@MISC{Terraf906existentiallydefinable, author = {Pedro Sánchez Terraf}, title = {Existentially Definable Factor Congruences}, year = {906} }

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Abstract

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,