Demiquantifiers on $\ell$-groups

Abstract

In this paper we introduce two kinds of unary operations on abelian $\ell$-groups with a positive distinguished element u. One of them, called demiquantifier of type I, behaves like an existential quantifier (in the sense of Cimadamore and Varela) in the negative cone, and like a universal quantifier in the positive cone. The other kind of unary operation we introduce, called demiquantifier of type II, satisfies analogous properties to demiquantifiers of type I via a translation of the negative cone, by means of the element u. These unary operations are interdefinable with the usual existential quantifiers, provided the distinguished element u is a strong unit. Moreover, if G is an abelian $\ell$-group, then the restriction of a demiquantifier of type II to the MV-algebra $\Gamma (G,u)$ yields a different type of quantifier, where $\Gamma$ is Mundici’s functor. These quantifiers are interdefinable with the usual existential quantifiers on MV-algebras given by Rutledge, provided that the involution of the corresponding MV-algebras have a fixed point.