Novel Operational Properties and Inductive Relationships of TL-Fuzzy Filters and TL-Fuzzy Congruences in Residuated Lattices

Filters and congruences are fundamental concepts in residuated lattices for characterizing their structure. In this paper, using a complete lattice L as the truth-value set and based on a triangular norm T and its induced operator νT, we investigate new operational properties of TL-fuzzy filters and TL-fuzzy congruences. We first define the ⊗T and νT-operations on L-fuzzy sets and study their effects on TL-fuzzy filters. Next, we examine the congruence-preserving properties and operational rules of TL-fuzzy congruences under ∗T and ⋄νT-compositions. Finally, leveraging the correspondence between TL-fuzzy filters and TL-fuzzy congruences, we explore the interplay and internal relationships among their respective operations.