The finite basis problem for the endomorphism semirings of finite semilattices

TL;DR

This work resolves the finite basis problem for endomorphism semirings of finite semilattices by showing that End($\mathcal{A}$) is finitely based if and only if $|\mathcal{A}|\le 2$, with stronger nonfinite basability arising in larger semilattices. The authors combine three methods—inherent, strong, and containment in a $\mathcal{B}_2^1$-variety—using embeddings of End($\mathcal{C}_3$) and rook semirings to establish nonfinite basability, and to connect endomorphism semirings to universal properties of ai-semi-rings. The results reveal a sharp contrast with finite rings, highlighting complex identity behavior in ai-semirings and advancing the broader finite basis program. The paper also outlines directions for future work on chains and specific six-element ai-semi-rings, contributing to a deeper understanding of how structural parameters like size and height govern finite basis properties.

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 $|A|\le 2$.

Figures (7)

• Figure 1: The only compatible upper semilattice order on $B_2^1$
• Figure 2: A compatible upper semilattice order on $A_2^1$
• Figure 3: Representing the elements of ${A}^1_2$ by the endomorphisms of the chain $0<1<2$ that fix 2
• Figure 4: Another compatible upper semilattice order on $A_2^1$
• Figure 5: Representing the elements of ${A}^1_2$ by the endomorphisms of the chain $0<1<2$ that fix 0

Theorems & Definitions (24)

• Proposition 1: Dolinka09a
• Proposition 2: JackRenZhao22
• Proposition 3: GusVol23b
• Proposition 4: GusVol23b
• Theorem 1
• Claim 1
• Claim 2
• Claim 3
• Claim 4
• proof : Proof of Claim \ref{['claim:if']}