Profinite MV-algebras and Multisets

Abstract

Simple algebraic and topological characterizations of profinite MV-algebras (i.e., MV-algebras that are inverse limits of finite MV-algebras) are obtained. It is shown that these are the direct products of finite Łukasiewicz chains. We also prove that the category M of multisets is dually equivalent to the category P of profinite MV-algebras and complete homomorphisms. This duality extends the well-known duality between finite MV-algebras, and finite multisets on one hand, and the duality between sets with functions and atomic complete Boolean algebras with complete homomorphisms on the other hand.

Stone MV-algebras and strongly complete MV-algebras

“…The third follows from Theorem 3.2(2-3) and the remaining isomorphisms follow from the definition of the Boolean center. Recall [15,Theorem 2.5] that if A is a profinite MV-algebra, then A ∼ = x∈X L nx for some set X and n x ≥ 2 are integers. Let P f (X) be the set of finite subsets of X.…”

“…Recall that if A is a profinite MV-algebra, then A ∼ = x∈X L nx for some set X, and some integers n x ≥ 2. In addition, by [15,Corollary 3.4], the set η(A) := {n x : x ∈ X} is uniquely determined by A.…”

“…During the author's BLAST 2014's presentation on "profinite MV-algebras and multisets" [15], several attendees asked about the position of the category of MV-algebras with respect to the above problems. The primary goal of the present paper is to provide a complete answer to those questions and study related problems.…”

Profinite completions and MacNeille completions of MV-algebras

“…A detailed treatment of topological MV-algebras can be found in [14,20] Unlike bounded distributive lattices, Heyting algebras, Boolean algebras, or orthomodular lattices, where the topic of profiniteness has been well investigated (see for e.g., [3,4,5,6]), MV-algebras have not yet received the same level of attention. To continue our study on the theme of profiniteness in MV-algebras, which has been initiated in [18,19], we focus in this article on profinite completions and applications.…”

“…For basic MV-algebra terminologies, the reader can consult [8,17], and for basic facts about profinite MV-algebras, the reader can consult [18,19].…”