On subsemigroup lattices without non-trivial identities

Abstract

A latticeL is called discriminating if for any free latticeF and for any finite number of elementsu1,u2, ...,un∈F, there exists a homomorphismf:F→L such thatf(ui)≠f(uj) wheneverui ≠ uj (1≤i, j≤n). In this paper it is proved that the subsemigroup lattice SubS of a commutative semigroupS does not satisfy a non-trivial identity if and only if SubS is discriminating. In particular, in this case every finite projective lattice can be embedded into SubS. It should be noted that the most important examples of semigroups whose subsemigroup lattices satisfy no non-trivial identity and therefore have the discriminating property are the following: the infinite cyclic semigroup, the free semilattice of countable rank, any commutative nilsemigroup which is not nilpotent and so on.