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Zidong Gao

Publications and source records attributed to Zidong Gao.

10 recordsLinked to original sources
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 nonfinitely based additively idempotent semiring that is not strongly nonfinitely based and generates a limit variety

We present an explicit infinite equational basis for the six-element additively idempotent semiring $TR_6$ and prove that $TR_6$ is nonfinitely based. We also give a complete description of the subvariety lattice of the variety generated by $TR_6$, showing that it forms a four-element chain. Our results demonstrate that the variety generated by $TR_6$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. This provides a new limit variety of additively idempotent semirings, distinct from all previously known ones. In fact, $\mathsf{V}(TR_6)$ is the first explicit limit subvariety of the variety generated by the max-plus algebra $\mathbf{N}$. Moreover, $TR_6$ is not strongly nonfinitely based: it belongs to a finitely based variety generated by a finite additively idempotent semiring. Together with the six-element additively idempotent semiring $SR_6$, these are the first two finite additively idempotent semirings that are nonfinitely based but not strongly nonfinitely based. Finally, we study the variety generated by $SR_6$ and $TR_6$, showing that it is nonfinitely based and has exactly nine subvarieties, four of which are nonfinitely based and the remaining five are finitely based.

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

The finite basis problem for the flat semirings $S(W)$

We focus on the finite basis problem for flat semirings of the form $S(W)$, where $W$ is an arbitrary set of nonempty words. We prove that $S(W)$ generates a Cross variety (and hence is finitely based) whenever every word in $W$ has length at most $3$, whereas it is nonfinitely based whenever there exists $k \geq 3$ such that $W$ is $x^{k+2}$-free but not $x^{k+1}$-free. In particular, if $W_k$ denotes the set of all words of length $k$, then $S(W_k)$ is finitely based if and only if $k \leq 3$. Moreover, $S(W)$ is nonfinitely based whenever $W$ is finite and not $x^4$-free. These results provide a partial answer to an open problem raised by Jackson et al.~(J Algebra 611: 211--245, 2022).

math.CO↗
arXiv2026

Ultra-Large-Capacity Passive Quantum Access Network Powered By Single Thermal Source

Quantum Key Distribution (QKD) provides secure keys for classical communications through one-time-pad (OTP) encryption with physical-law security. Advanced PON-based Classical Access Networks (CANs) support up to 256 users with a total rate of 10 Gbps (10-Gbps @ 256-users). The equivalent rate demand of OTP encryption requires QKD Access Networks (QANs) to reach comparable performance, yet state-of-the-art PON-based QANs remain far from this standard. To address this gap, we propose a passive Thermal-State QAN (TS-QAN) distributing polychromatic quantum randomness from a single thermal source and supporting 304 users with an aggregate secret key rate (SKR) of 13 Gbps (13-Gbps @ 304-users). This performance is enabled by three features. First, broadband thermal states with Bose-Einstein statistics can be represented, through the Glauber-Sudarshan representation, as high-bandwidth Gaussian coherent-state ensembles across frequency modes, eliminating many active modulators and quantum random number generators (QRNGs). Second, Electro-Optic (EO) comb beacons provide time-varying polychromatic phase tracking, so each frequency-mode thermal signal can be coherently measured with a Local Local Oscillator (LLO) aided by its beacon, without large-scale phase-locking networks. Third, state broadcasting allows each user to obtain independent final keys via reverse reconciliation after accounting for residual broadcast-induced correlations, expanding network capacity with small SKR losses. Experimentally, we verify a 13-Gbps @ 304-users TS-QAN using Continuous-Variable QKD (CV-QKD) under covariance-matrix-based network security analysis including multimode Holevo leakage and broadcast correlations. This work meets the SKR and capacity demands from CAN to QAN: 13-Gbps @ 304-users satisfies the 10-Gbps @ 256-users benchmark and provides a scalable solution for modern telecommunication systems.

quant-ph↗
arXiv2026

The flat semirings with nilpotent multiplicative reducts

In this paper, we focus on the variety $\mathbf{NF}_3$ generated by all flat semirings with $3$-nilpotent multiplicative reducts. By introducing graph semirings, we characterize all subdirectly irreducible members of $\mathbf{NF}_3$. We prove that the variety $\mathbf{NF}_3$ has uncountably many subvarieties and show that every finitely generated subvariety of $\mathbf{NF}_3$ is a Cross variety. Moreover, we demonstrate that $\mathbf{NF}_3$ has a unique limit subvariety, which is generated by all acyclic graph semirings.

math.GR↗
arXiv2026

A finitely based finite semiring generates a variety with continuum many subvarieties

This paper establishes the existence of a finitely based finite semiring whose variety contains a continuum of subvarieties; such a variety is said to be of type \(2^{\aleph_0}\). Using the homomorphism theory of Kneser graphs, we prove that the 3-element semiring \(S_{53}\) is the first known example with this property. Moreover, \(S_{53}\) belongs to the variety of the max-plus semiring \((\mathbb{N},\max,+)\), which therefore is also of type \(2^{\aleph_0}\). For the finitely based 4-element semiring \(B_0\), we demonstrate that its variety contains infinitely many subvarieties and suggest that \(B_0\) could be another potential example of type \(2^{\aleph_0}\).

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

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

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.

math.GR↗