Infinite-dimensional quasigroups of finite orders

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 $fleft( {bar y} right) ne fleft( {bar z} right)$ 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^?, where ? are positive integers, generate Lebesgue nonmeasurable subsets of the interval [0, 1].