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Mengya Yue

Publications and source records attributed to Mengya Yue.

8 recordsLinked to original sources
arXiv2026

The finite basis problem for the power semirings of finite groups

For any group $G$, the set of all nonempty subsets of $G$ forms an additively idempotent semiring under set-theoretic union and elementwise multiplication, called the power semiring of $G$ and denoted by $\mathcal{P}(G)$. We prove that for a finite group $G$, $\mathcal{P}(G)$ has no finite basis for its identities if and only if $|G| \geq 3$. This completes the classification of the power semirings of finite groups with respect to the finite basis property.

math.GR↗
arXiv2026

A nonfinitely based additively idempotent semiring of order four

We first establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. As applications, we exhibit several examples of additively idempotent semirings satisfying this condition, including a $4$-element semiring $S_{(4,124)}$ whose additive reduct has two minimal elements and two coatoms. Consequently, these semirings have no finite basis for their identities.

math.GR↗
arXiv2026

A new limit variety of additively idempotent semirings

We establish a sufficient condition for an additively idempotent semiring to be nonfinitely based. Applying this condition, we prove that the six-element additively idempotent semiring $SR_6$ has no finite basis for its identity. Furthermore, we provide a complete description of the subvariety lattice of the variety $\mathsf{V}(SR_6)$ generated by $SR_6$, showing that it forms a four-element chain. Our results demonstrate that $\mathsf{V}(SR_6)$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. Moreover, $SR_6$ is the smallest known example of an additively idempotent semiring generating a limit variety.

math.RA↗
arXiv2026

Two nonfinitely based additively idempotent semirings of order four

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.

math.RA↗
arXiv2025

Every additively idempotent semiring satisfying $xy\approx xz$ is finitely based

We study the finite basis problem for additively idempotent semirings satisfying the identity $xy \approx xz$. Let $\mathbf{R}$ denote the variety of all such semirings. Yue et al. (2025, Algebra Universalis, DOI:10.1007/s00012-025-00908-5) established that $\mathbf{R}$ is finitely generated. In this paper, we show that the subvariety lattice of $\mathbf{R}$ forms a distributive lattice of order $10$. As a consequence, the variety $\mathbf{R}$ is a Cross variety, and every member of $\mathbf{R}$ is finitely based.

math.GR↗