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