Complexity problems associated with matrix rings, matrix semigroups and Rees matrix semigroups

完全简单半群的交换图

Commuting graphs of completely simple semigroups

本研究刻画群上Rees矩阵半群的交换图及其多项性质，推导完全简单半群交换图的性质与对应图类，还限定了群交换图部分性质的取值范围。

Primitive representations of finite semigroups I

Primitive representations of finite groups as well as primitive finite groups were classified in the O'Nan-Scott Theorem. In this paper we classify faithful finite primitive semigroup representations. To each finite primitive representation, we associate an invariant, a finite dimensional matirix with entries in a primitive finite groups representation. To a large extent, this matrix determines the representation. In later papers in this series the invariant is further explored. Our primitivity resoullts rely on two main ideas, one of which is a small but important part of the theory of tame a…

Computing finite semigroups

Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and $0$-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently co…

Graded von Neumann regularity of rings graded by semigroups

In this article, we give a complete characterization of semigroup graded rings which are graded von Neumann regular. We also demonstrate our results by applying them to several classes of examples, including matrix rings and groupoid graded rings.

On the upper nilradical and Jacobson radical of rings graded by t.u.p.-semigroups

Given a t.u.p.-semigroup (two unique product semigroup) $X$, we show if $R$ is an $X$-graded ring then both its nilradical and Jacobson radical are homogeneous. This partially answers questions of Smoktunowicz and Jespers.

Traces on Semigroup Rings and Leavitt Path Algebras

The trace on matrix rings, along with the augmentation map and Kaplansky trace on group rings, are some of the many examples of linear functions on algebras that vanish on all commutators. We generalize and unify these examples by studying traces on (contracted) semigroup rings over commutative rings. We show that every such ring admits a minimal trace (i.e., one that vanishes only on sums of commutators), classify all minimal traces on these rings, and give applications to various classes of semigroup rings and quotients thereof. We then study traces on Leavitt path algebras (which are quotie…

Noetherian properties of rings of differential operators of affine semigroup algebras

We consider the Noetherian properties of the ring of differential operators of an affine semigroup algebra. First we show that it is always right Noetherian. Next we give a condition, based on the data of the difference between the semigroup and its scored closure, for the ring of differential operators being anti-isomorphic to another ring of differential operators. Using this, we prove that the ring of differential operators is left Noetherian if the condition is satisfied. Moreover we give some other conditions for the ring of differential operators being left Noetherian. Finally we conject…

并非所有幂零幺半群都是有限相关的

Not all nilpotent monoids are finitely related

该研究探讨幂零幺半群的有限相关性问题，证明4-幂零幺半群均为有限相关，给出首个5-幂零非有限相关幺半群示例，还发现有限相关半群的子半群、同态像及Rees商可非有限相关。

关于Buchstaber–Rees“Frobenius n-同态”理论及其推广

On the Buchstaber--Rees theory of "Frobenius $n$-homomorphisms" and its generalization

本文综述了Buchstaber–Rees n-同态理论及其推广，提出以特征函数刻画交换环间线性映射的乘法性质，并由此引入p|q-同态新概念，简化了原有理论。

Étale groupoid algebras with coefficients in a sheaf and skew inverse semigroup rings

Given an action $φ$ of of inverse semigroup $S$ on a ring $A$ (with domain of $φ(s)$ denoted by $D_{s^*}$) we show that if the ideals $D_e$, with $e$ an idempotent, are unital, then the skew inverse semigroup ring $A\rtimes S$ can be realized as the convolution algebra of an ample groupoid with coefficients in a sheaf of rings. Conversely, we show that the convolution algebra of an ample groupoid with coefficients in a sheaf of rings is isomorphic to a skew inverse semigroup ring of this sort. We recover known results in the literature for Steinberg algebras over a field as special cases.

Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

Given a partial action $π$ of an inverse semigroup $S$ on a ring $\mathcal{A}$ one may construct its associated skew inverse semigroup ring $\mathcal{A} \rtimes_πS$. Our main result asserts that, when $\mathcal{A}$ is commutative, the ring $\mathcal{A} \rtimes_πS$ is simple if, and only if, $\mathcal{A}$ is a maximal commutative subring of $\mathcal{A} \rtimes_πS$ and $\mathcal{A}$ is $S$-simple. We apply this result in the context of topological inverse semigroup actions to connect simplicity of the associated skew inverse semigroup ring with topological properties of the action. Furthermore,…