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(a1,...,an) = c for some c Î A and all the a1,...,an Î A) or a projection (i.e., f(a1,...,an) = ai for some i and all the a1,...,an 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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