Algebras whose equivalence relations are congruences

• Published in: Journal of mathematical sciences (New York, N.Y.) Vol. 177; no. 6; pp. 886 - 907
• Main Authors: Kozhukhov, I. B., Reshetnikov, A. V.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 12-09-2011; Springer
• Subjects: Algebra; Mathematics; Mathematics and Statistics; Universal Algebra; Congruence Lattice; Equivalence Relation; Rectangular Band; Torsion Class
• ISSN: 1072-3374; 1573-8795
• DOI: 10.1007/s10958-011-0517-1

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.