Distance functions and filter topological residuated lattices 1

Abstract

In this paper, we firstly extend C. C. chang’s distance functions from MV-algebras into residuated lattices. But in general, the functions may not be a distance function on residuated lattices. We introduce weak involutory residuated lattices, in which Chang’s function is a pseduo distance function. Moreover we prove that the functions become distance functions on involutory residuated lattices. Secondly by use of the function and a lattice filter, we define F-ball on residuated lattices, and we prove that the set of all F-balls forms a base of a topology τ _LF on an involutory residuated lattice. Moreover we show that the topology τ _LF on an involutory residuated lattice L makes L to be a topological residuated lattice. At last, we use filters instead of lattice filters to set up filter topologies on an involutory residuated lattice, and study the properties of the filter topologies.