Polyhedral MV-algebras

Abstract

A polyhedron in R^n is a finite union of simplexes in R^n. An MV-algebra is polyhedral if it is isomorphic to the MV-algebra of all continuous I-valued piecewise linear functions with integer coefficients, defined on some polyhedron P in R^n. We characterize polyhedral MV-algebras as finitely generated subalgebras of semisimple tensor products of a simple MV-algebra and a finitely presented MV-algebra. We establish a duality between the category of polyhedral MV-algebras and the category of polyhedra with Z-maps. We prove that polyhedral MV-algebras are preserved under various kinds of operations, and have the amalgamation property. Strengthening the Hay-Wojcicki theorem, we prove that every polyhedral MV-algebra is strongly semisimple, in the sense of Dubuc-Poveda.