About Some Properties of Semilattices

Semilattices are, in fact, idempotent mono-semirings. The article shows that elements of any semilattice are separated by its prime ideals. It gives a multiplicative representation of semirings with semilattice multiplication. This implies that any semilattice is isomorphic to some lattice of sets of its prime spectrum. It is shown that finite chains are retracts in all semilattices containing them. Corollaries are obtained. Additional remarks have been made.