Folding Theory Applied to Residuated Lattices

Residuated lattices play an important role in the study of fuzzy logic based on -norms. In this paper, we introduce some notions of -fold filters in residuated lattices, study the relations among them, and compare them with prime, maximal and primary, filters. This work generalizes existing results in BL-algebras and residuated lattices, most notably the works of Lele et al., Motamed et al., Haveski et al., Borzooei et al., Van Gasse et al., Kondo et al., Turunen et al., and Borumand Saeid et al., we draw diagrams summarizing the relations between different types of -fold filters and -fold residuated lattices. 1. Introduction Since Hájek introduced his basic fuzzy logics, (BL-logics in short) in 1998 [1], as logics of continuous -norms, a multitude of research papers related to the algebraic counterparts of BL-logics, has been published. In [2–5], the authors defined the notions of -fold (implicative, positive implicative, Boolean, fantastic, obstinate, and normal) filters in BL-algebras and studied the relation among them. A close analysis of the situation reveals that the main drive in all the previously mentioned works resides in the existence of an adjoint pair of operations. Just as the foldness theory for filters in BL-algebras generalizes filters introduced by Hájek, our foldness theory for filters in residuated lattices builds on recently published works on filters in residuated lattices by Haveshki et al. [6], Van Gasse et al. [7], Kondo and Dudek [8], Kondo and Turunen [9], Borumand Saeid and Pourkhatoun in [10], and Zahiri and Farahani in [11]. More specifically, we introduce the notions of -fold (implicative, positive implicative, Boolean, fantastic, normal, integral, and involutive) filters and -fold Boolean filters of the second kind in residuated lattices, notions that naturally generalize the corresponding ones previously studied in BL-algebras. Concurrently, we introduce the same foldness concepts on residuated lattices. In each folding class, we tie together the two concepts by characterizing the corresponding residuated lattices using their filters. For instance, it is shown (Proposition 20) that a residuated lattice is -fold implicative if and only if its trivial filter is -fold implicative if and only if all its filters are -fold implicative. Examples are included not only to illustrate the newly introduced concepts but also to differentiate them from the existing ones. Finally, diagrams summarizing all the relationships between the above classes of filters and residuated lattices are given (see Figures 1 and 2) for quick