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Mat. Zametki, 2013, Volume 93, Issue 3, Pages 457–465 (Mi mz9141)  

This article is cited in 3 scientific papers (total in 3 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.

DOI: https://doi.org/10.4213/mzm9141

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English version:
Mathematical Notes, 2013, 93:3, 479–486

Bibliographic databases:

Document Type: Article
UDC: 512.548+519.143
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

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. A. Malyugin, “Sovershennye dvoichnye kody beskonechnoi dliny”, PDM. Prilozhenie, 2015, no. 8, 117–120  mathnet  crossref
    2. S. A. Malyugin, “Perfect binary codes of infinite length”, J. Appl. Industr. Math., 11:2 (2017), 227–235  mathnet  crossref  crossref  elib
    3. S. A. Malyugin, “Sovershennye dvoichnye kody beskonechnoi dliny s polnoi sistemoi troek”, Sib. elektron. matem. izv., 14 (2017), 877–888  mathnet  crossref
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