Algebras whose equivalence relations are congruences

Abstract

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 ∈ A and all the a 1 , . . . , a n ∈ A) or a projection (i.e., f(a 1 , . . . , a n ) = a i for some i and all the a 1 , . . . , a n ∈ A). All the equivalence relations of a groupoid G are its right congruences if and only if either |G| ≤ 2 or every element a ∈ G is a right unit or a generalized right zero (i.e., x a = y a for all x, y ∈ 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∪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 ∈ A, b ∈ 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.