n-fold filters in residuated lattice
A. Kadji, C. Lele, M. Tonga
TL;DR
The paper develops a comprehensive framework for $n$-fold filters in residuated lattices, extending BL-algebra concepts from fuzzy logic to a broader algebraic setting. It defines and analyzes $n$-fold implicative, $n$-fold positive implicative, $n$-fold boolean, $n$-fold fantastic, $n$-fold normal, and $n$-fold obstinate filters, providing multiple characterizations, quotient-structure correspondences, and extension theorems. Key results include the equivalence between $F$ being $n$-fold implicative and $L/F$ inheriting $n$-fold implicative residuated structure, the inclusion of $n$-fold positive implicative filters in $n$-fold implicative filters, and the characterization of $L/F$ being $n$-fold fantastic residuated lattices. The work also introduces semi-maximal and obstinate filters, demonstrates their relations to maximal, boolean, and prime-type filters, and presents diagrammatic summaries of the interrelations, thereby broadening the algebraic tools available for t-norm based fuzzy logics and their semantic interpretations.
Abstract
Residuated lattices play an important role in the study of fuzzy logic based of t-norm. In this paper, we introduced the notions of n-fold implicative filters, n-fold positive implicative filters, n-fold boolean filters, n-fold fantastic filters, n-fold normal filters and n-fold obstinate filters in residuated lattices and study the relations among them. This generalized the similar existing results in BL-algebra with the connection of the work of Kerre and all in [14], Kondo and all in [7], [11] and Motamed and all in [9]. At the end of this paper, we draw two diagrams; the first one describe the relations between some type of n-fold filters in residuated lattices and the second one describe the relations between some type of n-fold residuated lattices.