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This article is cited in 4 scientific papers (total in 4 papers)
Infinite-Dimensional Quasigroups of Finite Orders
V. N. Potapovab a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University
Abstract:
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 $[0,1]$.
Keywords:
$n$-ary quasigroup, separable quasigroup, Lebesgue nonmeasurable sets, semilinear quasigroup, Boolean function.
Received: 29.04.2011 Revised: 26.02.2012
Citation:
V. N. Potapov, “Infinite-Dimensional Quasigroups of Finite Orders”, Mat. Zametki, 93:3 (2013), 457–465; Math. Notes, 93:3 (2013), 479–486
Linking options:
https://www.mathnet.ru/eng/mzm9141https://doi.org/10.4213/mzm9141 https://www.mathnet.ru/eng/mzm/v93/i3/p457
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