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

TL;DR

This paper investigates the relationship between the reducibility of n-ary quasigroups and their smaller retracts, establishing conditions under which irreducible quasigroups have irreducible lower-arity retracts, with applications to quasigroups of prime order.

Contribution

It proves that every irreducible n-ary quasigroup has an irreducible (n-1)- or (n-2)-ary retract, especially for prime orders, and applies this to classify certain quasigroups of order 5 and 7.

Findings

Irreducible n-ary quasigroups have irreducible (n-1)- or (n-2)-ary retracts.

For prime order, irreducible quasigroups have an irreducible (n-1)-ary retract.

All n-ary quasigroups of order 5 or 7 with binary retracts isotopic to Z_5 or Z_7 are reducible for n>3.

Abstract

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 hypercubes