Generalization of filter theory of EQ-algebras based on lattices

Abstract

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