Generalization of filter theory of EQ-algebras based on lattices.

This article focuses on the development of lattice-valued (|${\mathcal{L}}$|-) prefilter and filter theory within EQ-algebras, algebraic structures related to fuzzy logic. It introduces |${\mathcal{L}}$|-prefilters and |${\mathcal{L}}$|-filters as generalizations of classical filters using complete lattices instead of the real unit interval, providing characterizations, equivalent conditions, and constructions of these filters generated by |${\mathcal{L}}$|-sets. The paper establishes that the lattices of |${\mathcal{L}}$|-prefilters and |${\mathcal{L}}$|-filters form complete Brouwerian (Heyting) lattices and semi-De Morgan algebras, with certain substructures forming Boolean algebras. It further explores |${\mathcal{L}}$|-congruences, showing a bijective correspondence between |${\mathcal{L}}$|-filters and relative |${\mathcal{L}}$|-congruences, and proves that factor EQ-algebras induced by |${\mathcal{L}}$|-prime filters are linearly ordered separated EQ-algebras. The article also presents several isomorphism theorems, including a Fundamental Homomorphism Theorem, extending classical universal algebra results to the lattice-valued setting in EQ-algebras.