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21
2025/09
Power monoids and their system of length sets
Let $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$ be the family of all non-empty finite subsets of $\mathbb{N}$ that contain $0$, where $\mathbb{N}$ is the set of non-negative integers. Equipped with the binary operation of set addition\[(A,B) \mapsto A+B := \{a+b: a \in A, \, b \in B \},\] this family forms a commutative monoid, called the reduced finitary power monoid of $\mathbb{N}$.The identity element is the singleton $\{0\}$. A non-identity element $A \in \mathcal{P}_{{\rm fin},0}(\mathbb{N})$ is called an atom if it cannot be expressed as the sum of two elements of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$, both distinct from $\{0\}$. For each $B \in \mathcal{P}_{{\rm fin},0}(\mathbb{N})$, we denote by $\mathsf{L}(B)$ the \textsf{length set} of $B$, that is, the set of all integers $n \ge 0$ such that $B$ can be written as the sum of $n$ atoms of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$. The collection\[\mathcal{L}(\mathcal{P}_{{\rm fin},0}(\mathbb{N})) := \{\mathsf{L}(B): B \in \mathcal{P}_{{\rm fin},0}(\mathbb{N}) \}\] is called the system of length sets of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$.It is not difficult to verify that every length set of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$, except for $\{0\}$ and $\{1\}$, is a non-empty finite subset $L$ of $\mathbb{N}$ whose minimum is larger than or equal to $2$. Fan and Tringali (2018) conjectured that the converse also holds, namely, every such set $L$ is indeed a length set of $\mathcal{P}_{{\rm fin},0}(\mathbb{N})$.In this talk, we present several results supporting this conjecture. In particular, we show that $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ is \textsf{fully elastic}, that is, for every rational number $r \ge 1$, there exists a set $C \in \mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$, distinct from $\{0\}$, such that $\max \mathsf{L}(C) = r \, \min \mathsf{L}(C)$.
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21
2025/09
On power monoids and their automorphisms
Let $H$ be a multiplicatively written monoid. The family $\mathcal{P}_{{\rm fin},1}(H)$ of non-empty finite subsets of $H$ containing the identity, endowed with the binary operation of setwise multiplication $(X,Y) \mapsto \{xy: x \in X, y \in Y\}$ induced by $H$, is called the reduced finitary power monoid of $H$. Recently, Tringali and Yan initiated the investigation of the automorphism group of these objects and showed that the reduced finitary power monoid of the monoid $\mathbb N_0$ of non-negative integers under addition has precisely two automorphisms, the identity and the so-called reversion map. The existence of the latter is interesting in the sense that it is not the canonical extension of an automorphism of the base monoid $\mathbb{N}_0$.In this talk, we give a complete description of the automorphism group of $\mathcal{P}_{{\rm fin},1}(H)$, where $H$ is either a finite abelian group or a submonoid of the additive group of rational numbers. More precisely, we show that there is a canonical isomorphism between the automorphism group of $\mathcal{P}_{{\rm fin},1}(H)$ and the automorphism group of $H$, except in certain special cases.
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21
2025/09
Isomorphism problems for ideals of numerical semigroups
Let $S$ be a numerical semigroup, that is, a cofinite submonoid of the non-negative integers under addition. A non-empty set of integers $I$ is said to be an ideal of $S$ if $I+S\subseteq I$ and $I$ has a minimum. The sum of two ideals $I$ and $J$, defined as $I+J=\{i+j : i\in I, j\in J\}$, is also an ideal of $S$. Thus, the set of ideals of $S$, denoted $\mathfrak{I}(S)$, is a commutative monoid under this operation, with neutral element $S$. If $S$ and $T$ are numerical semigroups and $\mathfrak{I}(S)$ is isomorphic to $\mathfrak{I}(T)$, then $S$ and $T$ must be the same numerical semigroup.If $I$ and $J$ are ideals of $S$, we write $I\sim J$ if $I=z+J$ for some $z \in \mathbb Z$. The ideal class monoid of $S$ is defined as the set of ideals of $S$ modulo this relation, where addition of two classes $[I]$ and $[J]$ is defined as $[I]+[J] = [I+J]$.An ideal $I$ is said to be normalized if $\min(I) = 0$. The set of normalized ideals of $S$, denoted by $\mathfrak{I}_0(S)$, is a monoid isomorphic to the ideal class monoid of $S$ \cite{icm}. It is known that if $S$ and $T$ are numerical semigroups for which $\mathfrak{I}_0(S)$ is isomorphic to $\mathfrak{I}_0(T)$, then $S$ and $T$ must be equal.The set $\mathfrak{I}_0(S)$ becomes a poset under inclusion. In \cite{iso-icm}, we also prove that if $S$ and $T$ are numerical semigroups such that the poset $(\mathfrak{I}_0(S),\subseteq)$ is isomorphic to the poset $(\mathfrak{I}_0(T),\subseteq)$, then $S = T$.On $\mathfrak{I}_0(S)$ we can define a partial order $\preceq$ as $I\preceq J$ if there exists $K \in \mathfrak{I}_0(S)$ such that $I+K = J$. We know that if $S$ and $T$ are numerical semigroups with multiplicity three such that the poset $(\mathfrak{I}_0(S),\preceq)$ is isomorphic to the poset $(\mathfrak{I}_0(T),\preceq)$, then $S = T$. However, if we remove the condition on the multiplicity, this isomorphism problem is still open.In recent work with Bonzio, we study the case when the poset $(\mathfrak{I}_0(S),\preceq)$ is a lattice. We show that this is the case if and only if the multiplicity of $S$ is at most four.During the talk, will give an overview of these recent results and present some open problems.
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21
2025/09
Power Monoids in a New Framework for Factorization
Let $H$ be any multiplicative monoid, and consider the collection of all its finite subsets that contain the identity element. This collection forms a monoid under setwise multiplication, known as the reduced power monoid of $H$ and denoted by $\mathcal P_{{\rm fin},1}(H)$. Since $\mathcal P_{{\rm fin},1}(H)$ is non-cancellative whenever $H$ is nontrivial, it serves as a central example in the study of a newly developed general theory of factorization. This theory, recently introduced by Cossu and Tringali, investigates decompositions into (almost) arbitrary factors within monoids that may admit nontrivial idempotents. Within this framework, we focus on minimal factorizations into irreducible elements in reduced power monoids. Among other results, we will discuss necessary and sufficient conditions on $H$ under which $\mathcal P_{{\rm fin},1}(H)$ admits unique minimal factorizations.
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21
2025/09
Maximal Common Divisors in Power Monoids
A commutative monoid or domain is said to possess the MCD property if every non-empty finite subset admits a maximal common divisor (MCD). It is well known that monoids satisfying the ascending chain condition on principal ideals (ACCP) necessarily have the MCD property. Although atomicity does not, in general, ascend to polynomial domains, it does ascend when restricted to the class of MCD domains (Roitman, 1993). The behavior of atomicity with respect to power monoids parallels this phenomenon, and we will discuss this in more detail during the first part of the talk.On the other hand, we say that a commutative monoid has the MCD-finite property if every non-empty finite subset admits only finitely many MCDs (up to associates). Similar to atomicity, the IDF property does not, in general, ascend to polynomial domains (Malcolmson-Okoh, 2009). Nevertheless, the IDF property does ascend to polynomial domains when restricted to the class of MCD-finite domains (Eftekhari-Khorsandi, 2018). In the second part of the talk, we will examine the ascent of the IDF property to power monoids.
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21
2025/09
Recent trends in the theory of power semigroups
Let $S$ be a multiplicatively written semigroup. Endowed with the operation of setwise multiplication induced from $S$ and defined by $(X, Y) \mapsto \{xy \colon x \in X,\, y \in Y\}$, the family of non-empty subsets of $S$ forms a semigroup in its own right, hereafter denoted by $\mathcal{P}(S)$.The term "power semigroup" is generically used for various subsemigroups of $\mathcal{P}(S)$ that, in a certain vague sense, lie between $S$ and $\mathcal{P}(S)$ itself. I will review recent progress on power semigroups and highlight some questions and open problems that are currently guiding the evolution of the theory.
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20
2025/09
第五届“文学与教育跨学科研究”学术研讨会 ——暨（中国）中外语言文化比较学会文学教育研究专业委员会2025年年会
人工智能浪潮袭来，不仅深刻影响着新时代文学教育的功能与担当，而且改变了文学教育的跨学科研究视角与文学教育范式。面对技术革新，我们亟需探索AI技术与文学教育互动的多种可能性，重新审视文学与教育跨学科研究的机遇和挑战。《外国文学研究》编辑部与(中国)中外语言文化比较学会文学教育研究专业委员会、河北师范大学外国语学院，拟于2025年9月19-21日在石家庄举办第五届“文学与教育跨学科研究”学术研讨会。本次会议的主题是“AI时代的文学教育跨学科范式研究”，旨在凝聚全国学界智慧，共同探讨AI时代文学与教育跨学科研究的新范式和新视野，助力建构AI时代中国的文学教育理论与实践的自主知识体系。
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22
2025/09
Neural combinatorial optimization for practical vehicle routing problems: Advances and challenges
Neural combinatorial optimization (NCO) has emerged as a promising paradigm for solving combinatorial optimization problems by leveraging the power of deep learning, which offers a data-driven, and adaptive approach that learns from problem instances and demonstrates strong generalization capabilities for effectively handling a wide range of problem instances. This talk focuses on the applications of NCO in solving practical vehicle routing problems (VRPs), which are fundamental in logistics and transportation. Several representative NCO models for VRPs will be reviewed firstly. I will then highlight how we developed NCO models to address VRPs with complex real-world constraints, such as time-dependent travel speeds and uncertain parking availability, which are challenging for traditional techniques. The results of performance comparisons between our NCO models and several benchmark models will be presented. Finally, significant challenges in applying NCO to practical VRPs will be discussed.
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22
2025/09
中国北方风沙活动的近期变化
中国北方风沙活动的近期变化
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22
2025/09
补偿机制延缓了全球粉尘释放的变化趋势
补偿机制延缓了全球粉尘释放的变化趋势
