Alexandre A. Panin
We consider the lattice of subsemigroups of the general linear group over an Artinian ring containing the group of diagonal matrices and show that every such semigroup is actually a group.
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Methods from additive number theory are applied to construct families of finitely generated linear semigroups with intermediate growth.
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.
We define and study the notion of a crossed module over an inverse semigroup and the corresponding $4$-term exact sequences, called crossed module extensions. For a crossed module $A$ over an $F$-inverse monoid $T$, we show that equivalence classes of admissible crossed module extensions of $A$ by $T$ are in a one-to-one correspondence with the elements of the cohomology group $H^3_\le(T^1,A^1)$.
We prove the following results: (1) Every group is a maximal subgroup of some free idempotent generated semigroup. (2) Every finitely presented group is a maximal subgroup of some free idempotent generated semigroup arising from a finite semigroup. (3) Every group is a maximal subgroup of some free regular idempotent generated semigroup. (4) Every finite group is a maximal subgroup of some free regular idempotent generated semigroup arising from a finite regular semigroup.
We present a construction for the holomorph of an inverse semigroup, derived from the cartesian closed structure of the category of ordered groupoids. We compare the holomorph with the monoid of mappings that preserve the ternary heap operation on an inverse semigroup: for groups these two constructions coincide. We present detailed calculations for semilattices of groups and for the polycyclic monoids.
To solve two problems of Bergman stated in 1981, we construct a group $G$ such that $G$ contains a free noncyclic subgroup (hence, $G$ satisfies no group identity) and $G$, as a group, is generated by its subsemigroup that satisfies a nontrivial semigroup identity.
Let $a$ be a non-invertible transformation of a finite set and let $G$ be a group of permutations on that same set. Then $\genset{G, a}\setminus G$ is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by $G$ and $a$. Likewise, the conjugates $a^g=g^{-1}ag$ of $a$ by elements $g\in G$ generate a semigroup denoted $\genset{a^g | g\in G}$. We classify the finite permutation groups $G$ on a finite set $X$ such that the semigroups $\genset{G,a}$, $\genset{G, a}\setminus G$, and $\genset{a^g | g\in G}$ are regular for all transformations of $X$. We also cla…
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 $m=k$.
In this paper we exhibit a type of semigroup presentations which determines a class of local groups. We show that the finite elements of this class generate the pseudovariety ${\bf LG}$ of all finite local groups and use them as test-semigroups to prove that ${\bf LG}$ and ${\bf S}$, the pseudovariety of all finite semigroups, verify the same $κ$-identities involving $κ$-terms of rank at most 1, where $κ$ denotes the implicit signature consisting of the multiplication and the $(ω-1)$-power.
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.
We study algebraic and topological properties of subsemigroups of the hyperspace exp(G) of non-empty compact subsets of a topological group G endowed with the Vietoris topology and the natural semigroup operation. On this base we prove that a compact Clifford topological semigroup S is topologically isomorphic to a subsemigroup of exp(G) for a suitable topological group G if and only if S is a topological inverse semigroup with zero-dimensional idempotent semilattice.
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.
The study of word hyperbolic groups is a prominent topic in geometric group theory; however word hyperbolic groups are defined by a geometric condition which does not extend naturally to semigroups. We propose a linguistic definition. Roughly speaking a semigroup is word hyperbolic if its multiplication table is a context free language. For groups this definition is equivalent to the original geometric one. We also briefly consider word problems of semigroups.
We study the scale function, space of directions and scale-multiplicative semigroups for restricted Burger-Mozes groups. We relate these general notions to intrinsic properties of the group. Among other things, we give a formula for the scale function; relate the space of directions to both the action on the tree and an action on a $\operatorname{CAT}(0)$ cube complex; and construct maximal scale-multiplicative semigroups from the space of directions.
Group-Joined-Semigroups and their structures
本文研究群-e-半群与同群、类群等代数结构的关系，证明每个群-e-半群都是群-e-同群，并给出其成为群-e-类群的充要条件及若干刻画和例子。
An involution on a semigroup S (or any algebra with an underlying associative binary operation) is a function f:S->S that satisfies f(xy)=f(y)f(x) and f(f(x))=x for all x,y in S. The set I(S) of all such involutions on S generates a subgroup C(S)=<I(S)> of the symmetric group Sym(S) on the set S. We investigate the groups C(S) for certain classes of semigroups S, and also consider the question of which groups are isomorphic to C(S) for a suitable semigroup S.
In this article, we show that a group $G$ is the union of two proper subsemigroups if and only if $G$ has a nontrivial left-orderable quotient. Furthermore, if $G$ is the union of two proper semigroups, then there exists a minimum normal subgroup $N\unlhd G$ for which $G/N$ is left-orderable and nontrivial.
Given an arbitrary group $G$ we construct a semigroup of idempotents (band) $B_G$ with the property that the free idempotent generated semigroup over $B_G$ has a maximal subgroup isomorphic to $G$. If $G$ is finitely presented then $B_G$ is finite. This answers several questions from recent papers in the area.
We answer the question by Matt Brin on the structure of diagram groups over semigroup presentation ${\mathcal P}=\langle a,b,c\mid a=bc,b=ca,c=ab\rangle$. In the talk on Oberwolfach workshop, Brin conjectured that the diagram group over $\mathcal P$ with base $a$ is isomorphic to the generalized Thompson's group $F_9$. We confirm this conjecture and consider some generalizations of this fact.
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…
For a semigroup $S$, the covering number of $S$ with respect to semigroups, $σ_s(S)$, is the minimum number of proper subsemigroups of $S$ whose union is $S$. This article investigates covering numbers of semigroups and analogously defined covering numbers of inverse semigroups and monoids. Our three main theorems give a complete description of the covering number of finite semigroups, finite inverse semigroups, and monoids (modulo groups and infinite semigroups). For a finite semigroup that is neither monogenic nor a group, its covering number is two. For all $n\geq 2$, there exists an invers…
The semigroup $\mathbf{I}\mathbb{N}_{\infty}$ of all partial co-finite isometries of positive integers is studied. We describe Green's relations on the semigroup $\mathbf{I}\mathbb{N}_{\infty}$, its band and proved that $\mathbf{I}\mathbb{N}_{\infty}$ is a simple $E$-unitary $F$-inverse semigroup. We described the least group congruence $\mathfrak{C}_{\mathbf{mg}}$ on $\mathbf{I}\mathbb{N}_{\infty}$ and proved that the quotient-semigroup $\mathbf{I}\mathbb{N}_{\infty}/\mathfrak{C}_{\mathbf{mg}}$ is isomorphic to the additive group of integers. An example of a non-group congruence on the semigr…
For an arbitrary group $G$, it is shown that either the semigroup rank $G{\rm rk}S$ equals the group rank $G{\rm rk}G$, or $G{\rm rk}S = G{\rm rk}G+1$. This is the starting point for the rest of the article, where the semigroup rank for diverse kinds of groups is analysed. The semigroup rank of relatively free groups, for any variety of groups, is computed. For a finitely generated abelian group~$G$, it is proven that $G{\rm rk}S = G{\rm rk}G+1$ if and only if $G$ is torsion-free. In general, this is not true. Partial results are obtained in the nilpotent case. It is also proven that if $M$ is…
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…
We describe how the SgpDec computer algebra package can be used for composing and decomposing permutation groups and transformation semigroups hierarchically by directly constructing substructures of wreath products, the so called cascade products.
In the paper we study the semigroup $\mathscr{C}_{\mathbb{Z}}$ which is a generalization of the bicyclic semigroup. We describe main algebraic properties of the semigroup $\mathscr{C}_{\mathbb{Z}}$ and prove that every non-trivial congruence $\mathfrak{C}$ on the semigroup $\mathscr{C}_{\mathbb{Z}}$ is a group congruence, and moreover the quotient semigroup $\mathscr{C}_{\mathbb{Z}}/\mathfrak{C}$ is isomorphic to a cyclic group. Also we show that the semigroup $\mathscr{C}_{\mathbb{Z}}$ as a Hausdorff semitopological semigroup admits only the discrete topology. Next we study the closure $\oper…
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…
For each group G which decomposes into a finitary direct product of free groups of finite rank we construct a regular band B such that the free idempotent generated semigroup over B contains a maximal subgroup isomorphic to G. In particular, there exists a (regular) band B_0 with the property that any idempotent generated semigroup whose biordered set is isomorphic to that of B_0 must have all its subgroups abelian.
In this paper we prove two new results about closed semigroups in the family of solvable groups H_{mn} that are semidirect products of R^m and R^n, and for which the structure homomorphism maps nontrivially into the center of Aut(R^n). The first result states that the closure of a semigroup generated by a set in H_{mn} that is not included in a maximal semigroup with nonempty interior is actually a group. The second result states that among the subsets of H_{mn} that are not included in a maximal proper semigroup, those that generate H_{mn} as a closed semigroup are dense. Results of this natu…
We introduce a preorder on an inverse semigroup $S$ associated to any normal inverse subsemigroup $N$, that lies between the natural partial order and Green's ${\mathscr J}$-relation. The corresponding equivalence relation $\simeq_N$ is not necessarily a congruence on $S$, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on $S$. We give an appli…
Ideal series of semigroups play an important role in the examination of semigroups which have proper two-sided ideals. But the corresponding theorems cannot be used when left simple (or right simple or simple) semigroups are considered. So it is a natural idea that we try to use the group theoretical methods (instead of the ring theoretical ones) for the examination of these semigroups. The purpose of this paper is to find a suitable type of subsemigroups of left simple semigroups which makes possible for us to generalize some notions (the notion of the normal series and the composition series…
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…
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.
The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup $\mathsf{IG}(\mathcal{E})$ - the `free-est' semigroup with a given biordered set $\mathcal{E}$ of idempotents. We show that when $\mathcal{E}$ is finite, the word problem for $\mathsf{IG}(\mathcal{E})$ is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of ma…
A regular ordered semigroup $S$ is called right inverse if every principal left ideal of $S$ is generated by an $\mathcal{R}$-unique ordered idempotent. Here we explore the theory of right inverse ordered semigroups. We show that a regular ordered semigroup is right inverse if and only if any two right inverses of an element $a\in S$ are $\mathcal{R}$-related. Furthermore, different characterizations of right Clifford, right group-like, group like ordered semigroups are done by right inverse ordered semigroups. Thus a foundation of right inverse semigroups has been developed.
In an earlier work, the author observed that Boolean inverse semi-groups, with semigroup homomorphisms preserving finite orthogonal joins, form a congruence-permutable variety of algebras, called biases. We give a full description of varieties of biases in terms of varieties of groups: (1) Every free bias is residually finite. In particular, the word problem for free biases is decidable. (2) Every proper variety of biases contains a largest finite symmetric inverse semigroup, and it is generated by its members that are generalized rook matrices over groups with zero. (3) There is an order-pres…
For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et. al, we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra.
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.
The structure of the automorphism group of the sandwich semigroup IS_n is described in terms of standard group constructions.
In 1959, P. Hall introduced the locally finite group $\mathcal{U}$, today known as Hall's universal group. This group is countable, universal, simple, and any two finite isomorphic subgroups are conjugate in $\mathcal{U}$. It can be explicitly described as a direct limit of finite symmetric groups. It is homogeneous in the model-theoretic sense since it is the Fraisse limit of the class of all finite groups. Since its introduction Hall's group, and several natural generalisations, have been widely studied. In this article we use a generalisation of Fraisse theory to construct a countable, univ…
We give a survey on results regarding self-similar and automaton presentations of free groups and semigroups and related products. Furthermore, we discuss open problems and results with respect to algebraic decision problems in this area.
The question of computing the group complexity of finite semigroups and automata was first posed in K. Krohn and J. Rhodes, \textit{Complexity of finite semigroups}, Annals of Mathematics (2) \textbf{88} (1968), 128--160, motivated by the Prime Decomposition Theorem of K. Krohn and J. Rhodes, \textit{Algebraic theory of machines, {I}: {P}rime decomposition theorem for finite semigroups and machines}, Transactions of the American Mathematical Society \textbf{116} (1965), 450--464. Here we provide an effective lower bound for group complexity.
Semigroups, groups, and algebras of dynamical origin
本文综述了与具丰富部分对称性对象相关的动力起源代数结构的最新成果，重点详述收缩自相似逆半群的定义构造，辅以黄金分割旋转半群、Penrose铺砌半群等示例说明。
This article studies the properties of word-hyperbolic semigroups and monoids, i.e. those having context-free multiplication tables with respect to a regular combing, as defined by Duncan & Gilman. In particular, the preservation of word-hyperbolicity under taking free products is considered. Under mild conditions on the semigroups involved, satisfied e.g. by monoids or regular semigroups, we prove that the semigroup free product of two word-hyperbolic semigroups is again word-hyperbolic. Analogously, with a mild condition on the uniqueness of representation for the identity element, satisfied…
An example of an extension of a completely simple semigroup U by a group H is given which cannot be embedded into the wreath product of U by H. On the other hand, every central extension of U by H is shown to be embeddable in the wreath product of U by H, and any extension of U by H is proved to be embeddable in a semidirect product of a completely simple semigroup V by H where the maximal subgroups of V are direct powers of those of U.
We study algebraic and topological properties of the convolution semigroups of probability measures on a topological groups and show that a compact Clifford topological semigroup $S$ embeds into the convolution semigroup $P(G)$ over some topological group $G$ if and only if $S$ embeds into the semigroup $\exp(G)$ of compact subsets of $G$ if and only if $S$ is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup $S$ embeds into the functor-semigroup $F(G)$ over a suitable compact topological group $G$ for each weakly normal monadic func…
Semigroup algorithmic problems in metabelian groups
本文证明有限生成亚阿贝尔群的有限生成子半群上恒等元、群和逆元三个问题均可判定，并结合多学科工具建立对应关系，补全相关可判定性图景。
We show that double cosets of the infinite symmetric group with respect to some special subgroups admit natural structures of semigroups. We interpret elements of such semigroups in combinatorial terms (chips, colored graphs, two-dimensional surfaces with polygonal tiling) and describe explicitly the multiplications.
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.
Locally involutive semigroups
引入局部对合半群并嵌入有序广群范畴，推广了逆半群与归纳广群之间的经典ESN对应，并研究了左对合半群的性质及其与逆半群分类拓扑的关系。
Streaming algorithms for groups and semigroups
研究群与半群中字问题的确定性和随机性流算法，提出区分器概念，构造低空间复杂度的区分器，适用于线性半群和某些构造得到的半群，并给出下界证明部分半群不支持亚线性空间区分器。
We describe the $C^*$-algebra of an $E$-unitary or strongly 0-$E$-unitary inverse semigroup as the partial crossed product of a commutative $C^*$-algebra by the maximal group image of the inverse semigroup. We give a similar result for the $C^*$-algebra of the tight groupoid of an inverse semigroup. We also study conditions on a groupoid $C^*$-algebra to be Morita equivalent to a full crossed product of a commutative $C^*$-algebra with an inverse semigroup, generalizing results of Khoshkam and Skandalis for crossed products with groups.
A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. I…
In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups, which are generated by partial, yet invertible automata. We show that there is an automaton-inverse semigroup and, thus, an automaton semigroup with a PSPACE-complete word problem. We also show that there is an automaton group for which the word problem with a single rational constraint is PSPACE-complete. Additionally, we provide simpler constructions for …
Semigroup intersection problems in the Heisenberg groups
本文研究了Heisenberg群中的半群交集问题和轨道交集问题，证明了这两个问题在特定群结构下的可判定性。
The large scale geometry of inverse semigroups and their maximal group images
研究反半群与其最大群像的几何关系，揭示其几何扭曲，并证明E-单位反半群在特定条件下具有Yu的Property A。
Semiring arising as Lattice of Groupsemirings
本文研究由分配格标引的群半环类构造半环，证明若群半环构成强分配格，则所得半环的乘法半群为Clifford半群、加法半群为左正规带，且所有组成群彼此同构。
We show that an automaton group or semigroup is infinite if and only if it admits an $ω$-word (i. e. a right-infinite word) with an infinite orbit, which solves an open problem communicated to us by Ievgen V. Bondarenko. In fact, we prove a generalization of this result, which can be applied to show that finitely generated subgroups and subsemigroups as well as principal left ideals of automaton semigroups are infinite if and only if there is an $ω$ -word with an infinite orbit under their action. The proof also shows some interesting connections between the automaton semigroup and its dual. F…
Quotient of Topological Ternary Semigroup
该研究通过定义同余引入拓扑三元半群的商结构，推导了其成为拓扑三元半群及基础结构为拓扑三元群时商为拓扑三元群的条件。
We introduce the notion of an introverted Boolean algebra $\cal B$ of closed-and-open subsets of a topological group $G$, show that the associated Stone space $(ν_{\cal B} G, ν_{\cal B})$ is a totally disconnected semigroup compactification of $G$, and show that every totally disconnected semigroup compactification of $G$ takes this form. We identify and study the universal totally disconnected semigroup compactification, the universal totally disconnected semitopological semigroup compactification and the universal totally disconnected group compactification of $G$. Our main results are obtai…
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