The finite basis problem for matrix semirings over a two-element additively idempotent semiring

TL;DR

The paper solves the finite basis problem for matrix semirings over two-element ai-semirings by classifying when $\mathbf{M}_n(S)$ is finitely based for $n \ge 2$. It shows that this holds precisely when $S$ is not a distributive lattice, using Dolinka's results for the distributive case and constructing explicit equational bases for the non-distributive two-element ai-semirings, including detailed analyses for $\mathbf{M}_n(L_2)$, $\mathbf{M}_n(R_2)$, $\mathbf{M}_n(N_2)$, and $\mathbf{M}_n(T_2)$, as well as a dedicated treatment of $\mathbf{M}_n(M_2)$ via the finite base for a subsemiring $\mathbf{SR}_6$. The work integrates universal algebra techniques (varieties, free algebras, and derivability of identities) with concrete matrix-semifield constructions to deliver a complete dichotomy and clarifies how matrix-extensions interact with finite basis properties. The results advance understanding of how the finite basis property behaves under matrix construction and connect to prior classifications of ai-semirings and endomorphism semirings, offering a foundation for further exploration of matrix semirings over small bases.

Abstract

We provide a complete classification of matrix semirings $\mathbf{M}_n(S)$ over two-element additively idempotent semirings $S$ with respect to the finite basis property.Our main theorem shows that for every integer $n \geq 2$,the semiring $\mathbf{M}_n(S)$ is finitely based if and only if $S$ is distinct from a distributive lattice.

Figures (2)

• Figure 1: The additive order of $\mathbf{M}_2(M_2)$
• Figure 2: The additive order of $\mathbf{SR}_6$

Theorems & Definitions (24)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof