6 S ep 2 00 8 REPRESENTATION THEORY OF MV - ALGEBRAS EDUARDO

In this paper we develop a general representation theory for mv-algebras. We furnish the appropriate categorical background to study this problem. Our guide line is the theory of classifying topoi of coherent extensions of universal algebra theories. Our main result corresponds, in the case of mv-algebras and mv-chains, to the representation of commutative rings with unit as rings of global sections of sheaves of local rings. We prove that any mv-algebra is isomorphic to the mv-algebra of all global sections of a sheaf of mv-chains on a compact topological space. This result is intimately related to McNaughton’s theorem, and we explain why our representation theorem can be viewed as a vast generalization of McNaughton’s. On spite of the language utilized in this abstract, we wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory. preface We wrote this paper in a way that, we hope, could be read without much acquaintance with either sheaf theory or mv-algebra theory. Our basic reference on mv-algebras is the book ’Algebraic Foundations of Many-valued Reasoning’ [1], and we refer to this book and not to the original sources for the known specific results we utilize. Of sheaf theory we need only some basic general facts, the reader may consult the book ’Sheaves in Geometry and Logic’ [9]. The basic reference for category theory is of course the classical ’Categories for the working mathematician’ [8].