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Two nonfinitely based additively idempotent semirings of order four

Mengya Yue, Miaomiao Ren, Zidong Gao

Abstract

We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific $4$-element additively idempotent semirings, $S_{(4,545)}$ and $S_{(4,634)}$, whose additive reducts are chains, have no finite basis for their identities. Furthermore, we show that the interval $[\mathsf{V}(S_{(4,545)}),\mathsf{V}(S_{(4,634)})]$ in the lattice of semiring varieties contains \(2^{\aleph_0}\) distinct varieties. Consequently, the join of two finitely based additively idempotent semiring varieties is not necessarily finitely based. Moreover, we obtain the smallest example of a finitely based additively idempotent semiring $S$ whose extension $S^0$ (obtained by adjoining a new element) is nonfinitely based.

Two nonfinitely based additively idempotent semirings of order four

Abstract

We establish two sufficient conditions for an additively idempotent semiring to be nonfinitely based. As applications, we prove that two specific -element additively idempotent semirings, and , whose additive reducts are chains, have no finite basis for their identities. Furthermore, we show that the interval in the lattice of semiring varieties contains distinct varieties. Consequently, the join of two finitely based additively idempotent semiring varieties is not necessarily finitely based. Moreover, we obtain the smallest example of a finitely based additively idempotent semiring whose extension (obtained by adjoining a new element) is nonfinitely based.
Paper Structure (6 sections, 24 theorems, 80 equations, 6 tables)

This paper contains 6 sections, 24 theorems, 80 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equational basis for $S_{(4,545)}$
  4. Equational basis for $S_{(4, 634)}$
  5. The interval $[\mathsf{V}(S_{(4, 545)}), \mathsf{V}(S_{(4, 634)})]$
  6. Conclusion

Key Result

Lemma 2.1

Let $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$ be terms. If $\mathbf{v}$ is $\mathbf{u}$-free and $\mathbf{u}$ is a subterm of $\mathbf{w}$, then $\mathbf{v}$ is also $\mathbf{w}$-free.

Theorems & Definitions (71)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Claim 3.1
  • proof : Proof of Claim $\ref{['claim01']}$.
  • ...and 61 more