On the geometric theory of local MV-algebras

Abstract

Abstract. We investigate the geometric theory of local MV-algebras and its quotients axiomatizing the local MV-algebras in a given proper variety of MV-algebras. We show that, whilst the theory of local MValgebras is not of presheaf type, each of these quotients is a theory of presheaf type which is Morita-equivalent to an expansion of the theory of lattice-ordered abelian groups. Di Nola-Lettieri's equivalence is recovered from the Morita-equivalence for the quotient axiomatizing the local MV-algebras in Chang's variety, that is, the perfect MV-algebras. We establish along the way a number of results of independent interest, including a constructive treatment of the radical for MV-algebras in a fixed proper variety of MV-algebras and a representation theorem for the finitely presentable algebras in such a variety as finite products of local MV-algebras.

General affine adjunctions, Nullstellensätze, and dualities

“…Recall that a geometric theory is said to be of presheaf type if it is classified by a presheaf topos. This class of theories includes all finitary algebraic (or, more generally, cartesian) theories and many other interesting cases, such as the theory of total orders, the theory of algebraic extensions of a base field, the theory of lattice-ordered abelian groups with strong unit [11], the theory of perfect MV-algebras or more generally of local MV-algebras in a proper variety of MV-algebras (see [12] and [13]), etc. In fact, theories of presheaf type represent the 'logical counterpart' of small categories: every small category is, up to idempotent-splitting completion, the category of finitely presentable models of a theory of presheaf type (cf.…”