Algorithms and Computations in BL-Algebras

In this paper, the authors study the notion of n-fold implicative and fuzzy n-fold implicative filters, which correspond to various sets of provable formulaes, in BL-algebras. Several characterizations of fuzzy n-fold implicative filters are given and the authors analyse some relationships with various n-fold and fuzzy n-fold filters in BL-algebras, while constructing some algorithms for studying the foldness of filters in BL-algebras. The authors describe the relation between fuzzy n-fold implicative filters and fuzzy (n+1)-fold implicative filters and establish the extension property for fuzzy n-fold implicative filters in BL-algebras. DOI: 10.4018/jalr.2010100103 30 International Journal of Artificial Life Research, 1(4), 29-47, October-December 2010 Copyright © 2010, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. then we prove that every fuzzy n-fold implicative filter is a fuzzy filter, but the converse is not true. Thanks to a level set of a fuzzy set in a BL-algebra, we give a characterization of fuzzy n-fold implicative filters in BL-algebra. Furthermore, we describe the relation between fuzzy n-fold implicative filters and fuzzy (n+1)-fold implicative filters and we establish the extension property for fuzzy n-fold implicative filters in BL-algebras. The results of are used in the rest of the paper to analyse some relationships with various n-fold and fuzzy n-fold filters in BL-algebras. We obtain that every fuzzy n-fold positive implicative filter is also a fuzzy n-fold implicative filter. One of the difficult task while studying BL-algebras, is to construct some examples. Therefore, in the appendix of the paper, we give some algorithms for the computations in BL-algebras. All the results in this paper are the natural generalization of the notion of filters and fuzzy filters (namely deductive and fuzzy deductive systems) in BL-algebras (Zhan & Yang, 2008; Haveshki, Saied, & Eslami, 2006; Hajek, 1988; Dumitru & Dana, 2003; Jun & Miko, 2004; Liu & Li, 2005a; Liu & Li, 2005b).