半群与单向函数

Involution matchings, the semigroup of orientation-preserving and orientation-reversing mappings, and inverse covers of the full transformation semigroup

We continue the study of permutations of a finite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a finite cycle. Finally, by way of application, we prove that finite full transformation semigroups have no cover by inverse semigroups that is closed under intersection.

Actions of $E-$dense semigroups and an application to the discrete log problem

We describe the structure of $E-$dense acts over $E-$dense semigroups in an analogous way to that for inverse semigroup acts over inverse semigroups. This is based, to a large extent, on the work of Schein on representations of inverse semigroups by partial one-to-one maps. We consider an application to the discrete log problem in cryptography as well as an application to the same problem using completely regular semigroups.

奇异变换半群的交叉联络

Cross-connections of the singular transformation semigroup

交叉连接是正则半群的一种构造，它使用某些称为正规范畴的范畴，这些范畴是半群的主左（右）理想的偏序集的抽象。本文描述了集合$X$上所有不可逆变换的半群$Sing(X)$中的交叉联络。所涉及的范畴被表征为幂集范畴$\mathscr{P}(X)$和划分范畴$Π(X)$。我们描述了这些类别，并显示如何置换$X$引起的交叉连接。进一步证明了它们之间的每一个交叉连接都是由置换诱导的，并构造了由交叉连接产生的正则半群。我们证明了这样产生的每个交叉联络半群同构于$Sing(X)$。我们还刻画了主左理想范畴同构于$\mathscr{P}(X)$的$Sing(X)$的右约化子半群。这项研究揭示了更一般的交叉连接理论，也提供了一种研究$Sing(X)$结构的替代方法。

$Γ$ -半群的泛半群

The universal semigroup of a $Γ$-semigroup

给定一个$Γ$ -半群$S$，我们构造了一个半群$Σ$，使得$S$的单侧理想和拟理想可以分别看作$Σ$的单侧理想和拟理想.这种对应和其他性质的$Σ$，使我们能够获得几个结果$S$，而不需要直接与它的工作，但只采用众所周知的结果半群理论。例如，我们得到了在\cite{PT}中发现的$Γ$ -半群的格林定理，作为半群中通常的格林定理的推论。证明了若$S$是$Γ$ -半群且$γ_{0} \in Γ$使得$S_{γ_{0}}$是完全单半群，则对每个$γ\in Γ$，$S_γ$也是完全单的.

Permutations of a semigroup that map to inverses

We investigate the question as to when the members of a finite regular semigroup may be permuted in such a way that each member is mapped to one of its inverses. In general this is not possible. However we reformulate the problem in terms of a related graph and, using an application of Hall's Marriage Lemma, we show in particular that the finite full transformation semigroup does enjoy this property.

半群嵌入的长度函数

Length Functions for Semigroup Embeddings

继Olshanskii所做的工作组，我们描述，对于一个给定的半群$S$，其中的功能$l : S \rightarrow \mathbb{N}$可以实现等价的长度功能$g \mapsto |g|_{H}$嵌入$S$到一个生成的半群$H$。在Olshanskii和Sapir的工作基础上，我们给出了一个在一个可表示半群内具有可表示关系集的可表示生成半群的长度函数的完整描述.