Bisemilattice-valued fuzzy sets

Abstract

Bisemilattice-valued fuzzy set (B-fuzzy set) is defined to be a mapping from a nonempty set to bisemilattice (an algebraic structure with two binary operations, both commutative, associative and idempotent). To each B-fuzzy set corresponds two collections of subsets, levels (cuts) of that fuzzy set. These two families are determined by two ordering relations on a bisemilattice. Theorems of decomposition of B-fuzzy sets into levels are formulated and proved in the paper. Starting from two families of subsets of a set X, we give conditions for synthesis of a B-buzzy set. Some other properties of B-fuzzy sets are also formulated.