Anti Fuzzy Deductive Systems of BL-Algebras

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 of type (2; 2; 2; 2; 0; 0) that satisfies the following conditions:

• BL-1: is a bounded lattice;

• BL-2:is an commutative monoid, i.e., is commutative and associative with ;

• BL-3: iff (Residuation);

• BL-4: (Divisibility);

• BL-5: (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 and by: and for all respectively. Then is a BL-algebra called the power BL-algebra of X.

  • 2.

    where is the residuum of a continuous t-norm is a BL-algebra.