On lifting quasi-filters and strong lifting quasi-filters in MV-algebras

Abstract

A lifting inference rule is a many-valued inference rule that manipulates with evaluated formulas, it has played an important role in Pavelka-style fuzzy propositional logic. In this paper, we consider the algebraic characterization of Pavelka-style fuzzy logic deductive system based on lifting rules in MV-algebras from the algebraic semantical point of view, which can be a tool to analyze the provability in fuzzy logic with evaluated syntax based on the lifting inference rules. We introduce the notions of lifting quasi-filter and strong lifting quasi-filter in MV-algebras, they are the algebraic abstractions of the sets of provable formulae which are closed with respect to lifting inference rules in the corresponding formal logic system, and some important properties are presented. We also characterize two kinds of extended lifting quasi-filters and several kinds of generated lifting quasi-filters, and prove that the class F _α  (L) of all lifting quasi-filters with respect to α is an MV-algebra, the class Fs _α  (L) of all strong lifting quasi-filters with respect to α is a complete Heyting algebra.