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.

Location: http://mech.math.msu.su/~fpm/eng/k10/k103/k10309h.htm

Last modified: March 24, 2011