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**1 - 3**of**3**### Boolean Factor Congruences and Property (*)

, 809

"... A variety V has Boolean factor congruences (BFC) if the set of factor congruences of every algebra in V is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniq ..."

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A variety V has Boolean factor congruences (BFC) if the set of factor congruences of every algebra in V is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniqueness of direct product representations of algebras, since it is a strengthening of the refinement property. We provide an explicit Mal’cev condition for BFC. With the aid of this condition, it is shown that BFC is equivalent to a variant of the definability property (*), an open problem in R. Willard’s work [8]. 1

### Existentially Definable Factor Congruences

, 906

"... A variety V has definable factor congruences if and only if factor congruences can be defined by a first-order formula Φ having central elements as parameters. We prove that if Φ can be chosen to be existential, factor congruences in every algebra of V are compact. We study factor congruences in ord ..."

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A variety V has definable factor congruences if and only if factor congruences can be defined by a first-order formula Φ having central elements as parameters. We prove that if Φ can be chosen to be existential, factor congruences in every algebra of V are compact. We study factor congruences in order to understand direct product representations in varieties. It is known that in rings with identity and bounded lattices, factor congruences are characterized, respectively, by central idempotent elements and neutral complemented elements. D. Vaggione [4] generalized these concepts to a broader context. A variety with ⃗0 & ⃗1 is a variety V in which there exist unary terms 01(w),..., 0l(w), 11(w),..., 1l(w) such that V | = ⃗0(w) = ⃗1(w) → x = y, where w, x and y are distinct variables, ⃗0 = (01,...,0l) and ⃗1 = (11,...,1l). If λ ∈ A ∈ V, we say that ⃗e ∈ A l is a λ-central element of A if there exists an isomorphism A → A1 ×A2 such that λ ↦ → 〈λ1, λ2〉, ⃗e ↦ → [⃗0(λ1),⃗1(λ2)]. where we write [⃗a, ⃗ b] in place of (〈a1, b1〉,..., 〈al, bl〉) ∈ (A × B) l for ⃗a ∈ A l and ⃗ b ∈ B l. It is clear from the above definitions that if the language of V has a constant symbol c, the terms ⃗0 and ⃗1 can be chosen closed, and we can define a central element of A to be just a c A-central element. We will work heretofore under this assumption. In [3], Vaggione and the author introduced the following concept: Definition 1. V has Definable Factor Congruences (DFC) iff there exists a first order formula Φ(x, y,⃗z) in the language of V such that for all A, B ∈ V, and a, c ∈ A, b, d ∈ B,

### Directly Indecomposables in Semidegenerate Varieties of Connected po-Groupoids

, 810

"... We study varieties with a term-definable poset structure, po-groupoids. It is known that connected posets have the strict refinement property (SRP). In [7] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecom ..."

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We study varieties with a term-definable poset structure, po-groupoids. It is known that connected posets have the strict refinement property (SRP). In [7] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegenerate varieties of connected po-groupoids and show its quantifier complexity is bounded in general. 1