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Matematicheskie Zametki, 2013, Volume 93, Issue 3, Pages 457–465
DOI: https://doi.org/10.4213/mzm9141
(Mi mzm9141)
 

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
Full-text PDF (505 kB) Citations (4)
References:
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
English version:
Mathematical Notes, 2013, Volume 93, Issue 3, Pages 479–486
DOI: https://doi.org/10.1134/S0001434613030152
Bibliographic databases:
Document Type: Article
UDC: 512.548+519.143
Language: Russian
Citation: V. N. Potapov, “Infinite-Dimensional Quasigroups of Finite Orders”, Mat. Zametki, 93:3 (2013), 457–465; Math. Notes, 93:3 (2013), 479–486
Citation in format AMSBIB
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  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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