Green's relations on the deformed transformation semigroups
G. Y. Tsyaputa
给出了变形有限逆对称半群$\mathcal{IS}_n$和变形有限对称半群$\mathcal{T}_n$上的格林关系.
Green's relations on the deformed finite inverse symmetric semigroup $\mathcal{IS}_n$ and the deformed finite symmetric semigroup $\mathcal{T}_n$ are described.
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Transformation Semigroups with the Deformed Multiplication
利用Ljapin提出的变形乘法，对由有限逆对称半群$\mathcal{IS}_n ,$、有限对称半群$\mathcal{T}_n$和双循环半群得到的成对非同构半群进行了分类.
Deformed multiplication in the semigroup PT_n
对由所有部分变换的半群PT_n通过Ljapin提出的变形乘法得到的成对非同构半群进行了分类。
On the paper "On Green's relations, $2^0$-regularity and Quasi-ideals in $Γ$-semigroups"
这是关于K. Hila和J. Dine，《数学学报》，英文丛书，29，第3期，（2013），609- 624。
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.
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…
Deformed Convolution, Cumulant Transforms, and Semigroup Generators in Hurwitz Series Rings
该研究在赫维茨级数环上引入单参数卷积，推导其性质与应用，得到相关半群生成元的显式公式，拓展了自由概率与代数结构的相关理论。
Green's J-order and the rank of tropical matrices
研究了热带半环上n阶矩阵半群的Green J-序和J-等价。我们给出了一个确切的表征的J-顺序，在热带凸集之间的态射。我们建立J-秩序，等距的热带凸集，和各种概念的热带矩阵的秩之间的联系。Izhakian和Margolis已经观察到对于热带半环上的所有3 × 3矩阵半群的$D \neq J$与$-\infty$，但是相反，我们证明了对于有限热带半环上的所有满矩阵半群的$D = J$.
On semigroups of endomorphisms of a chain with restricted range
令$X$为有限或无限链，令$O(X)$为$X$的所有自同态的么半群。本文给出了$O(X)$的最大正则子半群和$O(X)$上的格林关系.事实上，更一般地，如果$Y$是$X$的非空子集，而$O(X,Y)$是$O(X)$的值域包含在$Y$中的所有元素的子半群，我们刻画了$O(X,Y)$的最大正则子半群和$O(X,Y)$上的Green关系.此外，对于有限链，我们确定了两个$O(X,Y)$型半群何时同构，并计算了它们的秩。
We prove that if $S$ is a $le$-semigroup in which left ideal elements commute (condition which is called $\mathbfΛ$), then any $\mathcal{J}$-class satisfying the Green condition is a subsemigroup of $S$. As a corollary of this we show that semisimple $le$-semigroups satisfying $\mathbfΛ$ are precisely those that decompose as a semilattice of left simple $\vee e$-semigroups which are in addition semisimple, intra-regular and satisfy $\mathbfΛ$.
This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb categories. If $\mathcal K$ denotes any of these categories, and if $σ\in\mathcal K_{nm}$ is a fixed morphism, then an associative operation $\star_σ$ may be defined on $\mathcal K_{mn}$ by $α\star_σβ=ασβ$. The resulting semigroup $\mathcal K_{mn}^σ=(\mathcal K_{mn},\star_σ)$ is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green's relati…
We give an efficient algorithm for the enumeration up to isomorphism of the inverse semigroups of order n, and we count the number S(n) of inverse semigroups of order n<=15. This improves considerably on the previous highest-known value S(9). We also give a related algorithm for the enumeration up to isomorphism of the finite inverse semigroups S with a given underlying semilattice of idempotents E, a given restriction of Green's D-relation on S to E, and a given list of maximal subgroups of S associated to the elements of E.
Let $\mathcal M_{mn}=\mathcal M_{mn}(\mathbb F)$ denote the set of all $m\times n$ matrices over a field $\mathbb F$, and fix some $n\times m$ matrix $A\in\mathcal M_{nm}$. An associative operation $\star$ may be defined on $\mathcal M_{mn}$ by $X\star Y=XAY$ for all $X,Y\in\mathcal M_{mn}$, and the resulting \emph{sandwich semigroup} is denoted $\mathcal M_{mn}^A=\mathcal M_{mn}^A(\mathbb F)$. These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. In this article, we study $\mathcal M_{mn}^A$ as well as its subsemigro…
The semigroup of endomorphisms with restricted range of an independence algebra
本文研究了独立性代数中像位于子代数内的自同态半群，在统一框架下描述了其Green关系与理想，并完整刻画了其扩展Green关系。
A class of core inverses associated with Green's relations in semigroups
本文在 $*$-幺半群中定义了推广的 $(b,c)$-core-EP 逆，建立了其判定准则，阐明了多种核逆之间的关系，并给出了复矩阵中 $(B,C)$-core-EP 逆的判定准则及矩阵方程组的解。
For $n \in \mathbb{N}$, let $X_n = \{a_1, a_2, \ldots, a_n\}$ be an $n$ - element set and let $\textbf{F} = (X_n; <_f)$ be a fence, also called a zigzag poset. As usual, we denote by $I_n$ the symmetric inverse semigroup on $X_n$. We say that a transformation $α\in I_n$ is \textit{fence-preserving} if $x <_f y$ implies that $xα<_f yα$, for all $x, y$ in the domain of $α$. In this paper, we study the semigroup $PFI_n$ of all partial fence-preserving injections of $X_n$ and its subsemigroup $IF_n = \{α\in PFI_n : α^{-1}\in PFI_n\}$. Clearly, $IF_n$ is an inverse semigroup and contains all regula…
On semigroups of orientation-preserving partial permutations with restricted range
本文研究了有限链上像集受限的保向单射部分变换半群 $\mathcal{POPI}_{n}(Y)$，描述了其 Green 关系与正则性，计算了秩，并给出了此类半群同构的判定条件。
Tropical matrix duality and Green's D relation
本文给出了n阶热带矩阵乘法半群的格林D关系的完整刻画。我们的主要工具是一个新的变体之间的对偶行和列空间的热带矩阵（研究科恩，Gaubert和Quadrat和分别由Cherin和Sturmfels）。与已有的对偶定理不同，我们的版本允许一个逆，从而给出了两个热带凸集是矩阵的行和列空间的一个充要条件。我们还证明了矩阵对偶映射在任意热带矩阵的射影行空间和射影列空间之间诱导出一个等距（关于Hilbert射影度量），并建立了关于Green的其它关系的一些基本结果.
We study the algebraic properties of binary relations whose underlying digraph is smooth, that is has no source or sink. Such objects have been studied as surjective hyper-operations (shops) on the corresponding vertex set, and as binary relations that are defined everywhere and whose inverse is also defined everywhere. In the latter formulation, they have been called multipermutations. We study the lattice structure of sets (monoids) of multipermutations over an n-element domain. Through a Galois connection, these monoids form the algebraic counterparts to sets of relations closed under defin…
Semigroups of linear transformations whose restrictions belong to a general linear group
本文研究限制在固定子空间上属于一般线性群的线性变换半群，刻画其Green关系、理想、极小理想与极小幂等元，并在有限域有限维情形给出同构定理和生成集。
One-sided Noncommutative Groebner Bases with Applications to Green's Relations
标准非交换Gröbner基程序用于计算域上自由非交换多项式环的理想。本文描述了域上非交换代数中单边理想的Gröbner基程序。将$K$ -代数$A$定义为自由$K$ -代数的商的多项式与定义$A$的单侧理想$I$的多项式通过使用标记符号组合。然后，标准的非交换Gröbner基技术可以应用于混合多项式集，从而在自由结构中计算$A/I$，避免了$A$中计算的复杂性。最后，本文展示了如何将结果可以应用到可完成的介绍半群，使计算的格林关系。
Fix sets $X$ and $Y$, and write $\mathcal{PT}_{XY}$ for the set of all partial functions $X\to Y$. Fix a partial function $a:Y\to X$, and define the operation $\star_a$ on $\mathcal{PT}_{XY}$ by $f\star_ag=fag$ for $f,g\in\mathcal{PT}_{XY}$. The sandwich semigroup $(\mathcal{PT}_{XY},\star_a)$ is denoted $\mathcal{PT}_{XY}^a$. We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $\mathcal{PT}_{XY}^a$, as well as its regular and idempotent-generated subsemigroups, Reg$(\mathcal{PT}_{XY}^a)$ and $\mathbb E(\mathcal{PT}_{XY}^a)$. After describ…
Semigroups of transformations whose characters belong to a given semigroup
本文研究保持划分的全变换半群中特征变换属于给定半群的子半群，刻画其正则元、幂等元、Green关系及单位正则性，并应用于 $T(X,\mathcal{P})$。
Computing finite index congruences of finitely presented semigroups and monoids
本文提出两种计算有限展示半群或幺半群有限指数同余的算法，通过推广Sims低指数子群算法及利用相对Green关系理论减少主同余生成数量，并在GAP包和C++库中实现。
The primary contribution of this thesis is to introduce and examine the planar modular partition monoid for parameters $m, k \in \mathbb{Z}_{>0}$, which has simultaneously and independently generated interest from other researchers as outlined within. Our collective understanding of related monoids, in particular the Jones, Brauer and partition monoids, along with the algebras they generate, has heavily influenced the direction of research by a significant number of mathematicians and physicists. Examples include Schur-Weyl type dualities in representation theory along with Potts, ice-type and…
The regular part of transformation semigroups that preserve double direction equivalence relation
该研究针对保持双向等价关系的变换子半群$T_{E^*}(X)$，证明其正则部分是最大正则子半群，刻画了其格林关系与理想，确定了可表为对称群并的右群核，且证明所有右群都可嵌入该核。
Congruences on completely inverse $AG^{**}$-groupoids
完全逆$AG^{**}$ -广群是指满足恒等式$xx^{-1}=x^{-1}x$的逆$AG^{**}$ -广群$A$，其中$x^{-1}$表示$A$的唯一元素，使得$x=(xx^{-1})x$和$x^{-1}=(x^{-1}x)x^{-1}.$我们证明了这样的广群的所有幂等元的集合形成半格，并且格林关系$\mathcal{H,L, R,D}$和$\mathcal{J}$在$A$上重合.本文的主要结果是：任何完全逆的$AG^{**}$ -广群满足正则半群的著名的Lallement引理。最后证明了格林关系$\mathcal{H}$是任意完全逆$AG^{**}$ -广群上的最小半格同余和最大幂等元分离同余.
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