The variety of modal weak Godel algebras

Abstract

An algebra 〈A,∧,∨,→,□,0,1〉 of type (2,2,2,1,0,0) is said to be a modal weak Heyting algebra if 〈A,∧,∨,→,0,1〉 is a weak Heyting algebra and the following conditions are satisfied for every a,b∈A: M1) □(1)=1, M2) □(a∧b)=□(a)∧□(b) and M3) □(a→b)≤□(a)→□(b). If this algebra satisfies the inequality a∧(a→b)≤b then it is called modal RWH-algebra. In this paper we study the variety of modal RWH-algebras, which is denoted by KRWH, and some of its subvarieties. We focus our attention on the study of the lattice of congruences of any member of KRWH and some related properties. In particular, we give an equational basis for the subvariety of KRWH generated by the class of their totally ordered members.