The Interval $[\mathsf{V}(S_7),\mathsf{V}(B_2^1)]$ of Semiring Varieties Has the Cardinality of the Continuum

TL;DR

This paper resolves the size of the interval $[\mathsf{V}(S_7),\mathsf{V}(B_2^1)]$ within additively idempotent semiring (ai-semiring) varieties by showing it has the cardinality $2^{\aleph_0}$. The authors employ flat ai-semirings $S(\mathbf{w}_n)$ derived from words that incorporate Zimin words, and establish that the corresponding isoterms $\mathbf{w}_n$ remain valid for $B_2^1$, enabling a continuum of distinct subvarieties between $\mathsf{V}(S_7)$ and $\mathsf{V}(B_2^1)$. A standard embedding criterion for lattices of subvarieties is invoked to convert these isoterms into $2^{\aleph_0}$ distinct varieties, with parallel implications for semigroup varieties via $\mathsf{V}_s$. The result deepens understanding of the complexity of ai-semiring variety lattices and connects to related continuum intervals in adjacent frameworks, while indicating future directions on finitely based finite ai-semirings with similar behavior.

Abstract

We prove that the interval $[\mathsf{V}(S_7),\mathsf{V}(B_2^1)]$ in the lattice of additively idempotent semiring (ai-semiring) varieties has the cardinality of the continuum,where $S_7$ is the smallest nonfinitely based ai-semiring (a three-element algebra), and $B_2^1$ is the ai-semiring whose multiplicative reduct is the six-element Brandt monoid.

Figures (1)

• Figure 1: The additive order of $B_2^1$

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Proposition 1.3
• proof
• Lemma 2.1: rjzl
• Proposition 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof