Anti Fuzzy Deductive Systems of BL-Algebras

Anti Fuzzy Deductive Systems of BL-Algebras

Cyrille Nganteu Tchikapa (Department of Mathematics, GHS of Batchenga, Batchenga, Cameroon)
Copyright: © 2012 |Pages: 9
DOI: 10.4018/jalr.2012070103
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The aim of this paper is to introduce the notion of anti fuzzy (prime) deductive system in BL-algebra and to investigate their properties. It is shown that the set of all deductive systems (with the empty set) of a BL-algebra X is equipotent to a quotient of the set of all anti fuzzy deductive systems of X. The anti fuzzy prime deductive system theorem of BL-algebras is also proved.
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2. Preliminaries

We recollect some definitions and results which will be used in the following and weshall not cite them every time they are used.

  • Definition 2.1: A BL-algebra is an algebra jalr.2012070103.m09of type (2; 2; 2; 2; 0; 0) that satisfies the following conditions:

  • BL-1: jalr.2012070103.m10is a bounded lattice;

  • BL-2:jalr.2012070103.m11is an commutative monoid, i.e., jalr.2012070103.m12 is commutative and associative with jalr.2012070103.m13;

  • BL-3: jalr.2012070103.m14iff jalr.2012070103.m15 (Residuation);

  • BL-4: jalr.2012070103.m16 (Divisibility);

  • BL-5: jalr.2012070103.m17 (Prelinearity).

  • Example 2.1:

    • 1.

      Let X be a nonempty set and let P(X) be the family of all subsets of X. Define operations jalr.2012070103.m18 and jalr.2012070103.m19by: jalr.2012070103.m20and jalr.2012070103.m21for all jalr.2012070103.m22respectively. Then jalr.2012070103.m23is a BL-algebra called the power BL-algebra of X.

    • 2.

      jalr.2012070103.m24where jalr.2012070103.m25 is the residuum of a continuous t-norm jalr.2012070103.m26 is a BL-algebra.

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