On representations of variants of semigroups
Ganna Kudryavtseva, Victor Maltcev
我们构造了一个家庭的表示的任意变体$S$的半群$S$，诱导由一个给定的表示的$S$，并调查这样的表示和他们的内核的性质。
We construct a family of representations of an arbitrary variant $S_a$ of a semigroup $S$, induced by a given representation of $S$, and investigate properties of such representations and their kernels.
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A weakly U abundant is a class of semigroups characterized using some generalized Green' relations. In this paper we discuss the variants of weakly U - abundant semigroups and it is shown that the idempotent variants of these semigroups are again weakly U abundant. Further we also discuss the natural partial order on variants of a weakly U abundant semigroup.
For an element a in a semigroup S the local subsemigroup of S with respect to a is the subsemigroup aSa of S and the variant of S with respect to a is a semigroup with underlying set S with a sandwich operation xy = xay for all x, y in S. In this paper we discuss the structures of local subsemigroups of full transformation semigroups and symmetric inverse monoids. It is also shown that the set of all local subsemigroups of finite symmetric inverse monoids and the set of all variants of all finite symmetric inverse monoids are same upto isomorphism.
Cross-connection theory propounded by K. S. S. Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant $\mathscr{T}_X^θ$ of the full transformation semigroup $(\mathscr{T}_X,\cdot)$ for an arbitrary $θ\in \mathscr{T}_X$ is the semigroup $\mathscr{T}_X^θ= (\mathscr{T}_X,\ast)$ with the binary operation $α\ast β= α\cdotθ\cdotβ$ where $α, β\in \mathscr{T}_X$. In this article, we describe the ideal structure of the regular part $Reg(\mathscr{T}_X^θ)$ of the variant of the full transformation semigroup using cross-connections…
The variant of a semigroup S with respect to an element a in S, denoted S^a, is the semigroup with underlying set S and operation * defined by x*y=xay for x,y in S. In this article, we study variants T_X^a of the full transformation semigroup T_X on a finite set X. We explore the structure of T_X^a as well as its subsemigroups Reg(T_X^a) (consisting of all regular elements) and E_X^a (consisting of all products of idempotents), and the ideals of Reg(T_X^a). Among other results, we calculate the rank and idempotent rank (if applicable) of each semigroup, and (where possible) the number of (idem…
In a semigroup $S$ with fixed $c\in S$, one can construct a new semigroup $(S,\cdot_c)$ called a \emph{variant} by defining $x\cdot_c y:=xcy$. Elements $a,b\in S$ are \emph{primarily conjugate} if there exist $x,y\in S^1$ such that $a=xy, b=yx$. This coincides with the usual conjugacy in groups, but is not transitive in general semigroups. Araújo \emph{et al.} proved that transitivity holds in a variety $\mathcal{W}$ of epigroups containing all completely regular semigroups and their variants, and asked if transitivity holds for all variants of semigroups in $\mathcal{W}$. We answer this affir…
Isolated subsemigroups in the variants of T_n
本文将n元集的所有变换的半群T_n作为关于三明治运算的半群，对T_n中的所有孤立、完全孤立和凸子半群进行了分类。
In the paper we describe the group $\mathbf{Aut}\left(\mathscr{C}_{\mathbb{Z}}\right)$ of automorphisms of the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ and study the variants $\mathscr{C}_{\mathbb{Z}}^{m,n}$ of the extended bicycle semigroup $\mathscr{C}_{\mathbb{Z}}$, where $m,n\in\mathbb{Z}$. In particular, we prove that $\mathbf{Aut}\left(\mathscr{C}_{\mathbb{Z}}\right)$ is isomorphic to the additive group of integers, the extended bicyclic semigroup $\mathscr{C}_{\mathbb{Z}}$ and every its variant are not finitely generated, and describe the subset of idempotents $E(\mathscr{…
This is an elementary introduction to the representation theory of finite semigroups. We illustrate the Clifford-Munn correspondence between the representations of a semigroup and the representations of its maximal subgroups. The emphasis throughout is on naturally occurring examples.
As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in $\mathscr{I}_{X}$, a theory of representations of inverse semigroups by homomorphisms into complete atomistic inverse algebras is developed. This class of inverse algebras includes partial automorphism monoids of entities such as graphs, vector spaces and modules. A workable theory of decompositions is reached; however complete distributivity is required for results approaching those of the classical case.
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…
Isolated and nilpotent subsemigroups in the variants of $\IS$
描述了$n$ -元集的所有内射部分变换的半群$\IS$中的所有孤立、完全孤立和幂零子半群，并将其看作是一个具有夹心乘法的半群.
In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup. If $U$ is any semigroup, and $A$ is a subset of $U$, then we denote by $\langle A\rangle$ the least subsemigroup of $U$ containing $A$. If $B$ is any other subset of $U$, then, roughly speakin…
Transitive representations of inverse semigroups
虽然每个群同构于置换的传递群，但逆半群的类似性质失败：并非所有逆半群同构于集合的一对一部分变换的传递逆半群。我们描述那些逆半群。
We establish that, under certain closure assumptions on a pseudovariety of semigroups, the corresponding relatively free profinite semigroups freely generated by a non-singleton finite set act faithfully on their minimum ideals. As applications, we enlarge the scope of several previous join irreducibility results for pseudovarieties of semigroups, which turn out to be even join irreducible in the lattice of pseudovarieties of ordered semigroups, so that, in particular, they are not generated by proper subpseudovarieties of ordered semigroups. We also prove the stronger form of join irreducibil…
In this paper we focus on Rees $I\times Λ$ matrix semigroups without zero over a semigroup $S$ with $Λ\times I$ sandwich matrix $P$, where $I$ is a singleton, $Λ$ is the factor semigroup of $S$ modulo the kernel $θ_S$ of the right regular representation of $S$, and $P$ is a choice function on the collection of all $θ_S$-classes of $S$. We describe the kernel of the right regular representation of this type of Rees matrix semigroups, and prove embedding theorems on them. Motivated by one of embedding theorems, we show how right commutative right cancellative semigroups can be constructed. We de…
A variant of a semigroup $S$ with respect to an element $a\in S$ is the semigroup $S^{a}=(S,\star_{a})$, where $x\star_{a}y=xay$ for any $x,y\in S$. Here, $a$ is the sandwich element of $S^{a}$. In this article, we study variants of the partial Brauer monoid $\mathcal{PB}_{n}$ for $n\in\mathbb{N}$. We give the classification of these variants in the case when the rank of the sandwich element is nonzero.
On Kiselman's semigroup
受\cite{Ki}的启发，我们研究了半群的级数$\mathrm{K}_n$的代数性质。特别地，我们描述了$\mathrm{K}_n$上的Green关系，证明了$\mathrm{K}_n$存在非负整数系数的$n\times n$矩阵的忠实表示（甚至显式地构造了这样的表示），并证明了$\mathrm{K}_n$不允许较小尺寸的矩阵的忠实表示.我们还刻画了$\mathrm{K}_n$中的极大幂零子半群，$\mathrm{K}_n$的所有孤立子半群和完全孤立子半群，所有自同构和反自同构.最后，我们明确地构造了域上$\mathrm{K}_n$的所有不可约表示，并描述了半群代数中的本原幂等元（我们证明了这是基本的）。
Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups
本文研究有限全变换半群变体中极大正则子半群的同余格，通过将其分解为 $Cong(T_r)$ 与组合系统的等价关系格的次直积，给出同余的显式分类及格的高的公式。
We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite 0-simple semigroups and their mutually annihilating unions.
On the minimal faithful degree of Rhodes semisimple semigroups
本文研究Rhodes半单半群的最小忠实部分变换表示次数，将问题转化为极大子群置换表示的相关分析，推广了逆半群的已有结论，并计算了全变换幺半群的反幺半群的对应次数。
Minimum degrees of finite rectangular bands, null semigroups, and variants of full transformation semigroups
本文研究有限半群的最小变换次数，给出矩形带、矩形群和零半群的次数公式，并应用于全变换半群变体，证明其次数在特定条件下达到上界，同时分类3-幂零子半群。
On the irreducible representations of a finite semigroup
Clifford，Munn和Ponizovski{ \u}的工作用有限半群的极大子群的不可约表示参数化了有限半群的不可约表示.不可约表示的显式构造后来由罗兹和扎尔施泰因以及拉勒芒和佩特里奇独立地得到。所有这些方法都利用Rees定理刻画0-单半群直到同构。本文基于J.A. Green，它允许我们绕过0-单半群的理论。这种方法的新颖之处在于它适用于任何基环。
On Isometric Representations of the Perforated Semigroup
研究了半群$\mathbb{Z}_+\backslash \{1\}$的等距表示.逆表示的概念被引入，并给出了这样的表示的完整描述（直到酉等价）。同时，我们还研究了半群$\mathbb{Z}_+\backslash \{1\}$的一类非逆不可约表示-- $β$ -表示。
Representation Theory of Finite Semigroups, Semigroup Radicals and Formal Language Theory
本文刻画了任意域上有限半群的所有不可约表示的直和所对应的同余，推广了Rhodes在复数域上的结果.应用程序给出了获得许多新的结果，以及更容易证明的几个结果在文献中，涉及：三角化有限半群;其中半群有（分裂）基本半群代数，双边半直积分解有限monoid;明确产品的理性语言;产品的理性语言与计数器;和Černeste的猜想的一个重要类别的自动机。
Let $S$ be a semigroup, $Λ$ a non-empty set and $P$ a mapping of $Λ$ into $S$. The set $S\times Λ$ together with the operation $\circ _P$ defined by $(s, λ)\circ _P(t, μ)=(sP(λ)t, μ)$ form a semigroup which is denoted by $(S, Λ, \circ _P)$. Using this construction, we prove a common connection between the semigroups $S$, $S/θ$ and $S/θ^*=(S/θ)/(θ^*/θ)$, where $θ$ and $θ^*/θ$ are the kernels of the right regular representations of $S$ and $S/θ$, respectively. We also prove an embedding theorem for the semigroup $(S, S/θ, \circ _p)$, where $S$ is a left equalizer simple semigroup without idempot…
We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $σ$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).
A Groupoid Approach to Discrete Inverse Semigroup Algebras
设$K$是有单位的交换环，$S$是逆半群.证明了半群代数$KS$可描述为与Paterson的$S$相关联的泛Etale广群上的函数的卷积代数.这个结果同时推广了作者早期关于有限逆半群的工作和关于泛$C^*$ -代数的Paterson定理。它为理解$KS$的结构提供了一个方便的拓扑框架，包括中心和它何时有一个单元。在这个理论中，Gelfand对偶的作用被Stone对偶取代。利用这种方法，我们能够构造任意域上逆半群的有限维不可约表示作为伴随群的诱导表示，推广了已有研究的具有多个幂等元的逆半群的情形.更一般地说，我们描述的不可约表示的逆半群$S$，可以诱导从相关的群体，正是那些满足一定的“有限性条件”。这个“有限性条件”是满足的，例如，由逆半群的所有表示，其图像包含一个本原幂等元。
In 1978, Munn proved that a bounded complex representation of an inverse semigroup is semiunitary and completely reducible. We consider the converse question in the finite case. We provide some sufficient conditions from a finite regular semigroup to a finite inverse semigroup.
Given a locally constant linear cocycle over a subshift of finite type, we show that the existence of a uniform gap between the i-th and (i+1)-th Lyapunov exponents for all invariant measures implies the existence of a dominated splitting of index i. We establish a similar result for sofic subshifts coming from word hyperbolic groups, in relation with Anosov representations of such groups. We discuss the case of finitely generated semigroups, and propose a notion of Anosov representation in this setting.
Contractive representations of odometer semigroup
本文研究了里程计半群 $O_n$ 的有界线性算子表示，证明了其压缩表示可导出更优表示，并完整描述了 Fock 空间上的表示及对 Nica 协变表示进行了分类。
On Word Representations and Embeddings in Complex Matrices
研究词结构嵌入到低维复数矩阵半群中的方法，开发新的技术构建欧几里得比昂奇群的词表示，为分析2x2矩阵半群的基本决策问题提供框架。
Isometric Representations of Totally Ordered Semigroups
设S是阿贝尔无挠群G的子半群。若S是G的正锥，则由S的忠实等距非酉表示生成的C* 代数是标准同构的.墨菲证明，这一陈述推广了科伯恩和道格拉斯的著名定理。在本说明中，我们证明了相反的情况。若S的忠实等距非酉表示生成的C* 代数都是标准同构的，则S是G的正锥.考虑G = Z \times Z，证明了若S在G上诱导全序，则S至少存在两个酉不等价的不可约等距表示.如果序是字典序-积序，则所有这样的表示都是酉等价的。
Algebra in superextensions of twinic groups
给定一个群$X$，研究了由$X$上的极大链系构成的紧右拓扑半群$λ(X)$的代数结构.这个半群包含超滤子半群$β(X)$作为闭子半群。我们构造了半群$λ(X)$在$X$幂集的所有自映射半群中的一个忠实表示，并利用这个表示刻画了$X$的每一个孪生群的$λ(X)$的极小理想和极小左理想的结构.孪生群类包括所有顺从群和所有具有周期生成元的群，但不包括具有两个生成元的自由群。
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