Recently W. Holliday gave a choice-free construction of a canonical extension of a boolean algebra B as the boolean algebra of regular open subsets of the Alexandroff topology on the poset of proper filters of B. We make this construction point-free by replacing the Alexandroff space of proper filters of B with the free frame LB generated by the bounded meet-semilattice of all filters of B (ordered by reverse inclusion) and prove that the booleanization of LB is a canonical extension of B. Our main result generalizes this approach to the category baℓ of bounded archimedean ℓ-algebras, thus yielding a point-free construction of canonical extensions in baℓ. We conclude by showing that the algebra of normal functions on the Alexandroff space of proper archimedean ℓ-ideals of A is a canonical extension of A∈ baℓ, thus providing a generalization of the result of Holliday to baℓ.

A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean ℓ -Algebras / G. Bezhanishvili, L. Carai, P. Morandi. - In: ORDER. - ISSN 0167-8094. - 40:2(2023 Jul), pp. 257-287. [10.1007/s11083-022-09609-3]

A Point-Free Approach to Canonical Extensions of Boolean Algebras and Bounded Archimedean ℓ -Algebras

L. Carai
Secondo
;
2023

Abstract

Recently W. Holliday gave a choice-free construction of a canonical extension of a boolean algebra B as the boolean algebra of regular open subsets of the Alexandroff topology on the poset of proper filters of B. We make this construction point-free by replacing the Alexandroff space of proper filters of B with the free frame LB generated by the bounded meet-semilattice of all filters of B (ordered by reverse inclusion) and prove that the booleanization of LB is a canonical extension of B. Our main result generalizes this approach to the category baℓ of bounded archimedean ℓ-algebras, thus yielding a point-free construction of canonical extensions in baℓ. We conclude by showing that the algebra of normal functions on the Alexandroff space of proper archimedean ℓ-ideals of A is a canonical extension of A∈ baℓ, thus providing a generalization of the result of Holliday to baℓ.
Boolean algebra; Bounded archimedean ℓ-algebra; Canonical extension; Gelfand duality; Point-free topology; Stone duality;
Settore MAT/01 - Logica Matematica
Settore MAT/02 - Algebra
Settore MAT/03 - Geometria
lug-2023
set-2022
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1018312
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