In this paper we classify the maximal subsemigroups of the \emph{full transformation semigroup} $Ω^Ω$, which consists of all mappings on the infinite set $Ω$, containing certain subgroups of the symmetric group $\sym(Ω)$ on $Ω$. In 1965 Gavrilov showed that there are five maximal subsemigroups of $Ω^Ω$ containing $\sym(Ω)$ when $Ω$ is countable and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary cardinality.
We classify the maximal subsemigroups of $Ω^Ω$ on a set $Ω$ of arbitrary infinite cardinality containing one of the following subgroups of $\sym(Ω)$: the pointwise stabiliser of a non-empty finite subset of $Ω$, the stabiliser of an ultrafilter on $Ω$, or the stabiliser of a partition of $Ω$ into finitely many subsets of equal cardinality. If $G$ is any of these subgroups, then we deduce a characterisation of the mappings $f,g\in Ω^Ω$ such that the semigroup generated by $G\cup \{f,g\}$ equals $Ω^Ω$.
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…
We investigate four finiteness conditions related to residual finiteness: complete separability, strong subsemigroup separability, weak subsemigroup separability and monogenic subsemigroup separability. For each of these properties we examine under which conditions the property is preserved under direct products. We also consider if any of the properties are inherited by the factors in a direct product. We give necessary and sufficient conditions for finite semigroups to preserve the properties of strong subsemigroup separability and monogenic subsemigroup separability in a direct product.
On completely regular and Clifford ordered semigroups
Lee和Kwon [12]定义了一个序半群S是完全正则的，如果对任意a2 S，a2（a2 Sa 2].刻画了完全正则序半群为t-单子半群的并，Clifford序半群为t-单子半群的完备半格.建立了完全正则序半群的格林定理。在一个序半群S中，我们称一个元素e为序幂等元，如果它满足e？e2.利用序幂等元对正则、完全正则和Clifford序半群进行了不同的刻画。从而发展了完全正则序半群和Clifford序半群的基础
Let $a$ be an element of a semigroup $S$. The local subsemigroup of $S$ with respect to $a$ is the subsemigroup $aSa$ of $S$. The variant of $S$ with respect to $a$ is the semigroup with underlying set $S$ and operation $\star_a$ defined by $x\star_ay=xay$ for $x,y\in S$. We show that the following classes contain precisely the same semigroups, up to isomorphism: all local subsemigroups of all finite full transformation semigroups; and all variants of all finite full transformation semigroups. This result was discovered as a result of some experiments (and accidents) when working with the Semi…
The multiplicative semigroup $M_n(F)$ of $n\times n$ matrices over a field $F$ is well understood, in particular, it is a regular semigroup. This paper considers semigroups of the form $M_n(S)$, where $S$ is a semiring, and the subsemigroups $UT_n(S)$ and $U_n(S)$ of $M_n(S)$ consisting of upper triangular and unitriangular matrices. Our main interest is in the case where $S$ is an idempotent semifield, where we also consider the subsemigroups $UT_n(S^*)$ and $U_n(S^*)$ consisting of those matrices of $UT_n(S)$ and $U_n(S)$ having all elements on and above the leading diagonal non-zero. Our gu…
We prove new results on inheritance of Green's relations by subsemigroups in the presence of stability of elements. We provide counterexamples in other cases to show in particular that not all right-stable semigroups are embeddable in left-stable semigroups. This is carried out in the context of a survey of the various closely related notions of stability and minimality of Green's classes that have appeared in the literature over the last sixty years, and which have sometimes been presented in different forms.
设G是群，g在G中，我们定义了从G到G的映射pg（G）和lg（G）：pg（x）=[x，g]和lg（x）=[g，x].设P（G）和L（G）分别表示由{pg：g in G}和{lg：g in G}生成的从G到G的所有映射的集合的子半群. P（G）和L（G）分别称为G的右交换半群和左交换半群。本文给出了G为二面体群时P（G）和L（G）的阶的显式表达式。
Fix (not necessarily distinct) objects $i$ and $j$ of a locally small category $S$, and write $S_{ij}$ for the set of all morphisms $i\to j$. Fix a morphism $a\in S_{ji}$, and define an operation $\star_a$ on $S_{ij}$ by $x\star_ay=xay$ for all $x,y\in S_{ij}$. Then $(S_{ij},\star_a)$ is a semigroup, known as a sandwich semigroup, and denoted by $S_{ij}^a$. This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green's relations and stability, focusing on the relationships between these properties on $S_{ij…
