Algebras whose equivalence relations are congruences

I. B. Kozhukhov ; A. V. Reshetnikov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 3, pp. 161-192

Résumé

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., $f(a_1,\dots,a_n)=c$ for some $c\in A$ and all the $a_1,\dots,a_n\in A$) or a projection (i.e., $f(a_1,\dots,a_n)=a_i$ for some $i$ and all the $a_1,\dots,a_n\in A$). 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.