arXiv subjects

Jinxing Zhao

Publications and source records attributed to Jinxing Zhao.

4 recordsLinked to original sources

arXiv2025

On automorphism groups of power semigroups over numerical semigroups or over numerical monoids

A numerical semigroup $S$ is a cofinite subsemigroup of $ \mathbb{N}$, where $\mathbb{N}$ is the additive monoid of non-negative integers. Denote by $\mathcal{P}_{\rm fin} (S)$ the semigroup consisting of all non-empty finite subsets of $S$ endowed with the operation of setwise addition defined by $$X+Y=\{x+y:x\in X, y\in Y\}, \qquad\text{for all } X, Y \in \mathcal P_\text{fin}(S).$$ We call $\mathcal{P}_{\rm fin} (S)$ the finitary power semigroup of $S$. When $0\in S$ (and hence $S$ is a numerical monoid), the family $\mathcal P_{\text{fin},0}(S)$ of all finite subsets of $S$ containing $0$ is a submonoind of $\mathcal P_\text{fin}(S)$; we call $\mathcal{P}_{{\rm fin}, 0}(S)$ the reduced finitary power monoid of $S$ with the singleton $\{0\}$ as zero-element. For a non-empty finite subset $X$ of $\mathbb{N}$, we denote by $ \min X$ and $\max X $ the minimum and the maximum in $X$. Tringali and Yan have recently proved in [J.\ Combin.\ Theory Ser.\ A 209 (2025)] that the only non-trivial automorphism of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$ is the involution $X \mapsto \max X - X$. By applying Tringali-Yan's result, we in this article determined the automorphism group of the finitary power semigroup $\mathcal{P}_{\rm fin}(S)$ of an arbitrary numerical semigroup $S$. More precisely, if $S$ is the set of all integers larger than or equal to a fixed $k \in \mathbb N$, then the only non-trivial automorphism of $\mathcal{P}_{\rm fin}(S)$ is the involution $X \mapsto \max X - X+ \min X$; otherwise, $\mathcal{P}_{\rm fin}(S)$ has only the identity automorphism.

math.GR↗

arXiv2025

On oriented $m$-semiregular representations of finite groups about valency three

Let $G$ be a group and $m$ a positive integer. We say an $m$-Cayley digraph $\Gamma$ over $G$ is a digraph that admits a group of automorphisms isomorphic to $G$ acting semiregularly on the vertex set with $m$ orbits. The digraph $\Sigma$ is $k$-regular if there exists a non-negative integer $k$ such that every vertex has out-valency and in-valency equal to $k$. All digraphs considered in this paper are regular. We say that $G$ admits an oriented $m$-semiregular representation (abbreviated as OmSR) if there exists a regular $m$-Cayley digraph $\Gamma$ over $G$ such that $\Gamma$ is oriented and its automorphism group is isomorphic to $G$. In particular, an O1SR is called an ORR. Xia et al. \cite{x2} provided a classification of finite simple groups admitting an ORR of valency 2. Furthermore, in 2022, Du et al. \cite{du2} proved that most finite simple groups admit an OmSR of valency 2 for $m \geq 2$, except for a few exceptional cases. In this paper, we classify the finite groups generated by at most two elements that admit an OmSR of valency 3 for $m \geq 2$.

math.GR↗

arXiv2025

Complete $k$-partite entanglement measure

The $k$-partite entanglement, which focus on at most how many particles in the global system are entangled but separable from other particles, is complementary to the $k$-entanglement that reflects how many splitted subsystems are entangled under partitions of the systems in characterizing multipartite entanglement. Very recently, the theory of the complete $k$-entanglement measure has been established in [Phys. Rev. A 110, 012405 (2024)]. Here we investigate whether we can define the complete measure of the $k$-partite entanglement. Consequently, with the same spirit as that of the complete $k$-entanglement measure, we present the axiomatic postulates that a complete $k$-partite entanglement measure should require. Furthermore, we present two classes of $k$-partite entanglement measures and show that one is complete while the other one is unified but not complete except for the case of $k=2$.

quant-ph↗

arXiv2025

Finite groups admitting a regular tournament $m$-semiregular representation

For a positive integer $m$, a finite group $G$ is said to admit a tournament $m$-semiregular representation (TmSR for short) if there exists a tournament $\Gamma$ such that the automorphism group of $\Gamma$ is isomorphic to $G$ and acts semiregularly on the vertex set of $\Gamma$ with $m$ orbits. Clearly, every finite group of even order does not admit a TmSR for any positive integer $m$, and T1SR is the well-known tournament regular representation (TRR for short). In 1986, Godsil \cite{god} proved, by a probabilistic approach, that the only finite groups of odd order without a TRR are $\mathbb{Z}_3^2$ and $\mathbb{Z}_3^3$ . More recently, Du \cite{du} proved that every finite group of odd order has a TmSR for every $m \geq 2$. The author of \cite{du} observed that a finite group of odd order has no regular TmSR when $m$ is an even integer, a group of order $1$ has no regular T3SR, and $\mathbb{Z}_3^2$ admits a regular T3SR. At the end of \cite{du}, Du proposed the following problem. \noindent{\sf\it Problem.} \ \ {\it For every odd integer $m\geq 3$, classify finite groups of odd order which have a regular TmSR.} The motivation of this paper is to give an answer for the above problem. We proved that if $G$ is a finite group with odd order $n>1$, then $G$ admits a regular TmSR for any odd integer $m\geq 3$.

math.GR↗