Tensor products of semilattices and fuzzy ideals

We study (complete) fuzzy ideals on semilattices from the point of view of tensor products of semilattices. We show that the lattice of all complete fuzzy ideals on a semilattice is an extension of tensor product. We define the notion of semicomplete bi-ideals, and show that the lattice of all fuzzy ideals on a distributive lattice is isomorphic to the lattice of all semicomplete bi-ideals. Moreover, we show that the lattice of all (complete) fuzzy ideals on a distributive (algebraic) lattice is distributive.