Lukasiewicz semantic MV-topology for MV-algebra and its application to Lukasiewicz propositional logic

Abstract

We introduce an MV-topology on the set of all valuations of MV-algebra and then establish Lukasiewicz semantic MV-topological space. We study the topological properties of Lukasiewicz semantic MV-topology, and prove that the Lukasiewicz semantic MV-topological space is a compact zero dimension Hausdorff MV-topological space and a N-compact space. We also establish a classical topology $\mathcal{D}$ on the valuations set of MV-algebra, and prove that topology $\mathcal{D}$ is finer than the cut topology $\mathcal{C}$ generated by Lukasiewicz semantic MV-topology. We prove that a σ-complete lattice is an MV-algebra if and only if it is isomorphic to an MV-clopen lattice of a Stone MV-space. As an application, we use the compactness of topology $\mathcal{D}$ to prove the compactness of Lukasiewicz semantic and Lukasiewicz propositional logic system.