The finite basis problem for the endomorphism semirings of finite semilattices

TL;DR

This paper investigates the algebraic structure of endomorphism semirings of finite semilattices, establishing that only those with at most two elements have a finite basis, thus characterizing their algebraic complexity.

Contribution

It provides a complete characterization of when the endomorphism semiring of a finite semilattice has a finite identity basis, revealing a sharp cutoff at two elements.

Findings

Endomorphism semirings of finite semilattices with more than two elements lack a finite basis.

Finite semilattices with one or two elements have endomorphism semirings with finite bases.

The result delineates the boundary between finite and infinite basis properties in this algebraic context.

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 $|A|\le 2$.