Generalization of filter theory of EQ-algebras based on lattices

Abstract In this article, considering the notion of a lattice-valued set (L-set, for short) we introduce the notions of L-prefilter and L-filter in EQ-algebras. We provide several characterizations and equivalent conditions for these concepts and also characterize the L-prefilter and L-filter generated by an L-set. Subsequently, we study the lattice structure of these filters and prove that in an EQ-algebra, the lattice of L-prefilters is a complete Brouwerian lattice, and hence it forms a Heyting lattice. In a residuated EQ-algebra, we show that the lattice of L-filters is also a Heyting lattice. They also form a semi-De Morgan algebra. Moreover, we demonstrate that the skeleton of an EQ-algebra under appropriate operations forms a Boolean algebra. Furthermore, we introduce (relative) L-congruences in EQ-algebras and investigate their properties. We also explore the relationships between L-prefilters/L-filters and L-congruences. We prove that L-prefilters induce a relative L-equivalence relation and L-filters correspond to relative L-congruences, and state and prove some isomorphism theorems.