NBII Disclaimer
This blog contains numerous links to data and information on the Web. Links and pointers to non-NBII sites are provided for information only and do not constitute endorsement, express or implied, by the NBII, its partner agencies, or the U.S. Government, of the referenced organizations, their suitability, content, products, services, usefulness, functioning, completeness, or accuracy; whether they are governmental, educational, or commercial.
Image Use and Permission
Images from this blog may be protected under Creative Commons or U.S. copyright laws. Please contact the source for permission for use.
"climate change" "Conservation" "endangered species" "Gap Analysis" "Threatened and Endangered Species" Aquatic species avian influenza BAMONA Bees Bioblitz 2010 Biodiversity bioremediation Bird Conservation Birds Botany butterflies Butterflies and Moths of North America climate change databases fish disease fish species Fisheries Habitat invasive species invasives Maps Marine Mammals Microbiology monitoring moths oceans pollinators Riparian ecosystems Sage Brush Ecosystem Sage grouse taxonomies Vegetation Characterization water quality Weeds Wildlife wildlife diseases Wildlife health
NBII Links
The National Biological Information Infrastructure (NBII) Homepage
Access: The Official Newsletter of the National Biological Information Infrastructure (NBII)
The NBII LIFE - Library of Images From the Environment
The NBII Metadata Clearinghouse
The NBII Kid's Page
NBII Feeds
Southeast Information Node
NBII Twitter Feeds


text:
The "Quit Your Job" Checklist: 8 Boxes You Must Tick Before Flipping Off Your Boss
February 28, 2018	Internet Marketing News 0 297
It's intoxicating, isn't' it? Your dream of entrepreneurial freedom. Every spare moment is consumed by your up-and-coming business as you toil, plot and plan. It's been a side-gig until now because you're still tethered to your day job. But you can't wait to march into your boss's office and plonk your resignation letter on his(...)
Repurposing What Works: The Journey, Episode 19
The Journey, a Social Media Examiner production, is an episodic video documentary that shows you what really happens inside a growing business. //www.youtube.com/watch?v=-GvAnICXe_I Watch The Journey: Episode 19 Episode 19 of The Journey follows Michael Stelzner, founder of Social Media Examiner, as he continues to pursue what many will see as an impossible goal: to [...]
This post Repurposing What Works: The Journey, Episode 19 first appeared on .
- Your Guide to the Social Media Jungle
How to Succeed at B2B Content Marketing with More Credible Content
Ask any B2B marketer about their top digital marketing challenges and one of the first answers you'll likely receive is: Getting in front of the right audience at the right time. In fact, Ytel's 2018 State of B2B Marketing Communications survey revealed that 55% of respondents agreed that they have a hard time getting their message and content in front of their target audience.
Why is that? Well, there's more content at buyers' fingertips than ever before, search engines are getting in touch with their human sides, and organic visibility on social media is effectively extinct. But, let's real talk for a minute, marketers—this is all old news. In today's digital marketing landscape, creating meaningful connections with your audience on any channel really comes down to trust and credibility.
Consumers are increasingly numb to advertising and marketing messages—and they're actively trying to avoid it all. Last year, PageFair reported that adblocking usage had grown by 30% globally. In addition, Facebook's recent decision to de-emphasize brand content was in response to users saying that posts from businesses, brands and media were crowding their News Feeds.
However, buyers are increasingly looking to those they know and those they think they know for insights, answers and recommendations. Multiple reports have shown that somewhere around 90% of consumers trust influencers or individuals over straight branded content. In addition, according to a recent report from CMI and SmartBrief, 40% of B2B decision-makers say that credibility trumps the source of the information.
This means B2B brands and marketers need to double-down on creating quality, credible content to drive marketing objectives and wins. But how? By infusing credible voices, perspectives and insights from influential sources—namely industry experts and thought leaders—into the content marketing game plan.
[bctt tweet="#B2B brands & marketers need to double-down on creating quality, #crediblecontent to drive #marketing objectives & wins." username="toprank"]
To highlight how creating more credible content with influencers can help your B2B brand get in front of interested buyers, create an engaging experience, and inspire action, take a look at these three examples from the TopRank Marketing playbook of successful client influencer content marketing programs.
Case Study #1 – Introhive: Reaching, engaging and inspiring a niche audience.
Introhive is a leading customer relationship management (CRM) solutions provider. Working across a variety of industries, Introhive aims to help their clients gain and effectively leverage customer intelligence in a way that can grow their business.
The Situation: The legal sector is one of Introhive's focus industries. However, it's an industry that's been reluctant to adopt CRM technologies. Why? Oftentimes business development isn't an established department within law practices, making it hard to justify investments in a "sales" technology. But law firms large and small have growth aspirations—and Introhive wanted to empower them to realize those opportunities.
Another challenge with attorneys and other legal professionals is that they often build their careers on evidence and witness testimony. Essentially, this industry is by nature hard to reach without offering credibility, authority and proof.
The Solution: With two unique challenges to overcome, our team knew that engaging other legal professionals to share their expertise and insights on business development, we could not only showcase the Introhive brand, but also needed to provide their audience with unique, relevant and trustworthy insights.
TopRank Marketing worked with the team at Introhive to develop an integrated influencer content program that began with a survey of legal community members. Conducting the survey helped facilitate building influencer relationships, while also collecting valuable data that could be used to further bolster campaign content. Other pieces of the integrated content marketing mix included an eBook—our anchor asset—blog posts, organic social amplification, paid social, and email.
The Results: For starters, we saw 15% more eBook downloads in the first month than the benchmark asset had in its lifetime. During the same time period, the accompanying blog content garnered over 600% more views compared to benchmarks for average blog content. Finally and without specifics available, the Introhive team reports that the program has delivered "medium to huge" marketing qualified leads (MQLs). Suffice it to say, this program leveraged credible content within influencers and research to generate substantial results.
Read the full Introhive integrated influencer campaign case study.
Case Study #2 – Cherwell: Increasing brand visibility and thought leadership in a competitive space.
Cherwell Software is a leading IT service management (ITSM) company with a mission to help their customers leverage intuitive technology to enable better, faster and more affordable innovation.
The Situation: Since its inception a little over a decade ago, Cherwell has been rapidly gaining traction in the competitive ITSM space—but they're still one of the newer kids on the block with other new competitors emerging rapidly. To continue their growth and fend off competition, Cherwell wanted to expand its marketing channels, increase brand awareness, engage industry thought leaders and—of course—eventually drive leads.
The Solution: Given Cherwell's position in the competitive ITSM space, the team at TopRank Marketing worked to design an influencer content campaign that was highly-targeted to key the decision-makers they wanted to reach. How? We knew in order to stand out in news feeds and build near-instant credibility with our content, we needed to understand what influences the target audience the most.
To uncover the people, publications, and content topics and types that "moved" our audience the most, as well as where they spent time on social media, we designed a new research tool—the RITHM report.
Using insights from the RITHM report to inform the content marketing approach, the resulting campaign included an eBook anchor asset, blog posts, an SEO-driven landing page, paid and organic social media.
The Results: According to Alison Munn, Social Media and Digital Marketing Lead at Cherwell: "Not only did this program meet the defined goals and objectives, but the results and process exceeded my expectations!"
With this campaign responsible for 22% of new revenue for Cherwell in 2017, it was a recent winner of the B2B Marketing Exchange "Killer Content Award".
You can learn more about this program in the case study video below:
[embed]https://youtu.be/cUf_a7RL9NY[/embed]
Case Study #3 – SAP SuccessFactors: Driving awareness and action by connecting to a specific audience pain point.
SAP SuccessFactors is a leading human capital management (HCM) suite that helps human resources (HR) professionals unleash the full potential of their workforce through transformation and engagement, and ultimately drive results across the business.
The Situation: For this niche human resources audience, employee wellness programs are part of the strategy to unleash the potential of their employees. However, the pain point for many organizations is finding a holistic solution in one place that also provides understanding of the true impact their efforts can have on the workplace. SAP wanted to drive awareness around their holistic solution, while also educating and engaging their audience.
The Solution: TopRank Marketing partnered with SAP SuccessFactors to craft an multi-pronged, influencer-driven content campaign that would not only raise awareness around their solution, but also provide their audience with credible, relevant, and actionable insights.
This campaign was anchored with an influencer eBook that featured insights from 10 top workplace culture, wellness, and technology experts, as well as internal experts from SAP SuccessFactors. In addition, other tactics such as a well-optimized landing page, social media promotion, and customized motion graphics were part of the mix.
The Results: For downloads, we saw a 272% increase over SAP's established benchmark. In addition, the accompanying landing page boasted a 68% conversion rate. Lastly, organic social promotion of the content—from the brand and influencers—drove 86% of overall views and 69% of overall conversions.
Read the full SAP SuccessFactors influencer-driven content campaign case study.
The Big B2B Takeaway for Credible Content
We're in an era of a distrust and indifference to B2B marketing messages—which means if buyers don't find your content credible and trustworthy, they'll move on.
From skepticism to standing out in a crowded and more seasoned field, each of the aforementioned brands were living the trends and looking for a way to capture the attention of their audiences.
By cleverly leveraging influencers to create more credible and authoritative content and more trusted amplification, these brands were able to deliver their audiences with thoughtful opinions and diverse insights, bolster brand authority and make more meaningful connections with their audiences. But perhaps the most exciting campaign result was that building credibility led to audience activation—or conversions in other words.
To put it simply, with the right strategy, insights and influencer infusion, credible content can help brands win over your audience at every stage of the buyer journey.
[bctt tweet="With the right strategy, insights & influencer infusion, #crediblecontent can absolutely help brands win over your audience at every stage of the buyer journey. @CaitlinMBurgess" username="toprank"]
Want to learn more creating more credible content? Check out our post on building credibility and authority with content marketing.
The post How to Succeed at B2B Content Marketing with More Credible Content appeared first on Online Marketing Blog - TopRank®.
14 Mobile Apps to Create Branded Visuals
Do you create visual images with your smartphone or tablet? Looking for tools to brand your images and video on the go? In this article, you'll discover 14 mobile apps to help you make branded social media visuals. #1: Pick Up Brand Colors Whether you're creating an Instagram story or using a design app, the [...]
This post 14 Mobile Apps to Create Branded Visuals first appeared on .
Unbounce Now More Secure Than Ever: Introducing Single Sign-on, Audit Logs, and 2FA Authentication
With heavy reliance on the internet to run businesses, security and data privacy threats involving SaaS tools are now more part of a marketer's world than ever. Similar to insurance, you hope you never have to worry, but you need to be prepared all the same. Today, learn about how security and data privacy differ, and what we've rolled out to help you focus on campaigns.
How to Use Snapchat Insights
Is your business on Snapchat? Looking for more robust analytics data on your snaps and stories? In this article, you'll discover how to navigate and use Snapchat Insights to learn more about your Snapchat marketing activities. Access Insights From the Home Screen Until now, Snapchat has limited its user-facing analytics to only views and screenshots, [...]
This post How to Use Snapchat Insights first appeared on .
13 Facebook Tools Marketers Use to Boost Engagement
Facebook's status as the go-to social platform for marketers isn't changing anytime soon. Despite its most recent update, Facebook usage among the everyday public Read more...
This post 13 Facebook Tools Marketers Use to Boost Engagement originally appeared on Sprout Social.
How to Use Facebook Collections to Manage Curated Content
Do you save content and ads on Facebook? Wondering how to organize the Facebook posts and ads you've saved? In this article, you'll learn how to use Facebook Collections to curate and organize saved content for easy access and sharing. Why Use Facebook Collections? Facebook lets you save content (posts, events, pages, and photos) directly [...]
This post How to Use Facebook Collections to Manage Curated Content first appeared on .
50 Social Media Marketing Influencers to Learn From in 2018
It's that time of year again for the conference extravaganza known as Social Media Marketing World. Each year I pull together a list of influential voices that are engaging on social networks around the topic social media marketing. My goal? To help showcase speakers for people in the industry to learn from and follow. Why are [...]
The post 50 Social Media Marketing Influencers to Learn From in 2018 appeared first on Online Marketing Blog - TopRank®.
Twitter Cracks Down on Bots and Automation
Welcome to this week's edition of the Social Media Marketing Talk Show, a news show for marketers who want to stay on the leading edge of social media. On this week's Social Media Marketing Talk Show, we explore Twitter cracking down on bots and automation with Madalyn Sklar, Instagram DM replay changes, and other breaking [...]
This post Twitter Cracks Down on Bots and Automation first appeared on .


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ABOUT GNU
= SOFTWARE =
GNU ART
GNU'S WHO?
GNU Kart
GNU Kart is a simple racing game for GNU. It uses OpenGL and SDL for graphics and features GNU characters. Its aim will be extensibility so that a general kind of racing game can be made, with any sorts of features.
GNU Kart has not had a stable release in years as it is being rewritten.
An alpha release is available from the GNU alpha server (HTTPS, HTTP, FTP).
GNU Kart has a mailing list, but currently doesn't use it. Please contact the maintainer for bug reports and development-related questions. The mailing list archives are still available.
Announcements about GNU Kart and most other GNU software are made on <info-gnu@gnu.org> (web interface).
Development of GNU Kart, and GNU in general, is a volunteer effort, and you can contribute. For information, please read How to help GNU. If you'd like to get involved, please contact the maintainer.
For development sources and other information, please see the GNU Kart project page at savannah.gnu.org.
GNU Kart is currently maintained by Sébastien Diaz <sebastien.diaz at gmail.com>. The previous maintainer was Eric P. Hutchins <hutchiep190@potsdam.edu>.
September 14, 2010: GNU Kart Alpha 0.3.0
GNU Kart is a kart racing game for GNU using characters based on GNU/Free Software stuff.
There are currently three characters that have a (incomplete) 3D model to them: GNU, guile, and icecat.
You have to run it from the command line and it only lets you go through one lap and then quits, but the computer characters will drive around the track nicely.
SpeedX is now GNU Kart. The "X" in the name "SpeedX" referred to the X window system, which especially made sense because it heavily used X11 libraries. GNU Kart doesn't use X11 libraries directly, and mostly uses OpenGL. If you check out the latest source in Bazaar, you will see that I am testing a track format that is based on spheres of different radii connected by edges that form tubes that tell which edge a player is traveling along and which sphere they are heading toward.
I hope to release something in the summer when I have more time. I tried to develop a voxel engine because at the time that I took over the project, OpenGL was not 100% free on gNewSense. Now that it is, I realize that my voxel engine is slow, cumbersome, and extremely limited. Also, I'm not sure how to extend it. So I've been hacking on OpenGL for a bit and I think I should have something good soon. Even without hardware acceleration (which I don't have, since there are very few free 3D graphics drivers), OpenGL seems fine as long as the scene stays simple, and the point of this game is to be a simple, easily playable game that anyone can enjoy quickly so it seems like it will work fine, even on a system where Neverball or SuperTuxKart are not playable.
GNU Kart is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.
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Please send general FSF & GNU inquiries to <gnu@gnu.org>. There are also other ways to contact the FSF. Broken links and other corrections or suggestions can be sent to <sebastien.diaz at gmail.com>.
Please see the Translations README for information on coordinating and contributing translations of this article.
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"Dacapo" is an planar radiator designed by Bernd Timmermanns equipped with a reasonably priced Ciare CH 250 full range driver boasting a high efficiency of 98 dB (2.83V, 1 metre).
To achieve this, Bernd Timmermanns investigated how wide the baffle had to be to ensure a frequency response that is as flat as possible. A 45 cm baffle resulted in a reasonably flat frequency response between 300 and 1000 Hz.
... "Despite the small distance between the two speakers, the two Ciare full range drivers were able to create an amazingly wide and deep sound stage. It is true that vocals and instruments were bigger than usual, but they were still realistic enough. Beethoven's "Ich liebe dich" performed by Hermann Prey (to be found on the Manger Test CD) was just terrific. We rarely had the opportunity to experience Hermann Prey, standing that tall in front of a concert grand. Even the piano was easy to locate and in its full size. Great!" ... "Tonality, however, was not as good as with well balanced multiple driver speakers. The minor shortcomings that are clearly noticeable in AB comparison are not bothering at all anymore when our ears have adjusted to these speakers. What's left is a phenomenally detailed performance with an almost infinitely fine resolution, that also has a drawback: even the smallest blunder by the recording engineer is audible. Really good recordings, however, like e.g. the Manger Test CD the listener will get sheer pleasure out of the Dacapo."
Upon request we supply a 220 x 155 mm universal PCB for mounting the crossover components.
We recommend to use Dacapo together with a subwoofer.