Smashed products of topological 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}.

线性（非线性）拟群及其仿旋的可解性

Orthogonality of linear (alinear) quasigroups and their parastrophes

给出了左（右）线性（非线性）拟群在各种组合下正交的充要条件。作为推论，我们得到了左（右）线性（线性）拟群的仿旋正交性条件。对称群S_n（$n\neq 2; 6$）上的任何线性拟群都不与它的（12）-仿射正交.

右积拟群与环

Right product quasigroups and loops

右群是右零半群与群的直积，在半群的半格分解理论中起着重要的作用。右群可以刻画为结合右拟群（其中左平移是双射的岩浆）。如果不假设结合性，则得到右拟群，它们不一定能表示为右零半群与拟群的直积。为了得到这样一个表示，我们需要更强的假设，这将我们引向\emph{right product quasigroup}的概念。如果拟群成分是一个（单侧）循环，则我们有一个\emph{right product (left, right) loop}。我们找到一个系统的身份公理化的右产品quasigroup，并使用这找到公理系统的右产品（左，右）循环;事实上，我们可以获得后者的每一个邻接只是一个适当的公理的右产品quasigroup公理系统。我们得到了右积拟群和环的其他性质，并证明了右积拟群的公理是独立的。

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.

有限代数系统对拓扑群的逼近

On approximation of topological groups by finite algebraic systems

已知局部紧群可被有限群逼近的条件是幺模的，但这个条件并不充分，例如单李群不能被有限群逼近为拓扑群.本文研究了局部紧群用更一般的有限代数系统的逼近。证明了有限半群对局部紧群的逼近等价于有限群的逼近，从而证明了不是所有的局部紧群都能被有限半群逼近.证明了任何局部紧群都可被有限左（右）拟群逼近，但局部紧群可被有限拟群（拉丁方）逼近蕴涵了它的单模性.局部紧群的幺模性是否蕴涵它可被有限拟群逼近的问题是公开的。我们只证明了离散群可由有限拟群逼近。

关于有限代数系统对拓扑群的逼近。II

On approximation of topological groups by finite algebraic systems. II

回想一下，如果G上的左Haar测度等于右Haar测度，则称局部紧群G为幺模群。本文证明了G是幺模的当且仅当它可由有限拟群（拉丁方）逼近。