Varieties with equationally deﬁnable
factor congruences II
Mariana Badano and Diego J. Vaggione
Abstract. We study four types of equational deﬁnability of factor congruences in
1. The paper completes the work of a previous paper on left
equational deﬁnability of factor congruences.
A variety with
1 is a variety V for which there are 0-ary terms
1 → x = y,
0 = (0
1 = (1
). (If a =(a
) we write a =
b to express
.) This condition is
equivalent to the fact that there is a nullary operation in the language of V
and no non-trivial algebra in V has a trivial subalgebra. Classical examples of
this type of varieties are the variety S
of bounded join semilattices and the
variety R of rings with identity (in both cases N = 1). If a ∈ A
b ∈ B
then we use [a ,
b] to denote the N-tuple ((a
)) ∈ (A × B)
If A ∈V, then we say that e ∈ A
is a central element of A if there exists an
isomorphism A → A
such that e → [
1]. Also, we say that e and
a pair of complementary central elements of A if there exists an isomorphism
A → A
such that e → [
f → [
0]. As is well known, the direct
product representations A → A
of an algebra A are closely related to
the concept of factor congruence. A pair of congruences (θ, δ) of an algebra A
is a pair of complementary factor congruences of A if θ ∩ δ = ∆ and θ ◦ δ = ∇
and in such a case θ and δ are called factor congruences.
Consider the following property.
(L) There is a ﬁrst order formula λ(z , x , y ) such that for every A, B ∈V,
A × B |= λ([
1], (a, b), (a
)) iﬀ a = a
Presented by J. Raftery.
Received October 15, 2015; accepted in ﬁnal form June 1, 2016.
2010 Mathematics Subject Classiﬁcation: Primary: 03C05; Secondary: 08B05, 08B10.
Key words and phrases: central element, equationally deﬁnable factor congruences,
Boolean factor congruences.
Algebra Univers. 78 (2017) 19–42
Published online February 25, 2017
© Springer International Publishing 2017