Cayley posets

Möbius function of semigroup posets through Hilbert series

In this paper, we investigate the M{ö}bius function $μ\_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. We introduce and develop a new approach to study $μ\_{\mathcal{S}}$ by using the Hilbert series of $\mathcal{S}$. The latter enables us to provide formulas for $μ\_{\mathcal{S}}$ when $\mathcal{S}$ belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given.

Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.

Distance labellings of Cayley graphs of semigroups

This paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. We show that certain general restrictions on the minimum spans are equivalent to the semigroup being combinatorial, and that other restrictions are equivalent to the semigroup being a right zero band. We obtain a description of the structure of all semigroups $S$ and their subsets $C$ such that $\Cay(S,C)$ is a disjoint union of complete graphs, and show that this description is also equivalent to several restrictions on the minimum span of $\Cay(S,C)$. We …

Numerical semigroups, polyhedra, and posets I: the group cone

Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first introduced by Kunz, has integer points in bijection with numerical semigroups of fixed multiplicity, and another, introduced by Hellus and Waldi, has integer points corresponding to oversemigroups of numerical semigroups with two generators. In this paper, we provide a combinatorial framework from which to study both families of polyhedra. We introduce a new family of polyhedra called group cones, each constructed from so…

数值半群、多面体与偏序集 IV：遍历 Kunz 锥的面

Numerical semigroups, polyhedra, and posets IV: walking the faces of the Kunz cone

本文引入 Kunz 扇并借助 Kunz 零半群刻画其组合性质，提出类似 Gröbner 遍历的 Kunz 锥面格遍历方法，并用于证明关于最小表示基数的猜想。

带符号偏序集的根半群环与权半群环

Root and weight semigroup rings for signed posets

本文研究带符号偏序集的根半群环与权半群环，给出其环面理想的生成元和Gröbner基，并刻画完全交集情形，推广了多个已有结果。

On Cayley graphs of basic algebraic structures

We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.

On planar right groups

In 1896 Heinrich Maschke characterized planar finite groups, that is groups which admit a generating system such that the resulting Cayley graph is planar. In our study we consider the question, which finite semigroups have a planar Cayley graph. Right groups are a class of semigroups relatively close to groups. We present a complete characterization of planar right groups.

Chomp on numerical semigroups

We consider the two-player game chomp on posets associated to numerical semigroups and show that the analysis of strategies for chomp is strongly related to classical properties of semigroups. We characterize, which player has a winning-strategy for symmetric semigroups, semigroups of maximal embedding dimension and several families of numerical semigroups generated by arithmetic sequences. Furthermore, we show that which player wins on a given numerical semigroup is a decidable question. Finally, we extend several of our results to the more general setting of subsemigroups of $\mathbb{N} \tim…

群在偏序集上的正则与半正则表示

Regular and semi-regular representations of groups by posets

本文研究群在偏序集上的正则与半正则表示，证明若群允许局部为自由群 Cayley 图的 Cayley 图，则其具有高度 1 且两轨道的半正则表示，并给出对有限单群、双曲群等群类的应用。

与半群上格林关系相关的偏序集的高度

Heights of posets associated with Green's relations on semigroups

该研究定义了半群上对应四类格林关系的偏序集高度，探讨了稳定半群、正则半群中各类高度的关系与上下界，构造例子验证了界的紧性，拓展了半群结构理论。

On monoid graphs

We investigate Cayley graphs of finite semigroups and monoids. First, we look at semigroup digraphs, i.e., directed Cayley graphs of semigroups, and give a Sabidussi-type characterization in the case of monoids. We then correct a proof of Zelinka from '81 that characterizes semigroup digraphs with outdegree $1$. Further, answering a question of Knauer and Knauer, we construct for every $k\geq 2$ connected $k$-outregular non-semigroup digraphs. On the other hand, we show that every sink-free directed graph is a union of connected components of a monoid digraph. Second, we consider monoid graphs…

Differential posets and restriction in critical groups

In recent work, Benkart, Klivans, and Reiner defined the critical group of a faithful representation of a finite group $G$, which is analogous to the critical group of a graph. In this paper we study maps between critical groups induced by injective group homomorphisms and in particular the map induced by restriction of the representation to a subgroup. We show that in the abelian group case the critical groups are isomorphic to the critical groups of a certain Cayley graph and that the restriction map corresponds to a graph covering map. We also show that when $G$ is an element in a different…