On the relationships between hybrid generalized Bosbach states and L-filters in non-commutative residuated lattices

Abstract

States and filters play the essential roles in studying logical algebras. In this paper, we study the relationships between hybrid generalized Bosbach states and L-filters and hybrid L-filters in non-commutative residuated lattices. Particularly, two types of L-filters and hybrid L-filters and their subclasses are defined, and some of their properties are obtained. Then, relationships between special types of (hybrid) L-filters and the hybrid generalized Bosbach states are considered where hybrid generalized Bosbach states are characterized by some type I or type II (hybrid) L-filters with additional conditions. Associated with these relationships, new subclasses of hybrid generalized Bosbach states such as implicative type IV-1, IV-2, V-1, V-2, VI-1, VI-2 states, Involution type IV-1, IV-2 states and Boolean type IV-1, IV-2 states are introduced, and the relationships between various types of hybrid generalized Bosbach states are investigated in detail.