Stone MV-algebras and Strongly complete MV-algebras

TL;DR

This paper characterizes compact Hausdorff and Stone MV-algebras, showing they are products of specific chains, and establishes conditions under which MV-algebras are strongly complete, linking algebraic and topological properties.

Contribution

It provides a complete characterization of compact Hausdorff and Stone MV-algebras as products of chains and identifies criteria for strong completeness in MV-algebras.

Findings

Compact Hausdorff MV-algebras are products of [0,1] and finite chains.

Stone MV-algebras are products of finite chains.

An MV-algebra is strongly complete iff it is profinite with principal finite-rank maximal ideals.

Abstract

Compact Hausdorff topological MV-algebras and Stone MV-algebras are completely characterized. We obtain that compact Hausdorff topological MV-algebras are product (both topological and algebraic) of copies $[0,1]$ with standard 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. We also prove that an MV-algebra is strongly complete (isomorphic to its profinite completion) if and only if it is profinite and its maximal ideals of finite ranks are principal.