Foldeness of fuzzy Filters in BL-algebras
by
Celestin Lele
University of Dschang, Mathematics Department, Box 67, Cameroon

Abstract

From the logical point of view, various filters correspond to various sets of provable formulaes. In [], Zadeh introduced the notion of fuzzy sets. At present, this concept has been applied to many algebraic structures such as semigroups, groups, rings, modules, vector spaces. L.Liu and K.Li [] introduced the notion of fuzzy filter in BL-algebras. In [] and [], we have studied the notion of n-fold and fuzzy n-fold positive implicative filters, n-fold and fuzzy n-fold fantastic filters in BL-algebras and established many important properties. All the above interesting results motivate us to further investigate the fuzzy foldness of others types of filters in BL-algebras.

In this paper, we study the notion of n-folds implicative and fuzzy n-fold implicative filters in BL-algebras. Several characterizations of fuzzy n-fold implicative filters are given, we show that every n-fold (fuzzy n-fold) implicative filter is a filter ( fuzzy filter), but the converse is not true. Using a level set of a fuzzy set in a BL-algebra, we give a characterization of fuzzy n-fold implicative filters and establish the extension property for fuzzy n-fold implicative filters in BL-algebras. We also analyse some relationships with various n-fold and fuzzy n-fold filters in BL-algebras. All the above results are the natural generalization of the notion of filters and fuzzy filters (namely deductive and fuzzy deductive systems) in BL-algebras [], [], [], [], [].

Key Words and phrases : BL-algebra, filter, fuzzy filter, n-fold implicative filter, fuzzy n-fold implicative filter
AMS Classifications: 06G10, 03E72, 03B52

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Date received: February 14, 2008