Bisemilattice-valued fuzzy set ($B$-fuzzy set) is defined to be a mapping from a nonempty set to a bisemilattice (an algebraic structure with two binary operations, both commutative, associative and idem-potent). 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 $B$-fuzzy set. Some other properties of $B$-fuzzy sets are also formulated.