Abstract
Fuzzy filters and their generalized types have been extensively studied in the literature. In this paper, a one-to-one correspondence between the set of all generalized fuzzy filters and the set of all generalized fuzzy congruences is established, a quotient residuated lattice with respect to generalized fuzzy filter is induced, and several types of generalized fuzzyn-fold filters such as generalized fuzzyn-fold positive implicative (fantastic and Boolean) filters are introduced; examples and results are provided to demonstrate the relations among these filters.
Highlights
Residuated lattices, introduced by Ward and Dilworth in [1], are very basic algebraic structures among algebras associated with logical systems
≤ max {f (x → z), α}, (2) a generalized fuzzy positive implicative filter if for all x, y, z ∈ L it holds that min {f (x → (y → z)), f (x → y), β} (2)
Generalized fuzzy filters have been extensively studied in the literature
Summary
Residuated lattices, introduced by Ward and Dilworth in [1], are very basic algebraic structures among algebras associated with logical systems. In [4, 5], some types of filters such as implicative filters, fantastic filters, and Boolean filters in BL-algebras were proposed and some characterizations of them with identity forms were given, and in [6], these filters were generalized to residuated lattices. In [6, 13, 14], fuzzy filters in MTLalgebras, BL-algebras, and residuated lattices were studied, respectively. This paper continues the study of generalized fuzzy nfold filters in residuated lattices.
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