Convergence with a Fixed Regulator in Perfect Mv-algebras

MV-algebras were introduced by Chang as an algebraic counterpart of the Lukasiewicz infinite-valued logic. D. Mundici proved that the category of MV-algebras is equivalent to the category of abelian l-groups with strong unit. A. Di Nola and A. Lettieri established a categorical equivalence between the category of perfect MV-algebras and the category of abelian l-groups. In this paper we investigate the convergence with a fixed regulator in perfect MV-algebras using Di Nola-Lettieri functors. The main result of the paper states that every locally Archimedean MV-algebra has a unique v-Cauchy completion. Introduction MV-algebras were defined by C.C. Chang in 1958 as algebraic models for the Lukasiewicz infinite-valued logic ([7]). Due to D. Mundici, MV-algebras can be viewed as intervals of abelian l-groups ([14]). A special subcategory of the category of MV-algebras is the class of perfect MV-algebras which are directly connected with the incompleteness of Lukasiewicz first order logic. A. Di Nola and A. Lettieri proved in [10] that the category of perfect MV-algebras is equivalent to the category of abelian l-groups. The order convergence in abelian l-groups is studied in [15] and [16], while α-convergence is presented in [1]. Š. Černák studied the convergence with a fixed regulator for abelian l-groups in [4] and for Archimedean lgroups in [5]. In the case of MV-algebras, the order convergence is presented in [12], α-convergence was investigated in [13] and various kinds of Cauchy completions are studied in [2]. Using the Mundici functor Γ, Š. Černák extended the convergence with a fixed regulator from abelian l-groups to MV-algebras ([6]). Order convergence in perfect MV-algebras has been presented in [11]. Using methods similar to those from [11], in this paper we