The finite basis problem for additively idempotent semirings that relate to S_7

TL;DR

The paper analyzes the finite basis problem for additively idempotent semirings related to $S_7$, proving a hypergraph-based sufficient condition for nonfinite baseness and applying it to varieties containing $S_7$. It then classifies the subvarieties of $\\mathsf{V}(S_7)$, showing only six are finitely based and that the rest are nonfinitely based, while establishing that the interval structure above $S_c(a)\dots$ contains continuum many subvarieties, i.e., type $2^{\aleph_0}$. A key contribution is the block hypergraph framework, which ties algebraic identities to hypergraph homomorphisms, enabling a detailed description of subdirectly irreducible members and the global subvariety lattice. These results significantly advance understanding of finiteness properties in ai-semirings and illuminate the rich combinatorial structure governing their varieties.

Abstract

The $3$-element additively idempotent semiring $S_7$ is a nonnitely based algebra of the smallest possible order. In this paper we study the nite basis problem for some additively idempotent semirings that relate to $S_7$. We present a su cient condition under which an additively idempotent semiring variety is nonnitely based and as applications, show that some additively idempotent semiring varieties that contain $S_7$ are also nonnitely based. We then consider the subdirectly irreducible members of the variety $\mathsf{V}(S_7)$ generated by $S_7$. We show that $\mathsf{V}(S_7)$ contains exactly $6$ finitely based subvarieties, all of which sit at the base of the subvariety lattice, then invoke results from the homomorphism theory of Kneser graphs to verify that $\mathsf{V}(S_7)$ contains a continuum of subvarieties.

Figures (1)

• Figure 1: The subvariety lattice of $\mathsf{V}(S_7)$.

Theorems & Definitions (81)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Example 2.1
• Theorem 2.2
• proof
• Remark 2.3
• Proposition 2.4
• proof
• Corollary 2.5