On connection between reducibility of an n-ary quasigroup and that of its retracts

An $n$-ary operation $Q:S^n\to S$ is called an $n$-ary quasigroup of order $|S|$ if in the equation $x_0=Q(x_1,...,x_n)$ knowledge of any $n$ elements of $x_0,...,x_n$ uniquely specifies the remaining one. An $n$-ary quasigroup $Q$ is (permutably) reducible if $Q(x_1,...,x_n)=P(R(x_{s(1)},...,x_{s(k)}),x_{s(k+1)},...,x_{s(n)})$ where $P$ and $R$ are $(n-k+1)$-ary and $k$-ary quasigroups, $s$ is a permutation, and $1<k<n$. An $m$-ary quasigroup $R$ is called a retract of $Q$ if it can be obtained from $Q$ or one of its inverses by fixing $n-m>0$ arguments. We show that every irreducible $n$-ary quasigroup has an irreducible $(n-1)$-ary or $(n-2)$-ary retract; moreover, if the order is finite and prime, then it has an irreducible $(n-1)$-ary retract. We apply this result to show that all $n$-ary quasigroups of order 5 or 7 whose all binary retracts are isotopic to $Z_5$ or $Z_7$ are reducible for $n>3$. Keywords: $n$-ary quasigroups, retracts, reducibility, latin hypercubesComment: English: 19pp; Russian: 20pp. V.2: case n=4 added, Russian translation added, title changed (old title: On reducibility of n-ary quasigroups, II

On a connection between the switching separability of a graph and that of its subgraphs

A graph of order $n>3$ is called {switching separable} if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the following: if removing any one or two vertices of a graph always results in a switching separable subgraph, then the graph itself is switching separable. On the other hand, for every odd order greater than 4, there is a graph that is not switching separable, but removing any vertex always results in a switching separable subgraph. We show a connection with similar facts on the separability of Boolean functions and reducibility of $n$-ary quasigroups. Keywords: two-graph, reducibility, separability, graph switching, Seidel switching, graph connectivity, $n$-ary quasigroupComment: english: 9 pages; russian: 9 page

Unsolved Problems in Group Theory. The Kourovka Notebook

This is a collection of open problems in group theory proposed by hundreds of mathematicians from all over the world. It has been published every 2-4 years in Novosibirsk since 1965. This is the 19th edition, which contains 111 new problems and a number of comments on about 1000 problems from the previous editions.Comment: A few new solutions and references have been added or update