The automorphism group of the sandwich inverse symmetric semigroup

Inverse semigroups with idempotent-fixing automorphisms

A celebrated result of J. Thompson says that if a finite group $G$ has a fixed-point-free automorphism of prime order, then $G$ is nilpotent. The main purpose of this note is to extend this result to finite inverse semigroups. An earlier related result of B. H. Neumann says that a uniquely 2-divisible group with a fixed-point-free automorphism of order 2 is abelian. We similarly extend this result to uniquely 2-divisible inverse semigroups.

Automorphism groups of endomorphism semigroups of free periodic groups

In this paper we describe the automorphism groups of the endomorphism semigroups of free Burnside groups $B(m,n)$ for odd exponents $n\ge1003$. We prove, that the groups $Aut(End(B(m,n)))$ and $Aut(B(m,n))$ are isomorphic. In particular, if the groups $Aut(End(B(m,n)))$ and $Aut(End(B(k,n)))$ are isomorphic, then $m=k$.

半群的偏内自同构逆单群

The Inverse Monoid of Partial Inner Automorphisms of a Semigroup

研究半群的偏内自同构逆单群结构，分析其在群、完全简单半群、全变换单群及有限Abel群作用下的有限G集的内单群中的表现，并提出开放问题。

Scale-multiplicative semigroups and geometry: automorphism groups of trees

A scale-multiplicative semigroup in a totally disconnected, locally compact group $G$ is one for which the restriction of the scale function on $G$ is multiplicative. The maximal scale-multiplicative semigroups in groups acting 2-transitively on the set of ends of trees without leaves are determined in this paper and shown to correspond to geometric features of the tree.

The structure of automorphism groups of semigroup inflations

It is proved that the automorphism group of a semigroup being an inflation of its proper subsemigroup decomposes into a semidirect product of two groups one of which is a direct sum of full symmetric groups.

Automorphism groups of superextensions of finite monogenic semigroups

A family $\mathcal L$ of subsets of a set $X$ is called linked if $A\cap B\ne\emptyset$ for any $A,B\in\mathcal L$. A linked family $\mathcal M$ of subsets of $X$ is maximal linked if $\mathcal M$ coincides with each linked family $\mathcal L$ on $X$ that contains $\mathcal M$. The superextension $λ(X)$ of $X$ consists of all maximal linked families on $X$. Any associative binary operation $* : X\times X \to X$ can be extended to an associative binary operation $*: λ(X)\timesλ(X)\toλ(X)$. In the paper we study automorphisms of the superextensions of finite monogenic semigroups and characterist…

Cross-connections and variants of the full transformation semigroup

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…

Classification of variants of partial Brauer monoids

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.

Automorphism tower problem and semigroup of endomorphisms for free Burnside groups

We have proved that the group of all inner automorphisms of the free Burnside group $B(m,n)$ is the unique normal subgroup in $Aut(B(m,n))$ among all its subgroups, which are isomorphic to free Burnside group $B(s,n)$ of some rank $s$ for all odd $n\ge1003$ and $m>1$. It follows that the group of automorphisms $Aut(B(m,n))$ of the free Burnside group $B(m,n)$ is complete for odd $n\ge1003$, that is it has a trivial center and any automorphism of $Aut(B(m,n))$ is inner. Thus, for groups $B(m,n)$ is solved the automorphism tower problem and is showed that it is as short as the automorphism tower…

On the Endomorphism Semigroups of Extra-special $p$-groups and Automorphism Orbits

For an odd prime $p$ and a positive integer $n$, it is well known that there are two types of extra-special $p$-groups of order $p^{2n+1}$, first one is the Heisenberg group which has exponent $p$ and the second one is of exponent $p^2$. In this article, a new way of representing the extra-special $p$-group of exponent $p^2$ is given. These representations facilitate an explicit way of finding formulae for any endomorphism and any automorphism of an extra-special $p$-group $G$ for both the types. Based on these formulae, the endomorphism semigroup $End(G)$ and the automorphism group $Aut(G)$ a…

A graph-theoretic description of scale-multiplicative semigroups of automorphisms

It is shown that a flat subgroup, $H$, of the totally disconnected, locally compact group $G$ decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, $P$, of a multiplicative semigroup in the quotient, $H/H(1)$, of $H$ by its uniscalar subgroup has a unique minimal generating set which determines a natural Cayley graph structure on $P$. For each compact, open subgroup $U$ of $G$, a graph is defined and it is shown that if $P$ is multiplicative over $U$ then this graph is a regular, rooted, strongly simple $P$-graph. This extends to higher rank…

当族$\boldsymbol{\nmathscr{F}}$由$\boldsymbol{ω}$的非空归纳子集构成时半群$\boldsymbol{B}_ω^{\nmathscr{F}}$的自同构研究

On automorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ in the case when the family $\mathscr{F}$ consists of nonempty inductive subsets of $ω$

本文研究当子集族$\nmathscr{F}$由$ω$的非空归纳子集构成时半群$B_ω^{\nmathscr{F}}$的自同构，给出单自同态为自同构的两个等价充要条件。

关于数值半群或数值单环的幂半群的自同构群

On automorphism groups of power semigroups over numerical semigroups or over numerical monoids

本文研究了数值半群的幂半群的自同构群，发现当半群为特定形式时，唯一的非平凡自同构是特定的映射，否则仅含恒等自同构。

Local subsemigroups and variants of some class of semigroups

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.

Tropical matrix groups

We study the subgroup structure of the semigroup of finitary tropical matrices under multiplication. We show that every maximal subgroup is isomorphic to the full linear automorphism group of a related tropical polytope, and that each of these groups is the direct product of the real numbers with a finite group. We also show that there is a natural and canonical embedding of each full rank maximal subgroup into the group of units of the semigroup of matrices over the tropical semiring with minus infinity. Our results have numerous corollaries, including the fact that every automorphism of a pr…

Several remarks on groups of automorphisms of free groups

Let $G$ be the group of automorphisms of a free group $F_\infty$ of infinite order. Let $H$ be the stabilizer of first $m$ generators of $F_\infty$. We show that the double cosets of $Γ$ with respect to $H$ admit a natural semigroup structure. For any compact group $K$ the semigroup $Γ$ acts in the space $L^2$ on the product of $m$ copies of $K$

The Rees-Suschkewitsch Theorem for simple topological semigroups

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product of a topological group over topological spaces with a continuous sandwich function. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $S\times S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or each power of $S$ is countably compact. The last item generalizes an…

Representations of inverse semigroups in complete atomistic inverse meet-semigroups

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.

CREAM: a Package to Compute [Auto, Endo, Iso, Mono, Epi]-morphisms, Congruences, Divisors and More for Algebras of Type $(2^n,1^n)$

The CREAM GAP package computes automorphisms, congruences, endomorphisms and subalgebras of algebras with an arbitrary number of binary and unary operations; it also decides if between two such algebras there exists a monomorphism, an epimorphism, an isomorphism or if one is a divisor of the other. Thus it finds those objects for almost all algebras used in practice (groups, quasigroups in their various signatures, semigroups possibly with many unary operations, fields, semi-rings, quandles, logic algebras, etc). As a one-size-fits-all package, it only relies on universal algebra theorems, wit…

On cross-sections of partial wreath product of inverse semigroups

We classify $\mathcal{R}$- and $\mathcal{L}$-cross-sections of partial wreath product of inverse semigroups. As a corollary, we get the description of $\mathcal{R}$- and $\mathcal{L}$-cross-sections of the semigroupof partial automorphisms of finite regular rooted tree and compute also the number of different $\mathcal{R}$- ($\mathcal{L}$-) cross-sections in this semigroup.

Semigroup Actions, Covering Spaces and Schutzenberger Groups

We associate a 2-complex to the following data: a presentation of a semigroup $S$ and a transitive action of $S$ on a set $V$ by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results on presentations of Schützenberger groups. We also give a geometric proof that a finitely generated regular semigroup with finitely many idempotents has polynomial growth if and only if all its maximal subgroups are virtua…

逆向表示问题

The Reverse Representation Problem

本文研究凯莱定理的逆向表示问题，在忠实半群框架下刻画给定变换群可表示的集合上所有群结构，发现其解构成堆结构，并进一步分析幺半群、群、逆半群和克利福德半群等具体情形。

Quasi-Automorphism Groups of Type F-infinity

The groups QF, QT, and QV are groups of quasi-automorphisms of the infinite binary tree. Their names indicate a similarity with Thompson's well-known groups F, T, and V. We will use the theory of diagram groups over semigroup presentations to prove that all of the above groups (and several generalizations) have type F-infinity. Our proof uses certain types of hybrid diagrams, which have properties in common with both planar diagrams and braided diagrams. The diagram groups defined by hybrid diagrams also act properly and isometrically on CAT(0) cubical complexes.

On the commuting graphs of Brandt semigroups

The commuting graph of a finite non-commutative semigroup S, denoted by Δ(S), is the simple graph whose vertices are the non-central elements of S and two distinct vertices x; y are adjacent if xy = yx. In the present paper, we study various graph-theoretic properties of the commuting graph Δ(B_n) of Brandt semigroup B_n including its diameter, clique number, chromatic number, independence number, strong metric dimension and dominance number. Moreover, we obtain the automorphism group Aut(Δ(Bn)) and the endomorphism monoid End(Δ(Bn)) of Δ(Bn). We show that Aut(Δ(Bn)) = S_n \times Z_2, where S_…

从代数构造拓扑

On constructing topology from algebra

本论文探讨了通过特定半群构造拓扑结构的方法，分析了不同半群的拓扑性质，并证明了Rubin定理，描述了Brin-Thompson群的自同构群。

Orbits of Primitive $k$-Homogenous Groups on $(n-k)$-Partitions with Applications to Semigroups

Let $X$ be a finite set such that $|X|=n$, and let $k< n/2$. A group is $k$-homogeneous if it has only one orbit on the sets of size $k$. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate $k$-homogeneous groups (for $k\ge 1$); in particular we show that $2$-homogeneous groups are $2$-generated. We also describe the orbits of $k$-homogenous groups on partitions with $n-k$ parts, classify the $3$-homogeneous groups $G$ whose orbits on $(n-3)$-partit…

On the group of automorphisms of the Brandt $λ^0$-extension of a monoid with zero

The group of automorphisms of the Brandt $λ^0$-extension $B^0_λ(S)$ of an arbitrary monoid $S$ with zero is described. In particular we show that the group of automorphisms $\mathbf{Aut}(B_λ^0(S))$ of $B_λ^0(S)$ is isomorphic to a homomorphic image of the group defines on the Cartesian product $\mathscr{S}_λ\times \mathbf{Aut}(S)\times H_1^λ$ with the following binary operation: \begin{equation*} [φ,h,u]\cdot[φ^{\prime},h^{\prime},u^{\prime}]= [φφ^{\prime},hh^{\prime},φu^{\prime}\cdot uh^{\prime}], \end{equation*} where $\mathscr{S}_λ$ is the group of all bijections of the cardinal $λ$, $\math…

On the geometry of the automorphism groups of affine varieties

This article is a survey on ind-varieties and ind-groups introduced by Shafarevich in 1965, with a special emphasis on automorphism groups of affine varieties and actions of ind-groups on ind-varieties. We give precise definitions and complete proofs, including several known results. The survey contains many examples and also some questions which came up during our work on the subject. Among the new results we show that for an affine variety X the automorphism group Aut(X) is always locally closed in the ind-semigroup End(X) of all endomorphisms, and we give an example of a strict closed subgr…

The Schützenberger category of a semigroup

In this paper we introduce the Schützenberger category $\mathbb D(S)$ of a semigroup $S$. It stands in relation to the Karoubi envelope (or Cauchy completion) of $S$ in the same way that Schützenberger groups do to maximal subgroups and that the local divisors of Diekert do to the local monoids $eSe$ of $S$ with $e\in E(S)$. In particular, the objects of $\mathbb D(S)$ are the elements of $S$, two objects of $\mathbb D(S)$ are isomorphic if and only if the corresponding semigroup elements are $\mathscr D$-equivalent, the endomorphism monoid at $s$ is the local divisor in the sense of Diekert a…

集合论Yang-Baxter方程、Kimura半群与函数图

The set-theoretic Yang-Baxter equation, Kimura semigroups and functional graphs

证明FS型集合论Yang-Baxter方程的解范畴等价于带基点Kimura半群范畴，分类了多种特殊解并给出同构类计数，同时刻画了其自同构群的结构。

Braided Diagram Groups and Local Similarity Groups

Hughes defined a class of groups that act as local similarities on compact ultrametric spaces. Guba and Sapir had previously defined braided diagram groups over semigroup presentations. The two classes of groups share some common characteristics: both act properly by isometries on CAT(0) cubical complexes, and certain groups in both classes have type F-infinity, for instance. Here we clarify the relationship between these families of groups: the braided diagram groups over tree-like semigroup presentations are precisely the groups that act on compact ultrametric spaces via small similarity str…

包络 Ellis 半群作为变换群的紧化

Enveloping Ellis semigroups as compactifications of transformations groups

本文引入真 Ellis 半群紧化，建立其与 Ellis 等一致性的对应，并刻画置换群、自同构群及酉群等情形。

拓扑群的 enveloping semigroups 作为紧致化的性质

Enveloping semigroups as compactifications of topological groups

本文研究了拓扑群的 enveloping semigroups 作为紧致化的性质，探讨了Ellis紧致化与Roelcke紧致化之间的比较，并给出了特定群在不同拓扑下的紧致化描述。

Universal sequences for the order-automorphisms of the rationals

In this paper, we consider the group Aut$(\mathbb{Q}, \leq)$ of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut$(\mathbb{Q}, \leq)$ is contained in an $N$-generated subgroup of Aut$(\mathbb{Q}, \leq)$ for some fixed $N\in\mathbb{N}$. We show that the least such $N$ is $2$. Moreover, for every countable subset of Aut$(\mathbb{Q}, \leq)$, we show that every element can be given as a prescribed product of two generators without using their inverses. More p…

Fraïssé structures and a conjecture of Furstenberg

We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between $S(G)$, the Samuel compactification, and $E(M(G))$, the enveloping semigroup of the universal minimal flow. We resolve Furstenberg's problem for several automorphism groups and give a detailed study in the case of $G = S_\infty$, leading us to define and investigate several new types of ultrafilter on a countable set.

On a new approach to the dual symmetric inverse monoid $I*_X$

We construct the \emph{inverse partition semigroup} $\mathcal{IP}_X$, isomorphic to the \emph{dual symmetric inverse monoid} $\mathcal{I}^{\ast}_X$, introduced in [6]. We give a convenient geometric illustration for elements of $\mathcal{IP}_X$. We describe all maximal subsemigroups of $\mathcal{IP}_X$ and find a generating set for $\mathcal{IP}_X$ when $X$ is finite. We prove that all the automorphisms of $\mathcal{IP}_X$ are inner. We show how to embed the symmetric inverse semigroup into the inverse partition one. For finite sets $X$, we establish that, up to equivalence, there is a unique …

实有理曲面自同构：正性与线性性

Real Rational Surface Automorphisms : Positivity and Linearity

本文研究保持尖点三次曲线的实有理曲面自同构，通过自由基本群中的正半群与本原转移矩阵证明其正拓扑熵，并在 m+n 为奇数时给出伪 Anosov 实现。

Using semidirect product of (semi)groups in public key cryptography

In this survey, we describe a general key exchange protocol based on semidirect product of (semi)groups (more specifically, on extensions of (semi)groups by automorphisms), and then focus on practical instances of this general idea. This protocol can be based on any group or semigroup, in particular on any non-commutative group. One of its special cases is the standard Diffie-Hellman protocol, which is based on a cyclic group. However, when this protocol is used with a non-commutative (semi)group, it acquires several useful features that make it compare favorably to the Diffie-Hellman protocol…

Graph automaton groups

In this paper we define a way to get a bounded invertible automaton starting from a finite graph. It turns out that the corresponding automaton group is regular weakly branch over its commutator subgroup, contains a free semigroup on two elements and is amenable of exponential growth. We also highlight a connection between our construction and the right-angled Artin groups. We then study the Schreier graphs associated with the self-similar action of these automaton groups on the regular rooted tree. We explicitly determine their diameter and their automorphism group in the case where the initi…