Stone MV-algebras and strongly complete MV-algebras

Abstract

Abstract. The article has two main objectives: characterize compact Hausdorff topological MV-algebras and Stone MV-algebras on one hand, and characterize strongly complete MV-algebras on the other hand. We obtain that compact Hausdorff topological MV-algebras are product (both topological and algebraic) of copies [0, 1] with the interval topology and finite Lukasiewicz chains with discrete topology. Going one step further we also prove that Stone MV-algebras are product (both topological and algebraic) of finite Lukasiewicz chains with discrete topology. In the second part we prove that an MV-algebra is isomorphic to its profinite completion if and only if it is profinite and its only maximal ideals of finite ranks are principal.To the memory of a great mentor, John V. Leahy (1937Leahy ( -2015.

Profinite completions and MacNeille completions of MV-algebras

“…The following result, which is a strengthened and more complete version of [19,Proposition 3.1], is crucial for our subsequent results. Proposition 2.1.…”

“…The next two results were established in [19], but we can now offer simpler proofs thanks to Theorem 2.4.…”

“…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].…”

“…Let PMax(A) denotes the set of principal maximal ideals of A. We also observe from [19,Theorem 2.3] that every isomorphic profinite MV-algebras are automatically homeomorphic. ⇒:) Suppose that A is isomorphic to its profinite completion.…”