A Unified Convex-Fuzzy Filter Theory in <i>BG</i> -Algebras

This paper develops a concise theory of [Formula: see text]-fuzzy filters in the setting of [Formula: see text]-algebras, emphasizing structural properties tied to a notion of fuzzy convexity. Here [Formula: see text] denotes a complete Heyting algebra that governs membership degrees, which extends classical filter concepts. We give a formal characterization of [Formula: see text]-fuzzy filters, describe homomorphisms that preserve convexity, and identify conditions under which the convex hull of a fuzzy set lies inside a filter. Principal results include monotonicity and closure properties, stability under convex combinations, and behavior under surjective homomorphisms: in particular, the image of an [Formula: see text]-fuzzy filter by a convexity-preserving map is again an [Formula: see text]-fuzzy filter. We also show closure-stability for sums, unions, and limits of filters, and illustrate the theory with representative examples and computations. Possible categorical formulations and extensions to related algebraic structures are noted as directions for future work.