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Mengya Yue

Publications and source records attributed to Mengya Yue.

12 recordsLinked to original sources
arXiv2026

The finite basis problem for all subvarieties of the variety $\mathsf{V}(S^0_7)$

We solve the finite basis problem for all subvarieties of $\mathsf{V}(S^0_7)$. By considering their intersections with $\mathsf{V}(S_7)$, we reduce the problem to six intervals. We show that $\mathsf{V}(S_7^0)$ has exactly fifteen finitely based subvarieties and that every other subvariety is nonfinitely based. Moreover, $\mathsf{V}(S_c(abc))$ and $\mathsf{V}(SR_6)$ are the only limit subvarieties of $\mathsf{V}(S_7^0)$.

math.GR↗
arXiv2026

The unique limit subvariety of the max-plus variety and finite basis properties

We prove that the variety generated by the max-plus algebra on the nonnegative integers has a unique limit subvariety, namely the variety generated by the six-element semiring constructed by Shao, Ren and Gao. More generally, we characterize hereditary finite basedness in a finitely defined variety of commutative additively idempotent semirings containing the max-plus variety. Its hereditarily finitely based subvarieties form a finite lattice and admit equational bases involving at most ten variables. We also prove that every proper subvariety of the max-plus variety is locally finite. A finite basis theorem for finite semirings with a multiplicative identity establishes finite basedness of all finite truncations. Finally, we determine the finite basis properties of two families of subvarieties defined by power identities and of subvarieties relatively defined by arbitrary families of identities making individual additive subterms greatest elements.

math.GR↗
arXiv2026

Explicit equational bases for the power semirings of $S_7$

For every semigroup $S$, the set $\mathcal{P}(S)$ of all subsets of $S$ and the set $\mathcal{P}^{+}(S)$ of all nonempty subsets of $S$ form additively idempotent semirings under set-theoretic union and elementwise multiplication, called the full and nonempty power semirings of $S$, respectively. We investigate the finite basis problem for the full and nonempty power semirings $\mathcal{P}(S_7)$ and $\mathcal{P}^{+}(S_7)$ of the multiplicative reduct of $S_7$, where $S_7$ is the unique nonfinitely based three-element additively idempotent semiring. We provide explicit infinite equational bases for both and prove that they are nonfinitely based. For $\mathcal{P}^{+}(S_7)$, we establish a new sufficient condition for an additively idempotent semiring to be nonfinitely based and apply it to obtain the required result. Moreover, we show that the interval $[\mathsf{V}(\mathcal{P}^{+}(S_7)), \mathsf{V}(\mathcal{P}(S_7))]$ in the lattice of additively idempotent semiring varieties has the cardinality of the continuum.

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

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

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↗