Ali Ghaffari
Let $M(S)$ be the Banach algebra of all bounded regular Borel measures on a locally compact Hausdorff semitopological semigroup $S$ with variation norm and convolution as multiplication. We obtain necessary and sufficient conditions for $M(S)^{*}$ to have a topologically left invariant mean.
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We study the connection between amenability, Følner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Følner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable i…
On left legal semigroups
该研究聚焦于满足恒等式$aba=ab$的半群，属于数学群论领域对半群代数结构性质的探讨。
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…
In this paper we characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence $\varrho$ on a semigroup $S$, let ${\mathbb F}[\varrho]$ denote the ideal of the semigroup algebra ${\mathbb F}[S]$ which determines the kernel of the extended homomorphism of ${\mathbb F}[S]$ onto ${\mathbb F}[S/\varrho]$ induced by the canonical homomorphism of $S$ onto $S/\varrho$. We examine the right colons $({\mathbb F}[\varrho]:_r{\mathbb F}[S])=\{ a\in {\mathbb F}[S]:\ {\mathbb F}[S…
In this paper, we have introduced the notion of (1,2)-ideal in an LA-semigroup and shown that (1,2)-ideal and two-sided ideal coincide in an intra-regular LA-semigroup. We have characterized an intra-regular LA-semigroup by using the properties of left and right ideals. Some natural examples of LA-semigroups have been given. Further we have investigated some useful conditions for an LA-semigroup to become an intra-regular LA-semigroup and given the counter examples to illustrate the converse inclusions. All the ideals (left, right, two-sided, interior, quasi, bi- generalized bi- and (1,2)) of …
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.
A left almost semigroup (LA-semigroup) or an Abel-Grassmann's groupoid (AG-groupoid) is investigated in several papers. In this paper we have discussed ideals in LA-semigroups. Specifically, we have shown that every ideal in an LA-semigroup S with left identity e is prime if and only if it is idempotent and the set of ideals of S is totally ordered under inclusion. We have shown that an ideal of S is prime if and only if it is semiprime and strongly irreducible. We have proved also that every ideal in a regular LA-semigroup S is prime if and only if the set of ideals of S is totally ordered un…
Left restriction monoids from left $E$-completions
该研究提出左$E$-完备化这一单侧幂等完备化构造，刻画其生成左限制半群的条件，并将其应用于分解三类半群结构，拓展了半群理论中幂等完备化的研究范畴。
The concept of an adequate transversal of an abundant semigroup was introduced by El-Qallali in [8] whilst in [7], he and Fountain initiated the study of quasi-adequate semigroups as natural generalisations of orthodox semigroups. In this work we provide a structure theorem for adequate transversals of certain types of quasi-adequate semigroup and from this deduce Saito's classic result on the structure of inverse transversals of orthodox semigroups. We also apply it to left ample adequate transversals of left adequate semigroups and provide a structure for these based on semidirect products o…
Embedding ample semigroups as (2,1,1)-subalgebras of inverse semigroups
本文研究了将特定类别的充分半群嵌入到逆半群中的(2,1,1)-型子代数问题，并给出了左(右)但非右(左)充分的半群例子。
A partial semigroup is a set with restricted binary operation. In this work we will extend a result due to V. Bergelson and N. Hindman concerning the rich structure presented in the product space of semigroups to partial semigroup. An $IP^{\star}$ set in a semigroup is a set that intersect every set of the form $\left\{ FS(x_{n})_{n=1}^{\infty}:x_{n}\in S\right\} $. V. Bergelson and N. Hindman proved that if $S_{1},S_{2},\ldots,S_{l}$ are finite collection of commutative semigroup, then under certain condition, an $IP^{\star}$ set in $S_{1}\times S_{2}\times\ldots\times S_{l}$ contains cartesi…
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{T}_{n}$ be the semigroup of full transformations on $[n]$. Let $\mathcal{CT}_{n}=\{α\in \mathcal{T}_{n}: (for ~all~x,y\in [n])~\left|xα-yα\right|\leq\left|x-y\right|\}$, then $\mathcal{CT}_{n}$ is a subsemigroup of $\mathcal{T}_{n}$. In this paper, we give a necessary and sufficient condition for an element to be regular and characterize all the Green's equivalences for the semigroup $\mathcal{CT}_{n}$. We further show that the semigroup $\mathcal{CT}_{n}$ is a left abundant semigroup.
Free left $h$-Ehresmann semigroups
本文研究左$h$-埃雷斯曼半群，证明其自由对象与自由左$h$-恰当半群一致，给出其有向树描述并研究相关有限性条件。
In this paper we show that if $Y$ is a subsemilattice of a finite semilattice indecomposable semigroup $S$ then $|Y|\leq 2\left\lfloor \frac{|S|-1}{4}\right\rfloor+1$. We also characterize finite semilattice indecomposable semigroups $S$ which contains a subsemilattice $Y$ with $|S|=4k+1$ and $|Y|=2\left\lfloor \frac{|S|-1}{4}\right\rfloor+1=2k+1$. They are special inverse semigroups. Our investigation is based on our new result proved in this paper which characterize finite semilattice indecomposable semigroups with a zero by only use the properties of its semigroup algebra.
In this article we study left I-orders in the bicyclic monoid $\mathcal{B}$. We give necessary and sufficient conditions for a subsemigroup of $\mathcal{B}$ to be a left I-oreder in $\mathcal{B}$. We then prove that any left I-order in $\mathcal{B}$ is straight.
A subsemigroup $S$ of an inverse semigroup $Q$ is a left I-order in $Q$, if every element in $Q$ can be written as $a^{-1}b$ where $a, b \in S$ and $a^{-1}$ is the inverse of $a$ in the sense of inverse semigroup theory. We study a characterisation of semigroups which have a primitive inverse semigroup of left I-quotients.
We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf's Theorem, stating that a group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.
On right $π$-inverse ordered semigroups
本文引入右 $\pi$-逆序半群等概念，给出左 $\pi$-$t$-单序半群的半格分解，并探讨广义格林关系与相关半格类的关系。
We introduce two natural notions of cogrowth for finitely generated semigroups --- one local and one global --- and study their relationship with amenability and random walks. We establish the minimal and maximal possible values for cogrowth rates, and show that non-monogenic free semigroups are exactly characterised by minimal global cogrowth. We consider the relationship with cogrowth for groups and with amenability of semigroups. We also study the relationship with random walks on finitely generated semigroups, and in particular the spectral radius of the associated Markov operators (when d…
Left reductive regular semigroups
本文研究了左约简正则半群的理想结构理论，建立了其与连接范畴的等价关系，并给出了多种相关半群的构造方法。
It is proved that, given a (von Neumann) regular semigroup with finitely many left and right ideals, if every maximal subgroup is presentable by a finite complete rewriting system, then so is the semigroup. To achieve this, the following two results are proved: the property of being defined by a finite complete rewriting system is preserved when taking an ideal extension by a semigroup defined by a finite complete rewriting system; a completely 0-simple semigroup with finitely many left and right ideals admits a presentation by a finite complete rewriting system provided all of its maximal sub…
The monoid of order isomorphisms between principal filters of $σ{\mathbb{N}^κ}$
本文推广双循环单子，研究集合$σ{\mathbb{N}^κ}$主滤子间序同构半群的代数性质，证明其为双单、$E$-酉、$F$-逆半群，并刻画其Green关系、单位群、极大子群、半直积结构及同余关系。
Given a semigroup S with zero, which is left-cancellative in the sense that st=sr \neq 0 implies that t=r, we construct an inverse semigroup called the inverse hull of S, denoted H(S). When S admits least common multiples, in a precise sense defined below, we study the idempotent semilattice of H(S), with a focus on its spectrum. When S arises as the language semigroup for a subsift X on a finite alphabet, we discuss the relationship between H(S) and several C*-algebras associated to X appearing in the literature.
On the diameter of semigroups of transformations and partitions
本文研究了变换与划分的无限半群的泛同余有限生成性，并确定了全变换、部分变换、单射及满射映射等相关半群的左右直径具体数值。
In the paper we introduce a notion of the Bruck-Reilly $λ$-polycyclic extension of a monoid $S$ with a homomorphism $θ$ which is an analogue of the Bruck-Reilly extension of a monoid $S$. We describe idempotens of the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$ and Green's relations on $\left(\mathscr{P}_λ(θ,S),*\right)$. It is proved that $\left(\mathscr{P}_λ(θ,S),*\right)$ is a $0$-simple semigroup for any semigroup $S$. We find necessary and sufficient conditions on a monoid $S$ and a homomorphism $θ$ under which the semigroup $\left(\mathscr{P}_λ(θ,S),*\right)$ is regular, inverse, $0$-b…
When a semigroup has a unary operation, it is possible to define two binary operations, namely, left and right division. In addition it is well known that groups can be defined in terms of those two divisions. The aim of this paper is to extend those results to other classes of unary semigroups. In the first part of the paper we provide characterizations for several classes of unary semigroups, including (a special class of) E-inversive, regular, completely regular, inverse, Clifford, etc., in terms of left and right division. In the second part we solve a problem that was posed elsewhere. The…
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。
Let $M$ be a cancellative monoid. It is known~\cite{Ta54} that if $M$ is left amenable then the monoid ring $K[M]$ satisfies Ore condition, that is, there exist nontrivial common right multiples for the elements of this ring. In~\cite{Don10} Donnelly shows that a partial converse to this statement is true. Namely, if the monoid $\mathbb Z^{+}[M]$ of all elements of $\mathbb Z[M]$ with positive coefficients has nonzero common right multiples, then $M$ is left amenable. He asks whether the converse is true for this particular statement. We show that the converse is false even for the case of gro…
On semigroups which admit only discrete left-continuous Hausdorff topology
本文研究了允许仅离散左连续Hausdorff拓扑的半群，构造了具有特定拓扑性质的子半群并探讨了其同构嵌入性。
For a numerical semigroup S $\subseteq$ N with embedding dimension e, conductor c and left part L = S $\cap$ [0, c -- 1], set W (S) = e|L| -- c. In 1978 Wilf asked, in equivalent terms, whether W (S) $\ge$ 0 always holds, a question known since as Wilf's conjecture. Using a closely related lower bound W 0 (S) $\le$ W (S), we show that if |L| $\le$ 12 then W 0 (S) $\ge$ 0, thereby settling Wilf's conjecture in this case. This is best possible, since cases are known where |L| = 13 and W 0 (S) = --1. Wilf's conjecture remains open for |L| $\ge$ 13.
Topological embeddings into transformation monoids
本文研究哪些拓扑半群能拓扑嵌入到全变换单形或对称逆单形中，刻画了属于特定类别的拓扑半群的特征，并构造了不嵌入到全变换单形的可数波兰拓扑半群的例子，回答了一个开放问题。
Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup $Sing(X)$ of all non-invertible transformations on a set $X$. The categories involved are characterized as the powerset category $\mathscr{P}(X)$ and the category of partitions $Π(X)$. We describe these categories and show how a permutation on $X$ gives rise to a cross-connection. Further we prove that every cross-connection between them …
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.
Post Clifford semigroups, the Yang-Baxter equation, relative Rota--Baxter Clifford semigroups and dual weak left braces
该研究拓展了Rota–Baxter Clifford半群的性质与结构，引入后Clifford半群等三类推广结构，证明相关范畴的等价关系，并得到它们可提供Yang–Baxter方程集合论解的结论。
Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{P}_{n}$ be the semigroup of partial transformations on $[n]$. Let $\mathcal{CP}_{n}=\{α\in \mathcal{P}_{n}: (for ~all~x,y\in Dom~α)~|xα-yα|\leq|x-y|\}$ be the subsemigroup of partial contraction mappings on $[n]$. We have shown that the semigroup $\mathcal{CP}_{n}$ and some of its subsemigroups are nonregular left abundant semigroups for all $n$ but not right abundant for $n\geq 4$.
In this paper we characterize left(right) ideals, bi-ideals and quasi-ideals of an ordered semigroup by an index $m$ and give some important interplays between these ideals. The concept of m-regularity of an ordered semigroups has been introduced. Moreover m-regular ordered semigroups are characterized by their m-quasi-ideals and the fact that for any m-regular ordered semigroups $A$, the set $Q_{A}$ of all m-quasi-ideals of $A$, with multiplication defined by: $Q_{1}\circ Q_{2}=(Q_{1}Q_{2}]$, for all $Q_{1},Q_{2}\in Q_{A}$, is a m-regular semigroup is obtained here.
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…
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$\left\langle\varpi\right\rangle^1$。
We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and compute normal forms in each of these, and hence also to test whether any given identity holds in the classes of left, right and/or two-sided adequate semigroups.
The left regular band structure on a hyperplane arrangement and its representation theory provide an important connection between semigroup theory and algebraic combinatorics. A finite semigroup embeds in a real hyperplane face monoid if and only if it is in the quasivariety generated by the monoid obtained by adjoining an identity to the two-element left zero semigroup. We prove that this quasivariety is on the one hand polynomial time decidable, and on the other minimally non-finitely based. A similar result is obtained for the semigroups embeddable in complex hyperplane semigroups.
We study the right and left commutation semigroups of finite metacyclic groups with trivial centre. These are presented \[G(m,n,k) = \left\langle {a,b;{a^m} = 1,{b^n} = 1,{a^b} = {a^k}} \right\rangle \quad (m,n,k\in\mathbb{Z}^+)\] with $(m,k - 1) = 1$ and $n = in{d_m}(k),$ the smallest positive integer for which ${k^n} = 1\,\pmod m,$ with the conjugate of $a$ by $b$ written ${a^b}( = {b^{ - 1}}ab).$ The \emph{right} and \emph{left commutation semigroups of} $G,$ denoted ${\rm P}(G)$ and $Λ(G),$ are the semigroups of mappings generated by $ρ(g):G \to G$ and $λ(g):G \to G$ defined by $(x)ρ(g) = …
In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar g…
We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroups with compact maximal subgroups. In particular, we show that a Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroup with a compact maximal subgroup is either compact or topologically isomorphic to the topological sum of its $\mathscr{H}$-classes. We describe the structure of Hausdorff locally compact semitopological $0$-bisimple inverse $ω$-semigroups with a monothetic maximal subgroups. In particular we prove the dichotomy for $T_1$ locally compact semitopological …
In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson's group $\mathcal{F}$. We introduce the notion of unique factorization semigroup which contains Thompson's semigroup $\mathcal{S}$ and the free semigroup $\mathcal{F}_n$ on $n$ generators ($\geq2$). Let $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal{S}$ and $\mathcal{F}_n$, respectively. It is proved that all derivations on $\mathfrak{B}(\mathcal{S})$ and $\mathfrak{B}(\mathcal{F}_n)$ are automatically c…
On non-topologizable semigroups
本文研究了bicyclic半群的非拓扑化性质，构造了具有特定拓扑性质的子半群，并探讨了其拓扑结构的连续性。
Probability inequalities for strongly left-invariant metric semigroups/monoids, including all Lie groups
本研究将Hoffmann-Jorgensen不等式等多个随机不等式及Levy等价性的适用范围从带双不变度量的半群拓展至带强左/右不变度量的半群，覆盖所有李群，并证明了该半群框架与对应幺半群框架等价。
We study the path category of an inverse semigroup admitting unique maximal idempotents and give an abstract characterization of the inverse semigroups arising from zigzag maps on a left cancellative category. As applications we show that every inverse semigroup is Morita equivalent to an inverse semigroup of zigzag maps and hence the class of Cuntz-Krieger $C^*$-algebras of singly aligned categories include the tight $C^*$-algebras of all countable inverse semigroups, up to Morita equivalence.
We study feebly compact shift-continuous $T_1$-topologies on the symmetric inverse semigroup $\mathscr{I}_λ^n$ of finite transformations of the rank $\leqslant n$. For any positive integer $n\geqslant2$ and any infinite cardinal $λ$ a Hausdorff countably pracompact non-compact shift-continuous topology on $\mathscr{I}_λ^n$ is constructed. We show that for an arbitrary positive integer $n$ and an arbitrary infinite cardinal $λ$ for a $T_1$-topology $τ$ on $\mathscr{I}_λ^n$ the following conditions are equivalent: $(i)$ $τ$ is countably pracompact; $(ii)$ $τ$ is feebly compact; $(iii)$ $τ$ is $d…
Let $S$ be a cancellative left-amenable semigroup and let $A$ be a finite set. We prove that every pre-injective cellular automaton $τ\colon A^S \to A^S$ is surjective.
Left regular bands with symmetry
本文研究了左正则带半群代数在有限群作用下的不变子代数表示论，通过等变组合拓扑方法分析其结构与性质。
In this note, we prove that a semigroup $S$ is left amenable if and only if every two nonzero elements of $\ell^1_+(S)$ have a common nonzero right multiple, where $\ell^1_+(S)$ is the positive part of the Banach algebra $\ell^1(S)$, or equivalently the semiring of finite measures on $S$. This characterization of amenability is new even for groups.
Left semibraces, near left semibraces and the Yang-Baxter equation
研究左半括号，证明了在特定条件下$(S, +)$是矩形群，通过右零左半括号和右可消左半括号的广义匹配积得到一般左半括号的结构定理，改进和丰富了相关结果。
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…
Frobenius numbers and the first Hilbert coefficients of certain numerical semigroup rings
本文研究了特定数值半群的弗罗贝尼乌斯数和其环的第一希尔伯特系数，提供了显式计算公式。
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