The finite basis problem for the endomorphism semirings of finite semilattices

Abstract

For every semilattice $\mathcal{A}=(A,+)$, the set $\mathrm{End}(\mathcal{A})$ of its endomorphisms forms a semiring under pointwise addition and composition. We prove that that if $\mathcal{A}$ is finite, then the endomorphism semiring $\mathrm{End}(\mathcal{A})$ has a finite identity basis if and only if $|A|\le 2$.