The finite basis problem for endomorphism semirings of finite semilattices with zero

If S is a join-semilattice with a distinguished least element, then all its endomorphisms form an additively idempotent semiring End(S); conversely, it is known that any additively idempotent semiring embeds into an endomorphism semiring of such kind. If vertical bar S vertical bar <= 2, then End(S) is readily seen to be finitely based. On the other hand, if S is finite and either contains the square of a two-element chain or is a chain with at least four elements, then End( S) is shown to be inherently nonfinitely based. This leaves only the case when S is a three-element chain as an open problem.