Craig Miller
We call a semigroup $S$ f-noetherian if every right congruence of finite index on $S$ is finitely generated. We prove that every finitely generated semigroup is f-noetherian, and investigate whether the properties of being f-noetherian and being finitely generated coincide for various semigroup classes.
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Inverse graph semigroups were defined by Ash and Hall in 1975. They found necessary and sufficient conditions for the semigroups to be congruence free. In this paper we give a description of congruences on a graph inverse semigroup in terms of the underlying graph. As a consequence, we show that the inverse graph semigroup of a finite graph is congruence Noetherian.
Congruences on a graph inverse semigroup were recently described in terms of the underline graph. Based on such descriptions, we show that the lattice of congruences on a graph inverse semigroup is upper semimodular but not lower semimodular.
We prove that every finite semigroup embeds in a finitely presented congruence-free monoid, and pose some questions around the Boone-Higman Conjecture.
In this paper we characterize the monoid congruences of commutative semigroups by the help of the notion of the separator of subsets of semigroups. We show that every monoid congruence of a commutative semigroup S can be constructed by the help of subsets A of S whose separator is not empty.
This paper enriches the list of known properties of congruence sequences starting from the universal relation and successively performing the operators lower $k$ and lower $t$. Two series of inverse semigroups, namely $\ker{α_n}$-is-Clifford semigroups and $β_n$-is-over-$E$-unitary semigroups, are investigated. Two congruences, namely $α_{n+2}$ and $β_{n+2}$, are found to be the least $\ker{α_n}$-is-Clifford and least $β_n$-is-over-$E$-unitary congruences on $S$, respectively. A new system of implications is established for the quasivarieties of inverse semigroups induced by the min network.
A subset A of a semigroup S is called a medial subset of S if xaby is in A if and only if xbay is in A for every elements x, y, a, b of S. In the paper we show how we can construct the commutative monoid congruences of a semigroup S by the help of medial subsets of S.
This paper enriches the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower $t$ and lower $k$. Three classes of completely regular semigroups, namely semigroups for which $\kerσ$ is a cryptogroup, semigroups for which $\kerν$ is a cryptogroup and semigroups for which $κ$ is over rectangular bands, are studied. $((ω_t)_k)_t$, $((\mathcal{D}_t)_k)_t$ and $((ω_k)_t)_k$ are found to be the least congruences on $S$ such that the quotient semigroups are semigroups for which $\kerσ$ is a cryptogroup, $\kerν$ is a cryp…
Let Γ be a directed graph and Inv(Γ) be the graph inverse semigroup of Γ. Luo and Wang [7] showed that the congruence lattice C(Inv(Γ)) of any graph inverse semigroup Inv(Γ) is upper semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri, Mesyan and Mitchell characterized the directed graph Γ for which C(Inv(Γ)) is lower semimodular [2]. In the present paper, we show that the lower semimodularity, modularity and distributivity in the congruence lattice C(Inv(Γ)) of any graph inverse semigroup Inv(Γ) are equivalent.
Some properties of congruence lattices of path semigroups
本文研究路径半群同余格的性质，建立同余与路径代数理想的一一对应，证明无圈箭图对应的同余格强上半模但未必下半模，并给出模格与分配格的等价条件。
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…
We provide a countable series of bisimple $\mathcal{H}$-trivial finitely presented congruence-free monoids.
In this paper we define a congruence $η^{\ast}$ on semigroups. For the finite semigroups $S$, $η^{\ast}$ is the smallest congruence relation such that $S/η^{\ast}$ is a nilpotent semigroup (in the sense of Malcev). In order to study the congruence relation $η^{\ast}$ on finite semigroups, we define a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup. We prove that, if $S$ is a $\textbf{CS}$-diagonal finite regular Rees matrix semigroup then $S/η^{\ast}$ is inverse. Also, if $S$ is a completely regular finite semigroup, then $S/η^{\ast}$ is a Clifford semigroup. We show that, for ever…
We exhibit an example of a finitely presented monoid that is congruence-free and simple but not bisimple.
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…
We prove that monoids $\mathrm{Mon}\langle a,b,c,d : a^nb=0, ac=1, db=1, dc=1, dab=1, da^2b=1, \ldots, da^{n-1}b=1\rangle$ are congruence-free for all $n\geq 1$. This provides a new countable family of finitely presented congruence-free monoids, bringing one step closer to understanding the Boone--Higman Conjecture. We also provide examples which show that finitely presented congruence-free monoids may have quadratic Dehn function.
A congruence $\varepsilon$ on a semigroup $S$ is perfect if for any congruence classes $x\varepsilon$ and $y\varepsilon$ their product as subsets of $S$ coincides (as a set) with the congruence class $(xy)\varepsilon$. Perfect congruences on the bicyclic semigroup were found in \cite{key7}. Using the structure of bisimple $ω$-semigroups determined in \cite{key25} and the description of congruences on these semigroups found in \cite{key20} and \cite{key1}, we obtain a complete characterization of perfect congruences on all bisimple $ω$-semigroups, substantially generalizing the above mentioned …
On group congruences on the semigroup $\boldsymbol{B}_ω^{\mathscr{F}}$ and its homomorphic retracts in the case when a family $\mathscr{F}$ consists of inductive non-empty subsets of $ω$
针对由 $\omega$ 的归纳非空子集构成的 $\omega$-闭族，本文刻画半群 $\boldsymbol{B}_\omega^{\mathscr{F}}$ 的群同余判定条件，并描述其所有非平凡同态收缩与同构。
In the paper we study inverse semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$ which are generated by partial monotone injective translations of a positive cone of a linearly ordered group $G$. We describe Green's relations on the semigroups $\mathscr{B}(G)$, $\mathscr{B}^+(G)$, $\bar{\mathscr{B}}(G)$ and $\bar{\mathscr{B}}\,^+(G)$, their bands and show that they are simple, and moreover the semigroups $\mathscr{B}(G)$ and $\mathscr{B}^+(G)$ are bisimple. We show that for a commutative linearly ordered group $G$ all non-trivial congruences …
For a given inverse semigroup, one can associate an étale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated étale groupoids. In this paper, we focus on congruences of inverse semigroups, which is a fundamental concept to consider quotients of inverse semigroup. We prove that a congruence of an inverse semigroup induces a closed invariant set and a normal subgroupoid of the universal groupoid. Then we show that the universal groupoid associated to a quotient inverse semigroup is described by the restriction and quo…
Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.
We show that every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) has linear growth. This implies that the the corresponding semigroup algebra is a PI algebra.
Every semigroup which is a finite disjoint union of copies of the free mono- genic semigroup (natural numbers under addition) has soluble word prob- lem and soluble membership problem. Efficient algorithms are given for both problems.
Given any directed graph E one can construct a graph inverse semigroup G(E), where, roughly speaking, elements correspond to paths in the graph. In this paper we study the semigroup-theoretic structure of G(E). Specifically, we describe the non-Rees congruences on G(E), show that the quotient of G(E) by any Rees congruence is another graph inverse semigroup, and classify the G(E) that have only Rees congruences. We also find the minimum possible degree of a faithful representation by partial transformations of any countable G(E), and we show that a homomorphism of directed graphs can be extend…
Abelianness and centrality in inverse semigroups
本文研究了逆半群中交换性和中心性的概念，证明了可解和 nilpotent 逆半群实际上为群
A congruence on an inverse semigroup $S$ is determined uniquely by its kernel and trace. Denoting by $ρ_k$ and $ρ_t$ the least congruence on $S$ having the same kernel and the same trace as $ρ$, respectively, and denoting by $ω$ the universal congruence on $S$, we consider the sequence $ω$, $ω_k$, $ω_t$, $(ω_k)_t$, $(ω_t)_k$, $((ω_k)_t)_k$, $((ω_t)_k)_t$, $\cdots$. The quotients $\{S/ω_k\}$, $\{S/ω_t\}$, $\{S/(ω_k)_t\}$, $\{S/(ω_t)_k\}$, $\{S/((ω_k)_t)_k\}$, $\{S/((ω_t)_k)_t\}$, $\cdots$, as $S$ runs over all inverse semigroups, form quasivarieties. This article explores the relationships amon…
Computing congruences of finite inverse semigroups
本文提出了一种高效算法，用于计算逆半群的同余，利用群论、自动机和逆半群理论中的多种技术，显著提升了计算效率。
Finite basis problem for involution semigroups of order four
本文证明了四阶反交换半群均为有限基，从而确定五阶为首个非有限基的反交换半群。
We prove that the semigroup generated by a finite state Mealy automaton $\mathcal{A}=(Q,A,τ)$ is infinite if and only if there exists some right-infinite word in the alphabet $A$ with infinite orbit.
Semigroup Congruences and Subsemigroups of the Direct Square
本文研究了DSC半群的性质，证明了除群外，其他广泛类别的半群中不存在DSC半群，但通过特定构造得到了非群的DSC半群例子。
An algebraic structure is said to be congruence permutable if its arbitrary congruences $α$ and $β$ satisfy the equation $α\circ β=β\circ α$, where $\circ$ denotes the usual composition of binary relations. For an arbitrary $G$-set $X$ with $G\cap X=\emptyset$, we define a semigroup $(G,X,0)$ with a zero $0$ ($0\notin G\cup X$), and give necessary and sufficient conditions for the congruence permutability of the $G$-set $X$ by the help of the semigroup $(G,X,0)$.
We built some congruences on semigroups, from where a decomposition of quasi-separative semigroups was obtained.
Howson property and finitely generated intersection problem for monogenic inverse semigroups
研究单演逆半群作为半群时的豪森性质，除单演自由逆半群外都具有该性质，且对于此特殊情况，判定两个有限生成子半群的交是否有限生成的问题可算法求解。
On Finite Presentability of Subsemigroups of the Monogenic Free Inverse Semigroup
本文推广了 Schein 关于单基因自由逆半群不可有限表现的经典结果，证明了其有限生成子半群有限可表现当且仅当仅含有限多个幂等元，并由此刻画了有限可表现的逆子半群恰为有限半格。
The congruence subgroup problem for a finitely generated group $Γ$ and $G\leq Aut(Γ)$ asks whether the map $\hat{G}\to Aut(\hatΓ)$ is injective, or more generally, what is its kernel $C\left(G,Γ\right)$? Here $\hat{X}$ denotes the profinite completion of $X$. In the case $G=Aut(Γ)$ we denote $C\left(Γ\right)=C\left(Aut(Γ),Γ\right)$. Let $Γ$ be a finitely generated group, $\barΓ=Γ/[Γ,Γ]$, and $Γ^{*}=\barΓ/tor(\barΓ)\cong\mathbb{Z}^{(d)}$. Denote $Aut^{*}(Γ)=\textrm{Im}(Aut(Γ)\to Aut(Γ^{*}))\leq GL_{d}(\mathbb{Z})$. In this paper we show that when $Γ$ is nilpotent, there is a canonical isomorphi…
Using Rees index, the subsemigroup growth of free semigroups is investigated. Lower and upper bounds for the sequence are given and it is shown to have superexponential growth of strict type $n^n$ for finite free rank greater than 1. It is also shown that free semigroups have the fastest subsemigroup growth of all finitely generated semigroups. Ideal growth is shown to be exponential with strict type $2^n$ and congruence growth is shown to be at least exponential. In addition we consider the case when the index is fixed and rank increasing, proving that for subsemigroups and ideals this sequen…
An algebra has the Howson property if the intersection of any two finitely generated subalgebras is finitely generated. A simple necessary and sufficient condition is given for the Howson property to hold on an inverse semigroup with finitely many idempotents. In addition, it is shown that any monogenic inverse semigroup has the Howson property.
Following the work done by Olshanskii for groups, we describe, for a given semigroup $S$, which functions $l : S \rightarrow \mathbb{N}$ can be realized up to equivalence as length functions $g \mapsto |g|_{H}$ by embedding $S$ into a finitely generated semigroup $H$. We also, following the work done by Olshanskii and Sapir, provide a complete description of length functions of a given finitely generated semigroup with enumerable set of relations inside a finitely presented semigroup.
Semigroups whose right ideals are finitely generated
本文研究右理想有限生成的半群（弱右诺特半群），给出其等价刻画，并探讨其在商、子半群、直积、正则半群及强半格等构造下的性质，最终证明交换半群弱右诺特的充要条件。
In the paper the notion of a star partial homeomorphism of a finite dimensional Euclidean space $\mathbb{R}^n$ is introduced. We describe the structure of the semigroup $\mathbf{PStH}_{\mathbb{R}^n}$ of star partial homeomorphisms of the space $\mathbb{R}^n.$ The structure of the band of $\mathbf{PStH}_{\mathbb{R}^n}$ and Green's relations on $\mathbf{PStH}_{\mathbb{R}^n}$ are described. We show that $\mathbf{PStH}_{\mathbb{R}^n}$ is a bisimple inverse semigroup and every non-unit congruence on $\mathbf{PStH}_{\mathbb{R}^n}$ is a group congruence.
For semigroup $S$, a commutative congruence $σ_{orient}$ on $S$ and a subsemigroup Orientable($S$) of $S$ were introduced in "Two cancellative commutative congruences and group diagrams", Semigroup Forum (2011) 82: 338-353. Here we demonstrate that when the semigroup is in fact a group $G$, then Orientable($G$) is the commutator subgroup $[G,G]$ and $ G / σ_{orient}$ is the abelian quotient group $G / [G,G]$.
A semigroup $S$ is right noetherian if every right congruence on $S$ is finitely generated. In this paper we present some fundamental properties of right noetherian semigroups, discuss how semigroups relate to their substructures with regard to the property of being right noetherian, and investigate whether this property is preserved under various semigroup constructions.
We study the semigroup $\mathscr{I}^{\mathrm{cf}}_λ$ of injective partial cofinite selfmaps of an infinite cardinal $λ$. We show that $\mathscr{I}^{\mathrm{cf}}_λ$ is a bisimple inverse semigroup and each chain of idempotents in $\mathscr{I}^{\mathrm{cf}}_λ$ is contained in a bicyclic subsemigroup of $\mathscr{I}^{\mathrm{cf}}_λ$, we describe the Green relations on $\mathscr{I}^{\mathrm{cf}}_λ$ and we prove that every non-trivial congruence on $\mathscr{I}^{\mathrm{cf}}_λ$ is a group congruence. Also, we describe the structure of the quotient semigroup $\mathscr{I}^{\mathrm{cf}}_λ/σ$, where $σ…
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…
Comparing Numbers of Diagonal Subsemigroups and Congruences for Semigroups
本文研究了半群中对角子半群与同余数量的关系，证明对于任意0<α≤1的有理数，存在半群使得DSC系数为α。
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。
This paper studies automatic structures for subsemigroups of Baumslag--Solitar semigroups (that is, semigroups presented by $\ < x,y \mid (yx^m, x^ny)\ >$, where $m$ and $n$ are natural numbers). A geometric argument (a rarity in the field of automatic semigroups) is used to show that if $m \gt n$, all of the finitely generated subsemigroups of this semigroup are [right-] automatic. If $m \lt n$, all of its finitely generated subsemigroups are left-automatic. If $m = n$, there exist finitely generated subsemigroups that are not automatic. An appendix discusses the implications of these results…
Taking residual finiteness as a starting point, we consider three related finiteness properties: weak subsemigroup separability, strong subsemigroup separability and complete separability. We investigate whether each of these properties is inherited by Schützenberger groups. The main result of this paper states that for a finitely generated commutative semigroup $S$, these three separability conditions coincide and are equivalent to every $\mathcal{H}$-class of $S$ being finite. We also provide examples to show that these properties in general differ for commutative semigroups and finitely gen…
In this paper, we introduce (Rees) artinian S-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that artinian S-acts are those acts whose all factor acts are finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian S-acts, namely, those acts whose (Rees) congruences are finitely generated. Moreover, as a useful tool for the investigation of chain conditions, we prove that the properties of being (Rees) noetherian and artinian are inherited in Rees short exact sequences. Next, we specifically consider the chain …
In this work, the lattice of varieties of semigroups and the lattice of varieties of I-semigroups (a common setting for both the variety of completely regular semigroups and the variety of inverse semigroups) are studied with respect to the following concepts: a variety V of (I-)semigroups is said to be locally K-finite, where K stands for any of the five Green's relations, if every finitely generated semigroup from V has only finitely many (distinct) K-classes.
A semigroup $S$ is called a permutable semigroup if $α\circ β=β\circ α$ is satified for all congruences $α$ and $β$ of $S$. A semigroup is called a Putcha semigroup if it is a semilattice of archimedean semigroups. In this paper we deal with finite permutable Putcha semigroups. We describe the finite permutable archimedean semigroups and finite permutable semigroups which are semilattices of a group and a nilpotent semigroup.
We introduce the space function $s(n)$ of a finitely presented semigroup $S =<A\mid R>.$ To define $s(n)$ we consider pairs of words $w,w'$ over $A$ of length at most $n$ equal in $S$ and use relations from $R$ for the transformations $w=w_0\to...\to w_t= w'$; $s(n)$ bounds from above the tape space (or computer memory) sufficient to implement all such transitions $w\to...\to w'.$ One of the results obtained is the following criterion: A finitely generated semigroup $S$ has decidable word problem of polynomial space complexity if and only if $S$ is a subsemigroup of a finitely presented semigr…
The index of a subgroup of a group counts the number of cosets of that subgroup. A subgroup of finite index often shares structural properties with the group, and the existence of a subgroup of finite index with some particular property can therefore imply useful structural information for the overgroup. A developed theory of cosets in inverse semigroups exists, originally due to Schein: it is defined only for closed inverse subsemigroups, and the structural correspondences between an inverse semigroup and a closed inverse subsemigroup of finite index are weaker than in the group case. Neverth…
A semigroup variety V is said to be locally K-finite, where K stands for any of Green's relations H, R, L, D, or J, if every finitely generated semigroup from V has only finitely many K-classes. We characterize locally K-finite varieties of finite axiomatic rank in the language of "forbidden objects".
This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley--Lieb and Jones partitions; linear and projec…
A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard algebraic and semigroup theoretic constructions: subsemigroups, homomorphic images, direct products, Rees matrix semigroups, including Brandt semigroups, and Bruck--Reilly extensions. We also investigate the relationship with the property of being weakly right noetherian, which requires all right ideals of $S$ to be finitely generated.
We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that the 9-element monoid $L_4^1$ is non-finitely based. The monoid $L_4^1$ was the only unsolved case in the finite basis problem for Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new …
The complex algebra of an inverse semigroup with finitely many idempotents in each $\mathcal D$-class is stably finite by a result of Munn. This can be proved fairly easily using $C^*$-algebras for inverse semigroups satisfying this condition that have a Hausdorff universal groupoid, or more generally for direct limits of inverse semigroups satisfying this condition and having Hausdorff universal groupoids. It is not difficult to see that a finitely presented inverse semigroup with a non-Hausdorff universal groupoid cannot be a direct limit of inverse semigroups with Hausdorff universal groupo…
Conjugacy and Least Commutative Congruences in Semigroups
本文将半群初级共轭推广为因子排列诱导的关系，证明其传递闭包是最小交换同余，并刻画其在多类半群中的等价类。
Not all nilpotent monoids are finitely related
该研究探讨幂零幺半群的有限相关性问题，证明4-幂零幺半群均为有限相关，给出首个5-幂零非有限相关幺半群示例，还发现有限相关半群的子半群、同态像及Rees商可非有限相关。
Congruences of maximum regular subsemigroups of variants of finite full transformation semigroups
本文研究有限全变换半群变体中极大正则子半群的同余格，通过将其分解为 $Cong(T_r)$ 与组合系统的等价关系格的次直积，给出同余的显式分类及格的高的公式。
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