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
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It is proved that all the equivalence relations of a universal
are its congruences if and only if either or every operation of the signature is
a constant (i.e.,
for some and all the
a projection (i.e.,
and all the ).
All the equivalence relations of a groupoid are its right congruences
if and only if either or every element
is a right unit or a generalized
right zero (i.e., for all ).
All the equivalence relations of a semigroup are right congruences if
and only if either or can be represented as
, where is an inflation of
a right zero semigroup, and is the empty set or
a left zero semigroup, and , for
a groupoid of or more elements and all
the equivalence relations of it are right or left congruences, then
either all the equivalence relations of the groupoid 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.
Last modified: March 24, 2011