The Quiver of the Semigroup Algebra of a Left Regular Band

Semigroups embeddable in hyperplane face monoids

The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.

The Face Semigroup Algebra of a Hyperplane Arrangement

This article presents a study of an algebra spanned by the faces of a hyperplane arrangement. The quiver with relations of the algebra is computed and the algebra is shown to be a Koszul algebra. It is shown that the algebra depends only on the intersection lattice of the hyperplane arrangement. A complete system of primitive orthogonal idempotents for the algebra is constructed and other algebraic structure is determined including: a description of the projective indecomposable modules; the Cartan invariants; projective resolutions of the simple modules; the Hochschild homology and cohomology…

On numerical semigroups with at most 12 left elements

For a numerical semigroup S $\subseteq$ N with embedding dimension e, conductor c and left part L = S $\cap$ [0, c -- 1], set W (S) = e|L| -- c. In 1978 Wilf asked, in equivalent terms, whether W (S) $\ge$ 0 always holds, a question known since as Wilf's conjecture. Using a closely related lower bound W 0 (S) $\le$ W (S), we show that if |L| $\le$ 12 then W 0 (S) $\ge$ 0, thereby settling Wilf's conjecture in this case. This is best possible, since cases are known where |L| = 13 and W 0 (S) = --1. Wilf's conjecture remains open for |L| $\ge$ 13.

On the intersection ideal graph of semigroups

The intersection ideal graph $Γ(S)$ of a semigroup $S$ is a simple undirected graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of $Γ(S)$. We show that if $Γ(S)$ is connected then $diam(Γ(S)) \leq 2$. Further we classify the semigroups such that the diameter of their intersection graph is two. Other graph invariants, namely perfectness, planarity, girth, dominance number, clique number, independence number etc. are also discussed. Finally, i…

On the Möbius function of the locally finite poset associated with a numerical semigroup

Let $S$ be a numerical semigroup and let $\left(\mathbb{Z},\leqslant\_S\right)$ be the (locally finite) poset induced by $S$ on the set of integers $\mathbb{Z}$ defined by $x \leqslant\_S y$ if and only if $y-x\in S$ for all integers $x$ and $y$. In this paper, we investigate the M{ö}bius function associated to $\left(\mathbb{Z},\leqslant\_S\right)$ when $S$ is an arithmetic semigroup.

具有对称性的左正则带

Left regular bands with symmetry

本文研究了左正则带半群代数在有限群作用下的不变子代数表示论，通过等变组合拓扑方法分析其结构与性质。

On the inclusion ideal graph of semigroups

The inclusion ideal graph $\mathcal{I}n(S)$ of a semigroup $S$ is an undirected simple graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if either $I \subset J$ or $J \subset I$. The purpose of this paper is to study algebraic properties of the semigroup $S$ as well as graph theoretic properties of $\mathcal{I}n(S)$. In this paper, we investigate the connectedness of $\mathcal{I}n(S)$. We show that diameter of $\mathcal{I}n(S)$ is at most $3$ if it is connected. We also obtain a necessary and sufficient condition of $S$ such…

Følner、Banach 与平移密度相等以及左 amenable 半群中关于密度的其他新结果

Følner, Banach, and translation density are equal and other new results about density in left amenable semigroups

本文证明在满足强 Følner 条件的半群中 Følner 密度、Banach 密度与平移密度相等，并由此解决超滤子双边理想问题，同时给出乘积密度公式等新结果。