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Algebras whose equivalence relations are congruences
I. B. Kozhukhov,
A. V. Reshetnikov Moscow State Institute of Electronic Technology
Abstract:
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.
UDC:
512.571+
512.548.2+512.533