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Colloquium Mathematicum

2002 | 93 | 2 | 209-235
Tytuł artykułu

Join-semilattices with two-dimensional congruence amalgamation

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
We say that a ⟨∨,0⟩-semilattice S is conditionally co-Brouwerian if (1) for all nonempty subsets X and Y of S such that X ≤ Y (i.e. x ≤ y for all ⟨x,y⟩ ∈ X × Y), there exists z ∈ S such that X ≤ z ≤ Y, and (2) for every subset Z of S and all a, b ∈ S, if a ≤ b ∨ z for all z ∈ Z, then there exists c ∈ S such that a ≤ b ∨ c and c ≤ Z. By restricting this definition to subsets X, Y, and Z of less than κ elements, for an infinite cardinal κ, we obtain the definition of a conditionally κ-co-Brouwerian ⟨∨,0⟩-semilattice.
We prove that for every conditionally co-Brouwerian lattice S and every partial lattice P, every ⟨∨,0⟩-homomorphism $φ: Con_{c} P → S$ can be lifted to a lattice homomorphism f: P → L for some relatively complemented lattice L. Here, $Con_{c} P$ denotes the ⟨∨,0⟩-semilattice of compact congruences of P.
We also prove a two-dimensional version of this result, and we establish partial converses of our results and various of their consequences in terms of congruence lattice representation problems. Among these consequences, for every infinite regular cardinal κ and every conditionally κ-co-Brouwerian S of size κ, there exists a relatively complemented lattice L with zero such that $Con_{c}L ≅ S$.
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Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
209-235
Opis fizyczny
Daty
wydano
2002
Twórcy
autor
• CNRS, UMR 6139, Département de Mathématiques, BP 5186, Université de Caen, Campus 2, 14032 Caen Cedex, France
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Bibliografia
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