The finite basis problem for endomorphism semirings of finite chains

For every semilattice [Formula: see text], the set [Formula: see text] of its endomorphisms forms a semiring under point-wise addition and composition. We prove that the semiring of all endomorphisms of the three-element chain has no finite identity basis. This, combined with earlier results by Dolinka [The finite basis problem for endomorphism semirings of finite semilattices with zero, Algebra Universalis 61 (2009) 441–448] gives a complete solution to the finite basis problem for semirings of the form [Formula: see text] where [Formula: see text] is a finite chain.