On subsemigroup lattices without non-trivial identities (Q1319057)

The author proves that the subsemigroup lattice \(\text{Sub }S\) of a commutative semigroup \(S\) does not satisfy a non-trivial identity iff \(\text{Sub }S\) is discriminating (a lattice \(L\) is called discriminating if for any finite number of elements \(u_ 1,\dots,u_ n\in F\), there exists a homomorphism \(f: F\mapsto L\) such that \(f(u_ i)\neq f(u_ j)\) whenever \(u_ i\neq u_ j\) \((1\leq i,j\leq n)\). In particular, in this case every finite projective lattice can be embedded into \(\text{Sub }S\).