The finite basis problem for additively idempotent semirings of order four, II

TL;DR

This work advances the finite basis problem for additively idempotent semirings of order four by exhaustively classifying many cases where the additive reduct is a quasi-antichain. It combines equational reasoning, subdirect product decompositions, dual multiplications, and known bases from related small algebras to establish finite basing for 151 of the $866$ ai-semirings, while isolating the unique nonfinitely based instance $S_{(4,435)}$. The authors provide explicit equational bases for numerous algebras tied to $S_{57}$, $S_{58}$, $S_{59}$, $S_{60}$ and $S^0$, along with detailed arguments linking these algebras to smaller, well-understood varieties. The results significantly advance understanding of the landscape of four-element ai-semirings and highlight the need for new techniques to finish the remaining cases, with potential implications for computational algebra and universal algebraic theory of idempotent semirings.

Abstract

We study the finite basis problem for $4$-element additively idempotent semirings whose additive reducts are quasi-antichains. Up to isomorphism, there are $93$ such algebras. We show that with the exception of the semiring $S_{(4, 435)}$, all of them are finitely based.

Figures (1)

• Figure 1: The additive order of $S_{(4, k)}$, $388\leq k \leq 480$

Theorems & Definitions (113)

• Theorem 1.1
• Lemma 1.2
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof