Varujan S. Atabekyan
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 $m=k$.
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Howie and Ribeiro \cite{a.Howie99,a.Howie00} introduced various ranks, viz. small rank, lower rank, intermediate rank, upper rank and the large rank of a finite semigroup. In this note, we investigate all these ranks of the semigroup of endomorphisms over Brandt semigroup.
Necessary and sufficient conditions are given for the endomorphism monoid of a profinite semigroup to be profinite. A similar result is established for the automorphism group.
It is proved that the semigroups $\mathrm{\mathbf{End}}(\boldsymbol{B}_ω)$ and $\mathrm{\mathbf{End}}(\boldsymbol{B}_{\mathbb{Z}})$ of the endomorphisms of the bicyclic semigroup $\boldsymbol{B}_ω$ and the endomorphisms of the extended bicyclic semigroup $\boldsymbol{B}_{\mathbb{Z}}$ are isomorphic to the semidirect products $(ω,+)\rtimes_φ(ω,*)$ and $\mathbb{Z}(+)\rtimes_φ(ω,*)$, respectively.
On the semigroup of monoid endomorphisms of the semigroup $\mathscr{C}_{+}(a,b)$
本文研究双循环幺半群子幺半群 $\mathscr{C}_{+}(a,b)$，刻画了由其全部同余和单射幺半群自同态共同生成的幺半群自同态。
On injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^4}$ with a four-element family $\mathscr{F}^4$ of inductive non-empty subsets of $ω$
该研究刻画了带有四元归纳非空子集族ℱ⁴的半群B_ω^ℱ⁴的单态射，证明其每个单幺半群态射为恒等变换，还刻画了所有单态射
A complete classification of endomorphisms of Kiselman's semigroup
本文完全分类了 Kiselman 半群 $K_n$ 的自同态，并构造了与其自同态幺半群同构的布尔矩阵幺半群，回答了 Kudryavtseva 和 Mazorchuk 提出的分类问题。
On the semigroup of monoid endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ with a two-element family $\mathscr{F}$ of inductive nonempty subsets of $ω$
本研究针对含两元素归纳子集族的半群$\boldsymbol{B}_ω^{\nmathscr{F}}$，分析其非单射幺半群自同态半群的结构，证明相关子半群的同构性质并刻画其上的格林关系。
We investigate the endomorphism monoids of certain finite $p$-groups of order $p^8$ first studied by Jonah and Konvisser in 1975 as examples for finite $p$-groups with abelian automorphism group, and we show some necessary conditions for a finite $p$-group to have commutative endomorphism monoid. As a by-product, apart from formulas for the number of conjugacy classes of endomorphisms of said groups, we will be able to derive the following: There exist nonabelian groups with commutative endomorphism monoid, and having commutative endomorphism monoid is a group property strictly stronger than h…
A {\it Schmidt group} is a non-nilpotent finite group in which each proper subgroup is nilpotent. Each Schmidt group G can be described by three parameters p, q and v, where p and q are different primes and v is a natural number, $v\ge 1$. Denote by ${\mathcal{S}}$ the class of all Schmidt groups which have the same parameters p, q and v. It is shown in this paper that the class ${\mathcal{S}}$ can be characterized by the properties of the endomorphism semigroups of the groups of this class. It follows from this characterization that if $G\in {\mathcal{S}}$ and H is another group such that the…
In this article we study automorphisms and endomorphisms of lacunary hyperbolic groups. We prove that every lacunary hyperbolic group is Hopfian, answering a question by Henry Wilton. In addition, we show that if a lacunary hyperbolic group has the fix point property for actions on $\mathbf R$-trees, then it is co-Hopfian and its outer automorphism group is locally finite. We also construct lacunary hyperbolic groups whose automorphism group is infinite, locally finite, and contains any locally finite group given in advance.
On the semigroup of injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}_n}$ which is generated by the family $\mathscr{F}_n$ of initial finite intervals of $ω$
本文描述了逆半群 $\boldsymbol{B}_ω^{\mathscr{F}}$ 在特定生成条件下的单射自同态，证明其同构于 $(ω,+)$，并刻画了 $λ{\times}λ$-矩阵单位半群 $\mathscr{B}_λ$ 的全自同态半群结构。
On injective endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three-element family $\mathscr{F}^3$ of inductive non-empty subsets of $ω$
该研究探讨了具有三个元素家族$\mathscr{F}^3$的半群$\boldsymbol{B}_ω^{\mathscr{F}^3}$的可注入内射映射，并发现了特定的内射$\varpi_3$和$\lambda$，用于描述所有可注入内射的关系。
On the semigroup of injective monoid endomorphisms of the monoid $\boldsymbol{B}_ω^{\mathscr{F}^3}$ with a three element family $\mathscr{F}^3$ of inductive nonempty subsets of $ω$
该研究刻画了带三元素归纳非空子集族的半群$\boldsymbol{B}_ω^{\u27E8\u27E9^3}$的单射幺半群自同态，并证明其所有单射自同态构成的幺半群同构于正整数乘法半群。
On the semigroup of endomorphisms of the semigroup $\boldsymbol{B}_ω^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$
本文研究了具有两个元素家族$\mathscr{F}^2$的半群$\boldsymbol{B}_ω^{\mathscr{F}^2}$的内射半群，并构造了其子单侧$\left\langle\varpi\right\rangle^1$。
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…
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 injective endomorphisms of the semigroup $\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$ with the two-element family $\mathscr{F}^2$ of inductive nonempty subsets of $ω$
本文针对带ω归纳非空子集二元族$\mathscr{F}^2$的半群$\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}^2}$，刻画其单自同态结构，证明其可分解为特定单自同态与自同构的乘积，并刻画了对应的特定单自同态。
Endomorphism monoid and automorphism group of residually finite and profinite quandles
该研究探讨剩余有限与投射quandle，证明有限生成及一大类剩余有限quandle的自同态幺半群、自同构群为剩余有限，给出投射quandle拓扑刻画并建立其自同态幺半群与自同构群为投射的充要条件。
We study automorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ with the family $\mathscr{F}$ of inductive nonempty subsets of $ω$ and prove that the group $\mathbf{Aut}(\boldsymbol{B}_{\mathbb{Z}}^{\mathscr{F}})$ of automorphisms of the semigroup $\boldsymbol{B}_{Z\mathbb{}}^{\mathscr{F}}$ is isomorphic to the additive group integers.
Irreducibility of Endomorphisms of Finitely Generated Free Semigroups
本文研究有限生成自由半群自同态的不可约性，引入素性概念，给出保秩自同态可约与不可约的判别条件，并分析分解的非唯一性及关联矩阵的应用。
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.
Determinant for automorphisms of semidirect product of groups
本文利用半直积自同态的 $2\ imes2$ 映射矩阵表示，引入并研究其行列式，进而用行列式工具刻画可逆自同态。
We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated and we do the same for periodic points. We also describe the automorphisms of such a group and solve the three Whitehead problems for automorphisms, monomorphisms and endomorphisms for groups in this class. We also obtain conditions for an endomorphism to be uniformly continuous when a suitable metric is considered and study infinite fixed points of the extension to the completion of the group.
An m-endomorphism of a free semigroup is an endomorphism that sends every generator to a word of length at most m. Two m-endomorphisms are combinatorially equivalent if they are conjugate under an automorphism of the semigroup. In this paper, we specialize an argument of N. G. de Bruijn to produce a formula for the number of combinatorial equivalence classes of m-endomorphisms on a rank-n semigroup. From this formula, we derive several little-known integer sequences.
We calculate the rank and idempotent rank of the semigroup $E(X,P)$ generated by the idempotents of the semigroup $T(X,P)$, which consists of all transformations of the finite set $X$ preserving a non-uniform partition $P$. We also classify and enumerate the idempotent generating sets of this minimal possible size. This extends results of the first two authors in the uniform case.
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。
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 rank of a semigroup is the cardinality of a smallest generating set. In this paper we compute the rank of the endomorphism monoid of a non-trivial uniform partition of a finite set, that is, the semigroup of those transformations of a finite set that leave a non-trivial uniform partition invariant. That involves proving that the rank of a wreath product of two symmetric groups is two and then use the fact that the endomorphism monoid of a partition is isomorphic to a wreath product of two full transformation semigroups. The calculation of the rank of these semigroups solves an open questio…
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…
The mapping torus of an endomorphism Φof a group G is the HNN-extension G*_G with bonding maps the identity and Φ. We show that a mapping torus of an injective free group endomorphism has the property that its finitely generated subgroups are finitely presented and, moreover, these subgroups are of finite type.
The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup.
We study endomorphisms of a free group of finite rank by means of their action on specific sets of elements. In particular, we prove that every endomorphism of the free group of rank 2 which preserves an automorphic orbit (i.e., acts ``like an automorphism" on one particular orbit), is itself an automorphism. Then, we consider elements of a different nature, defined by means of homological properties of the corresponding one-relator group. These elements (``generalized primitive elements"), interesting in their own right, can also be used for distinguishing automorphisms among arbitrary endomo…
We examine varieties of epigroups as unary semigroups, that is semigroups equipped with an additional unary operation of pseudoinversion. The article contains two main results. The first of them indicates a countably infinite family of injective endomorphisms of the lattice of all epigroup varieties. An epigroup variety is said to be a variety of finite degree if all its nilsemigroups are nilpotent. The second result of the article provides a characterization of epigroup varieties of finite degree in a language of identities and in terms of minimal forbidden subvarieties. Note that the first r…
An endomorphism of a free group is called primitivity preserving if it takes every primitive element to another primitive. In this paper we prove that every primitivity preserving endomorphism of a free group of a finite rank n > 2 is an automorphism.
We obtain sufficient criteria for endomorphisms of torsion-free nilpotent groups of finite rank to be automorphisms, by considering the induced maps on the torsion-free abelianisation and the centre. Whilst these results are known in the finitely generated case removing this assumption introduces several difficulties.
We investigate the structure of the monoid of endomorphisms of the ordered set $(\mathbb{Q},{\leq})$ of rational numbers. We show that for any countable linearly ordered set $Ω$, there are uncountably many maximal subgroups of $\operatorname{End}(\mathbb{Q},{\leq})$ isomorphic to the automorphism group of $Ω$. We characterise those subsets $X$ of $\mathbb{Q}$ that arise as a retract in $(\mathbb{Q},{\leq})$ in terms of topological information concerning $X$. Finally, we establish that a countable group arises as the automorphism group of a countable linearly ordered set, and hence as a maximal…
Elementary equivalence of endomorphism rings and automorphism groups of periodic Abelian groups
该研究探讨周期阿贝尔群的自同态环与自同构群的初等等价问题，给出了自同态环初等等价的充要条件，并针对特定周期阿贝尔群刻画了自同构群初等等价的等价条件。
Polish topologies on endomorphism monoids of linear orders
本文通过引入新结构条件性质XX，研究线性序的自同态幺半群End(N,≤)和End(Z,≤)上的Polish半群拓扑，证明了某些幺半群上Polish半群拓扑的唯一性或多样性。
Determinant for automorphisms of Zappa-Szép product of groups
本文利用映射的2×2矩阵描述，研究两个群Zappa-Szép积的自同态行列式，刻画可逆自同态，并计算交错群A5的行列式。
We survey recent results on endomorphisms and especially on automorphisms of the Cuntz algebras O_n, with a special emphasis on the structure of the Weyl group. We discuss endomorphisms globally preserving the diagonal MASA and their corresponding actions. In particular, we investigate those endomorphisms of O_n which restrict to automorphisms of the diagonal. We review a combinatorial approach to the study of permutative endomorphisms. All the presented material is put in context with current research topics.
Endomorphism and automorphism graphs of finite groups
本文定义有限群的有向/无向自同态图与自同构图，介绍其与变换幺半群的关联，探究其图论性质、判定对应图类的群类型并计算特殊群实例，最后以示例说明相关问题的复杂性。
In \cite{Ioana:vNsuperrigidity}, Ioana introduced three new invariants of type II$_1$ factors: the one-sided fundamental group, the endomorphism semigroup and the set of right-finite bimodules. In \cite{Ioana:vNsuperrigidity}, he does not provide many computations of these invariants. In particular, the question whether these invariants can be trivial is left open. We give an explicit example of a type II$_1$ factor for which all three invariants are trivial. More generally, for any countable left-cancellative semigroup $G$, we construct a type II$_1$ factor $M$ whose endomorphism semigroup is…
We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $Γ$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $Γ$. Further structural information about End $R$ is established including that Aut $Γ$ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.
Endomorphisms of Artin groups of type D
本文确定了 $n\ge 6$ 时 $D_n$ 型 Artin 群及其商群的自同态分类与自同构群，并利用拓扑论证证明了相关同态分类及商群的 co-Hopfian 性质。
The structure of the automorphism group of the sandwich semigroup IS_n is described in terms of standard group constructions.
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.
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…
The scale of an endomorphism, $α$, of a totally disconnected, locally compact group $G$ is the minimum index $[α(U) : α(U)\cap U]$, for $U$ a compact, open subgroup of $G$. A structural characterization of subgroups at which the minimum is attained is established. This characterization extends the notion of subgroup tidy for $α$ from previously understood case when $α$ is an automorphism to the case when $α$ is merely an endomorphism.
Polish topologies on endomorphism monoids of relational structures
本文提出了一种方法，用于刻画可数关系结构A的内射单形End(A)的最小和最大半群拓扑。结果显示，随机图、随机有向图和随机偏序等结构的内射单形具有唯一的波兰半群拓扑，即由Baire空间N^N的通常拓扑诱导的子空间拓扑。
Takahasi's theorem on chains of subgroups of bounded rank in a free group is generalized to several classes of semigroups. As an application, it is proved that the subsemigroups of periodic points are finitely generated and periodic orbits are bounded for arbitrary endomorphisms for various semigroups. Some of these results feature classes such as completely simple semigroups, Clifford semigroups or monoids defined by balanced one-relator presentations. In addition to the background on semigroups, proofs involve arguments over groups and finite automata.
We initiate a systematic investigation of endomorphisms of graph C*-algebras C*(E), extending several known results on endomorphisms of the Cuntz algebras O_n. Most but not all of this study is focused on endomorphisms which permute the vertex projections and globally preserve the diagonal MASA D_E of C*(E). Our results pertain both automorphisms and proper endomorphisms. Firstly, the Weyl group and the restricted Weyl group of a graph C*-algebra are introduced and investigated. In particular, criteria of outerness for automorphisms in the restricted Weyl group are found. We also show that the…
A uniform Tits alternative for endomorphisms of the projective line
本文推广了投影直线自同态半群的Tits二择性，通过扩展算术高度函数的乒乓球论证，证明了指数增长半群具有一致指数增长。
We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case $S=\mathbb N$. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a B…
We prove that the growth rate of an endomorphism of a finitely generated nilpotent group equals to the growth rate of induced endomorphism on its abelinization, generalizing the corresponding result for an automorphism in [14]. We also study growth rates of endomorphisms for specific solvable groups, lattices of Sol, providing a counterexample to a known result in [5] and proving that the growth rate is an algebraic number.
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