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Fundam. Prikl. Mat., 2010, Volume 16, Issue 3, Pages 161–192 (Mi fpm1326)  

This article is cited in 4 scientific papers (total in 4 papers)

Algebras whose equivalence relations are congruences

I. B. Kozhukhov, A. V. Reshetnikov

Moscow State Institute of Electronic Technology

Abstract: It is proved that all the equivalence relations of a universal algebra $A$ are its congruences if and only if either $|A|\le2$ or every operation $f$ of the signature is a constant (i.e., $f(a_1,…,a_n)=c$ for some $c\in A$ and all the $a_1,…,a_n\in A$) or a projection (i.e., $f(a_1,…,a_n)=a_i$ for some $i$ and all the $a_1,…,a_n\in A$). All the equivalence relations of a groupoid $G$ are its right congruences if and only if either $|G|\le2$ or every element $a\in G$ is a right unit or a generalized right zero (i.e., $xa=ya$ for all $x,y\in G$). All the equivalence relations of a semigroup $S$ are right congruences if and only if either $|S|\le 2$ or $S$ can be represented as $S=A\cup B$, where $A$ is an inflation of a right zero semigroup, and $B$ is the empty set or a left zero semigroup, and $ab=a$, $ba=a^2$ for $a\in A$, $b\in B$. If $G$ is a groupoid of 4 or more elements and all the equivalence relations of it are right or left congruences, then either all the equivalence relations of the groupoid $G$ are left congruences, or all of them are right congruences. A similar assertion for semigroups is valid without the restriction on the number of elements.

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English version:
Journal of Mathematical Sciences (New York), 2011, 177:6, 886–907

Bibliographic databases:

UDC: 512.571+512.548.2+512.533

Citation: I. B. Kozhukhov, A. V. Reshetnikov, “Algebras whose equivalence relations are congruences”, Fundam. Prikl. Mat., 16:3 (2010), 161–192; J. Math. Sci., 177:6 (2011), 886–907

Citation in format AMSBIB
\Bibitem{KozRes10}
\by I.~B.~Kozhukhov, A.~V.~Reshetnikov
\paper Algebras whose equivalence relations are congruences
\jour Fundam. Prikl. Mat.
\yr 2010
\vol 16
\issue 3
\pages 161--192
\mathnet{http://mi.mathnet.ru/fpm1326}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2786536}
\elib{http://elibrary.ru/item.asp?id=16350333}
\transl
\jour J. Math. Sci.
\yr 2011
\vol 177
\issue 6
\pages 886--907
\crossref{https://doi.org/10.1007/s10958-011-0517-1}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80052348584}


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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. A. V. Reshetnikov, “O kongruentsiyakh chastichnykh $n$-arnykh gruppoidov”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 11:3(2) (2011), 46–51  mathnet
    2. Reshetnikov A.V., “O chastichnykh gruppoidakh, u kotorykh kazhdoe otnoshenie ekvivalentnosti yavlyaetsya kongruentsiei”, Vestnik Moskovskoi gosudarstvennoi akademii delovogo administrirovaniya. Seriya: Filosofskie, sotsialnye i estestvennye nauki, 2011, no. 5, 166–170  elib
    3. A. R. Khaliullina, “Kongruentsii poligonov nad polugruppami pravykh nulei”, Chebyshevskii sb., 14:3 (2013), 142–146  mathnet
    4. A. V. Reshetnikov, “O chastichnykh $n$-arnykh gruppoidakh, u kotorykh kazhdoe otnoshenie ekvivalentnosti yavlyaetsya kongruentsiei”, Chebyshevskii sb., 17:1 (2016), 232–239  mathnet  elib
  • Фундаментальная и прикладная математика
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