Infinite-dimensional quasigroups of finite orders

Abstract

Let Σ be a finite set of cardinality k > 0, let $\mathbb{A}$ be a finite or infinite set of indices, and let $\mathcal{F} \subseteq \Sigma ^\mathbb{A}$ be a subset consisting of finitely supported families. A function $f:\Sigma ^\mathbb{A} \to \Sigma$ is referred to as an $\mathbb{A}$-quasigroup (if $\left| \mathbb{A} \right| = n$, then an n-ary quasigroup) of order k if $f\left( {\bar y} \right) \ne f\left( {\bar z} \right)$ for any ordered families $\bar y$ and $\bar z$ that differ at exactly one position. It is proved that an $\mathbb{A}$-quasigroup f of order 4 is reducible (representable as a superposition) or semilinear on every coset of $\mathcal{F}$. It is shown that the quasigroups defined on Σ^ℕ, where ℕ are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].