Nilpotent left quasigroups

Superconnected left quasigroups and involutory quandles

In this paper we study the classes of superconnected and superfaithful left quasigroups, that are relevant in the study of Mal'cev varieties of left quasigroups \cite{Maltsev_paper}. Then we focus on quandles and in particular to the involutory ones. We extend the main result of \cite{involutive_quandles_russo} to the infinite case and we offer a characterization of several classes of involutory quandles in terms of the properties of the canonical generators of the displacement group, improving the main results of \cite{Nobu}.

Orthogonality of linear (alinear) quasigroups and their parastrophes

Necessary and sufficient conditions of orthogonality of left (right) linear (alinear) quasigroups in various combinations are given. As corollary we obtain conditions of parastroph orthogonality of left (right) linear (alinear) quasigroups. Any linear (alinear) quasigroup over the symmetric group S_n ($n\neq 2; 6$) is not orthogonal to its (12)-parastrophe.

Smashed products of topological left quasigroups

In this article smashed products of topological left quasigroups are scrutinized. Quotients of topological fan quasigroups are investigated. Skew smashed products of fan quasigroups and their structure also are studied.

Units in quasigroups with classical Bol-Moufang type identities

We prolong Kunen research about existence of units (left, right, two-sided) in quasigroups with classical Bol-Moufang type identities. These identities were listed in Fenvesh article.

右群、左拟群和右堆

Right groups, left quasigroups, and right heaps

从右群而非群出发发展堆理论，定义右堆，并探索从左拟群出发的部分理论，为研究Yang-Baxter方程的左非退化集论解提供模拟斜左桁架的结构。

Idempotent solutions of the Yang-Baxter equation and twisted group division

Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity $(x*y)*(x*z)=(y*y)*(y*z)$. Using combinatorial properties of the Cayley kernel and the squaring mapping, we prove that a twisted Ward left quasigroup of prime order is either permutational or a quasigroup. Up to isomorphism, all twisted Ward quasigroups $(X,*)$ are obtained by twisting the left division operation in groups (that is, they are of the form $x*y=ψ(x^{-1}y)$ for a group $(X,\cdot)$ and its automor…

Complete graph decompositions and p-groupoids

We study P-groupoids that arise from certain decompositions of complete graphs. We show that left distributive P-groupoids are distributive, quasigroups. We characterize P-groupoids when the corresponding decomposition is a Hamiltonian decomposition for complete graphs of odd, prime order. We also study a specific example of a P-quasigroup constructed from cyclic groups of odd order. We show such P-quasigroups have characteristic left and right multiplication groups, as well as the right multiplication group is isomorphic to the dihedral group.

A left $3$-Engel element whose normal closure is not nilpotent

We give an example of a locally nilpotent group $G$ containing a left $3$-Engel element $x$ where $\langle x\rangle^{G}$ is not nilpotent.

Locally finite $p$-groups with a left 3-Engel element whose normal closure is not nilpotent

For any odd prime $p$, we give an example of a locally finite $p$-group $G$ containing a left 3-Engel element $x$ where $\langle x \rangle^G$ is not nilpotent.

On the Yang-Baxter equation and left nilpotent left braces

We study non-degenerate involutive set-theoretic solutions (X,r) of the Yang-Baxter equation, we call them simply solutions. We show that the structure group G(X,r) of a finite non-trivial solution (X,r) cannot be an Engel group. It is known that the structure group G(X,r) of a finite multipermutation solution (X,r) is a poly-Z group, thus our result gives a rich source of examples of braided groups and left braces G(X,r) which are poly-Z groups but not Engel groups. We also show that a finite solution of the Yang-Baxter equation can be embedded in a convenient way into a finite brace and into…

Non-affine latin quandles of order $2^k$

We prove that a non-affine latin quandle (also known as left distributive quasigroup) of order $2^k$ exists if and only if $k = 6$ or $k \geq 8$. The construction is expressed in terms of central extensions of affine quandles.

有限斜括号中的左幂零剩余及其补

The Left-Nilpotent Residual and Its Supplements in Finite Skew Braces

本文研究有限斜括号的左幂零剩余，证明其刻画最大左幂零商，并利用群论方法探讨其补的存在性，给出阶18的反例，最后联系KSV可解性。

Sylow理论与左幂零斜辫的幂零类

Sylow theory and the nilpotency class of left nilpotent skew braces

本文证明有限左幂零斜辫的乘法群中的Sylow p-子群是Sylow p-子辫，且每个p-子辫包含于某个Sylow p-子辫中，从而去掉了可解性假设，并给出了左幂零类的上界。

The Congruence Subgroup Problem for finitely generated Nilpotent Groups

The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphi…

关于有限生成的左幂零括号

On finitely generated left nilpotent braces

本文描述了类至多为2的有限生成左幂零括号，利用左幂零类至多为2则右幂零类至多为3的性质，并在该范畴中构造了自由对象。

Sandwich groups and (strong) left 3-Engel elements in groups

In this paper we prove a group theoretic analogue of the well known local nilpotence theorem for sandwich Lie algebras due to Kostrikin and Zel'manov. We introduce the notion of a strong left 3-Engel element of a group G and show that these are always in the locally nilpotent radical of G. This generalises a previous result of Jabara and Traustason that showed that a left 3-Engel element a of a group G is in the locally nilpotent radical of G whenever a is of odd order.

具有完全单理想的半群以及幺半群中的可解、幂零与超幂零性质

Solvable, nilpotent and supernilpotent semigroups with completely simple ideal and monoids

本文刻画含完全单理想半群在交换子理论下的可解、幂零及超幂零性，并给出幂零幺半群可嵌入幂零群的等价条件。

The Multiplication Group of an AG-group

We investigate the multiplication group of a special class of quasigroup called AG-group. We prove some interesting results such as: the multiplication group of an AG-group of order n is non-abelian group of order 2n and its left section is an abelian group of order n. The inner mapping group of an AG-group of any order is a cyclic group of order 2.

Skew left braces of nilpotent type

We study series of left ideals of skew left braces that are analogs of upper central series of groups. These concepts allow us to define left and right nilpotent skew left braces. Several results related to these concepts are proved and applications to infinite left braces are given. Indecomposable solutions of the Yang-Baxter equation are explored using the structure of skew left braces.

The space of bi-invariant orders on a nilpotent group

We prove a few basic facts about the space of bi-invariant (or left-invariant) total order relations on a torsion-free, nonabelian, nilpotent group G. For instance, we show that the space of bi-invariant orders has no isolated points (so it is a Cantor set if G is countable), and give examples to show that the outer automorphism group of G does not always act faithfully on this space. Also, it is not difficult to see that the abstract commensurator group of G has a natural action on the space of left-invariant orders, and we show that this action is faithful. These results are related to recen…

关于类2左幂零斜 braces

On left nilpotent skew braces of class 2

本文研究满足 $B^3=0$ 的类2左幂零斜 braces，证明幂零型必中心幂零并给出右幂零类界，阿贝尔型时类至多为3且界最优。

A new formula for Lazard's correspondence for finite braces and pre-Lie algebras

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence between finite Fp-braces and pre-Lie algebras proposed by Wolfgang Rump in 2014. As an application example, a classification of all right nilpotent Fp-braces generated by one element of cardinality p^4 is obtained, answering a question posed by Leandro Vendramin. It is also shown that the sum of a finite number of left nilpotent ideals in a left brace is a left…

Assuad-Nagata dimension of nilpotent groups with arbitrary left invariant metrics

Suppose $G$ is a countable, not necessarily finitely generated, group. We show $G$ admits a proper, left-invariant metric $d_G$ such that the Assouad-Nagata dimension of $(G,d_G)$ is infinite, provided the center of $G$ is not locally finite. As a corollary we solve two problems of A.Dranishnikov.

关于幂零群左序空间的Borel复杂性

On the Borel complexity of the space of left-orderings of nilpotent groups

本文首次构造非交换左序群，其序空间共轭轨道关系有无穷轨道但Borel光滑，并给出幂零群光滑性的充分条件，同时证明幂零性非充分条件。

论某些单生成元幂零 brace 的结构

On the structure of some one-generator nilpotent braces

该研究刻画了单元素生成的幂零左 brace 的结构，为理解此类代数对象提供了详细结果。

有限Morley秩的双可解与双幂零斜括号

Bisoluble and Binilpotent Skew Braces of Finite Morley Rank

研究有限Morley秩的连通斜括号，证明其加法与乘法群的幂零性或可解性可推出括号本身的左幂零性或弱可解性。

某些幂零3维莱布尼茨代数的自同构群

On the automorphism groups of some nilpotent 3-dimensional Leibniz algebras

本文研究具有幂零类2和一维中心的3维莱布尼茨代数的自同构群结构，给出了其完整描述。

左斜辫的中心幂零性与Yang-Baxter方程的解

Central nilpotency of left skew braces and solutions of the Yang-Baxter equation

本文研究中心幂零斜辫，引入指标、好的Fitting理想和Frattini理想，用吸收多项式刻画理想交换子，解决开放问题，并举例说明中心幂零理想乘积不封闭及理想化子一般不存在。