What is the difference between fuzzy set and L-fuzzy set?

A fuzzy set (class) A in X is characterized by a membership (characteristic)

function fA : X--> [0,1] which associates with each point in X a real

number in the interval [0, 1], with the value of fA(x) at x representing

the "grade of membership" of x in A. Thus, the nearer the value of

fA(x) to unity, the higher the grade of membership of x in A. When A

is a set in the ordinary sense of the term, its membership function can

take only two values 0 and 1, with fA(x) = 1 or 0 according as x

does or does not belong to A. Thus, in this case fA(x) reduces to the

familiar Characteristic function of a set A. (When there is a need to

differentiate between such sets and fuzzy sets, the sets with two-valued

characteristic functions will be referred to as ordinary sets or simply sets. )

On the other hand , an L-fuzzy set A in X is characterized by the membership function fA :L--> L , where L is a complete lattice with an involutive order preserving operation N : L--> L.