Infinite-Dimensional Quasigroups of Finite Orders

Let $\Sigma$ 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 $f\colon\Sigma^\mathbb{A}\to\Sigma$ is referred to as an $\mathbb{A}$-quasigroup (if $|\mathbb{A}|=n$, then an $n$-ary quasigroup) of order $k$ if $f(\overline{y})\neq f(\overline{z})$ for any ordered families $\overline{y}$ and $\overline{z}$ that differ at exactly one position. It is proved that an $\mathbb{A}$-quasigroup $f$ of order $4$ is separable (representable as a superposition) or semilinear on every coset of $\mathcal{F}$. It is shown that the quasigroups defined on $\Sigma^\mathbb{N}$, where $\mathbb{N}$ are positive integers, generate Lebesgue nonmeasurable subsets of the interval $[0,1]$.