### $3$-configurations with simple edge basis and their corresponding quasigroup identities

There is described a procedure which determines the quasigroup identity corresponding to a given 3-coloured 3-configuration with a simple edge basis.

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There is described a procedure which determines the quasigroup identity corresponding to a given 3-coloured 3-configuration with a simple edge basis.

Let $G$ be a finite group and ${C}_{2}$ the cyclic group of order $2$. Consider the $8$ multiplicative operations $(x,y)\mapsto {\left({x}^{i}{y}^{j}\right)}^{k}$, where $i$, $j$, $k\in \{-1,\phantom{\rule{0.166667em}{0ex}}1\}$. Define a new multiplication on $G\times {C}_{2}$ by assigning one of the above $8$ multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in {C}_{2}$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M(G,2)$.

We investigate loops defined upon the product ${\mathbb{Z}}_{m}\times {\mathbb{Z}}_{k}$ by the formula $(a,i)(b,j)=((a+b)/(1+t{f}^{i}\left(0\right){f}^{j}\left(0\right)),i+j)$, where $f\left(x\right)=(sx+1)/(tx+1)$, for appropriate parameters $s,t\in {\mathbb{Z}}_{m}^{*}$. Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If $s=1$, then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.

A pointed quasigroup is said to be semicentral if it is principally isotopic to a group via a permutation on one side and a group automorphism on the other. Convex combinations of permutation matrices given by the one-sided multiplications in a semicentral quasigroup then yield doubly stochastic transition matrices of finite Markov chains in which the entropic behaviour at any time is independent of the initial state.

In this note it is shown that the closure condition, X1Y2 = X2Y1, X1Y4 = X2Y3, X3Y3 = X4Y1 --> X4Y2 = X3Y4, (and its dual) is equivalent to the Thomsen condition in quasigroups but not in general. Conditions are also given under which groupoids satisfying it are principal homotopes of cancellative, abelian semigroups, or abelian groups.

In the present paper we construct the accompanying identity $\widehat{I}$ of a given quasigroup identity $I$. After that we deduce the main result: $I$ is isotopically invariant (i.e., for every guasigroup $Q$ it holds that if $I$ is satisfied in $Q$ then $I$ is satisfied in every quasigroup isotopic to $Q$) if and only if it is equivalent to $\widehat{I}$ (i.e., for every quasigroup $Q$ it holds that in $Q$ either $I,\widehat{I}$ are both satisfied or both not).

Let $Q$ be a loop such that $\left|Q\right|$ is square-free and the inner mapping group $I\left(Q\right)$ is nilpotent. We show that $Q$ is centrally nilpotent of class at most two.

A solvable primitive group with finitely generated abelian stabilizers is finite.