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 $|A|\le 2$.

15 pages, 7 figures. Changes in version 2 include fixing typos, removing two incorrect claims, adding a new reference, and other minor improvements