FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 3, PAGES 161-192

Algebras whose equivalence relations are congruences

I. B. Kozhukhov
A. V. Reshetnikov

Abstract

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It is proved that all the equivalence relations of a universal algebra $A$ are its congruences if and only if either $|A|$£ 2 or every operation $f$ of the signature is a constant (i.e., $f(a$₁,...,a_n) = c for some $c$Î A and all the $a$₁,...,a_n Î A) or a projection (i.e., $f(a$₁,...,a_n) = a_i for some $i$ and all the $a$₁,...,a_n Î A). All the equivalence relations of a groupoid $G$ are its right congruences if and only if either $|G|$£ 2 or every element $a$Î G is a right unit or a generalized right zero (i.e., $xa\; =\; ya$ for all $x,y$Î G). All the equivalence relations of a semigroup $S$ are right congruences if and only if either $|S|$£ 2 or $S$ can be represented as $S\; =\; A$È B, where $A$ is an inflation of a right zero semigroup, and $B$ is the empty set or a left zero semigroup, and $ab\; =\; a$, $ba\; =\; a2$ for $a$Î A, $b$Î 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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