Topological characterization of filter topological MV-algebras

Abstract

Motivated by the study of C-topology on lattice-ordered groups, Luan and Yang introduced the concept of filter topology on MV-algebras in a recent paper. In order to have a better perception of this concept, we study some basic facts and notable examples, including the standard MV-algebra and Chang's algebra, in this paper. Using the fact that an isomorphism between MV-algebras admits a topological isomorphism between filter topological MV-algebras, we show that every filter topology on a finite linearly ordered (finite simple) MV-algebra is discrete. In addition, we completely describe the shape of open sets in filter topology generated by a proper filter. Further, the description is applied to present the basic topological properties of a large class of filter topologies. As an application, we characterize an algebraic property of MV-algebras: an MV-algebra is bipartite if and only if it has a proper filter which generates a filter topology with exactly four open sets.