Robert D. Gray, Benjamin Steinberg
We give a topological proof that a free inverse monoid on one or more generators is neither of type left-$FP_2$ nor right-$FP_2$. This strengthens a classical result of Schein that such monoids are not finitely presented as monoids.
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There are well known relations between braid groups and symmetric groups, between Artin-Briskorn braid groups and Coxeter groups. Inverse braid monoid the same way is related to the inverse symmetric monoid. In the paper we show that similar relations exist between the inverse braid monoid of type $B$ and the inverse reflection monoid of type $B$. This gives a presentation of the last monoid.
Counting monogenic monoids and inverse monoids
本文研究了全变换单oid的单生成子单oid的数量，证明其等于对称群在1到n点上的循环子群数量之和，并扩展到有限全变换单oid、单生成子半群及有限对称逆单oid的单生成子逆单oid和子半群。
We show how the categorial approach to inverse monoids can be described as a certain endofunctor (which we call the partialization functor) of some category. In this paper we show that this functor can be used to obtain several recently defined inverse monoids, and use it to define a new object, which we call the inverse braid-permutation monoid. A presentation for this monoid is obtained. Finally, we study some abstract properties of the partialization functor and its iterations. This leads to a categorification of a monoid of all order preserving maps, and series of orthodox generalizations …
Braid groups and mapping class groups have many features in common. Similarly to the notion of inverse braid monoid inverse mapping class monoid is defined. It concerns surfaces with punctures, but among given $n$ punctures several can be omitted. This corresponds to braids where the number of strings is not fixed. In the paper we give the analogue of the Dehn-Nilsen-Baer theorem, propose a presentation of the inverse mapping class monoid for a punctured sphere and study the word problem. This shows that certain properties and objects based on mapping class groups may be extended to the invers…
For a topological monoid S the dual inverse monoid is the topological monoid of all identity preserving homomorphisms from S to the circle with attached zero. A topological monoid S is defined to be reflexive if the canonical homomorphism from S to its second dual inverse monoid is a topological isomorphism. We prove that a (compact or discrete) topological inverse monoid S is reflexive (if and) only if S is abelian and the idempotent semilattice of S is zero-dimensional. For a discrete (resp. compact) topological monoid its dual inverse monoid is compact (resp. discrete). These results unify …
$F$-congruences on $F$-inverse monoids
该研究刻画了F-逆幺半群上F-同余对应的核正规系与同余对，还完成了F-无同余的F-逆幺半群的分类，丰富了半群代数理论中F-逆幺半群的同余结构研究。
This note investigates and clarifies some connections between the theory of one-relator groups and special one-relation inverse monoids, i.e. those inverse monoids with a presentation of the form $\operatorname{Inv}\langle A \mid w=1 \rangle$. We show that every one-relator group admits a special one-relation inverse monoid presentation. We subsequently consider the classes ${\rm {\small ANY}}, {\rm {\small RED}}, {\rm {\small CRED}},$ and ${\rm {\small POS}}$ of one-relator groups which can be defined by special one-relation inverse monoid presentations in which the defining word is arbitrary…
The dual symmetric inverse monoid $\mathscr{I}_n^*$ is the inverse monoid of all isomorphisms between quotients of an $n$-set. We give a monoid presentation of $\mathscr{I}_n^*$ and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.
A note on Boolean inverse monoids and ample groupoids
本文研究布尔逆幺半群与 ample 群胚之间的联系，通过具体构造和性质分析，揭示了二者之间的对应关系。
On homological finiteness properties and free inverse monoids
基于幂等格作用构造充分条件，证明自由逆幺半群非FP_2型，并刻画FIM(1)的有限生成子幺半群为FP_2型当且仅当有限表示。
This paper considers the word problem for free inverse monoids of finite rank from a language theory perspective. It is shown that no free inverse monoid has context-free word problem; that the word problem of the free inverse monoid of rank $1$ is both $2$-context-free (an intersection of two context-free languages) and ET0L; that the co-word problem of the free inverse monoid of rank $1$ is context-free; and that the word problem of a free inverse monoid of rank greater than $1$ is not poly-context-free.
The growth of free inverse monoids
研究自由逆半群的指数增长速率，确定其上界，并发现幂等元数量的增长率表达式及极限。
We study the P versus NP problem through properties of functions and monoids, continuing the work of [3]. Here we consider inverse monoids whose properties and relationships determine whether P is different from NP, or whether injective one-way functions (with respect to worst-case complexity) exist.
We revisit our construction of the Thompson groups from the polycyclic inverse monoids in the light of new research. Specifically, we prove that the Thompson group $G_{n,1}$ is the group of units of a Boolean inverse monoid $C_{n}$ called the Cuntz inverse monoid. This inverse monoid is proved to be the tight completion of the polycyclic inverse monoid $P_{n}$. The étale topological groupoid associated with $C_{n}$ under non-commutative Stone duality is the usual groupoid associated with the corresponding Cuntz $C^{\ast}$-algebra. We then show that the group $G_{n,1}$ is also the group of auto…
On groups of units of special and one-relator inverse monoids
该研究探究特殊与单关系逆幺半群的单位群，提出求单位群生成表的新方法并给出其为单关系群的充分条件，同时构造反例证明相关经典结果对逆幺半群一般不成立。
Finite-above inverse monoids are a common generalization of finite inverse monoids and Margolis--Meakin expansions of groups. Given a finite-above $E$-unitary inverse monoid $M$ and a group variety $\mathit{U}$, we find a condition for $M$ and $\mathit{U}$, involving a construction of descending chains of graphs, which is equivalent to $M$ having an $F$-inverse cover via $\mathit{U}$. In the special case where $\mathit{U}=\mathit{Ab}$, the variety of Abelian groups, we apply this condition to get a simple sufficient condition for $M$ to have no $F$-inverse cover via $\mathit{Ab}$, formulated b…
We give a monoid presentation in terms of generators and defining relations for the partial analogue of the finite dual inverse symmetric monoid.
It is well known that under mild conditions on a connected topological space $\mathcal X$, connected covers of $\mathcal X$ may be classified via conjugacy classes of subgroups of the fundamental group of $\mathcal X$. In this paper, we extend these results to the study of immersions into 2-dimensional CW-complexes. An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between CW-complexes is a cellular map such that each point $y \in {\mathcal D}$ has a neighborhood $U$ that is mapped homeomorphically onto $f(U)$ by $f$. In order to classify immersions into a 2-dimensional CW-complex $\mathc…
We give sufficient conditions when a topological inverse $λ$-polycyclic monoid $P_λ$ is absolutely $H$-closed in the class of topological inverse semigroups. Also, for every infinite cardinal $λ$ we construct the coarsest semigroup inverse topology $τ_{mi}$ on $P_λ$ and give an example of a topological inverse monoid S which contains the polycyclic monoid $P_2$ as a dense discrete subsemigroup.
$F$-inverse monoids as algebraic structures in enriched signature
本文研究$F$-逆幺半群在富化签名下的代数结构，通过一元运算刻画其簇性质，并描述了自由$F$-逆幺半群及给定商群条件下的始对象。
We prove that the class of finitely presented inverse monoids whose Schützenberger graphs are quasi-isometric to trees has a uniformly solvable word problem, furthermore, the languages of their Schützenberger automata are context-free. On the other hand, we show that there is a finitely presented inverse monoid with hyperbolic Schützenberger graphs and an unsolvable word problem.
Free inverse monoids are co-context-free
本文利用文法证明了任意有限秩的自由逆幺半群的字问题属于上下文无关补语言，即其补字问题是上下文无关的。
This paper continues the study of Fourier transforms on finite inverse semigroups, with a focus on Fourier inversion theorems and FFTs for new classes of inverse semigroups. We begin by introducing four inverse semigroup generalizations of the Fourier inversion theorem for finite groups. Next, we describe a general approach to the construction of fast inverse Fourier transforms for finite inverse semigroups complementary to an approach to FFTs given in previous work. Finally, we give fast inverse Fourier transforms for the symmetric inverse monoid and its wreath product by arbitrary finite gro…
Maximal Subsemigroups of Infinite Symmetric Inverse Monoids
本文分类了无限集合上对称逆幺半群中包含对称群或其特定子群（如有限子集逐点稳定化子、超滤子稳定化子、等基数划分稳定化子）的所有极大子半群和极大逆子半群。
We prove that the equality problem is decidable for rational subsets of the monogenic free inverse monoid $F$. It is also decidable whether or not a rational subset of $F$ is recognizable. We prove that a submonoid of $F$ is rational if and only if it is finitely generated. We also prove that the membership problem for rational subsets of a finite $\mathcal{J}$-above monoid is decidable, covering the case of free inverse monoids.
Prefix monoids of groups and right units of special inverse monoids
本文完整刻画了有限呈现群的词首幺半群类，证明其与有限生成递归呈现群及自由积的关系，并分类了其单位群与Schützenberger群，同时给出特殊逆幺半群右单位幺半群的类似分类。
It is proved that the periodic point submonoid of a free inverse monoid endomorphism is always finitely generated. Using Chomsky's hierarchy of languages, we prove that the fixed point submonoid of an endomorphism of a free inverse monoid can be represented by a context-sensitive language but, in general, it cannot be represented by a context-free language.
On Special Inverse Monoids with the Strong $F$-Inverse Property
本文研究了具有强F-逆性质的特殊逆单oid，证明其与群G的关联，并提供了一种简化方法。
We curry the elementary arithmetic operations of addition and multiplication to give monotone injections on N, and describe & study the inverse monoids that arise from also considering their generalised inverses. This leads to well-known classic inverse monoids, as well as a novel inverse monoid (the 'arithmetic inverse monoid' A) that generalises these in a natural number-theoretic manner. Based on this, we interpret classic inverse semigroup theoretic concepts arithmetically, and vice versa. Composition and normal forms within A are based on the Chinese remainder theorem, and a minimal gener…
The aim of this paper is to see at what extent homological properties of an inverse monoid are determined from those of its maximum group image. We provide several evidences that the maximum group image contains vital homological information which can be used to study certain properties of the monoid itself. For instance, we prove that an inverse monoid $S$ is of type $FP_{\infty}$, if and only if it contains a minimal idempotent and its maximum group image is of the same type. Regarding cohomological dimensions, we show that the cohomological dimension of a free Clifford monoid and that of it…
We prove the following results: (1) There is a one-relator inverse monoid $\mathrm{Inv}\langle A\:|\:w=1 \rangle$ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The first of these results answers a problem originally posed by Margolis, Meakin and Stephen in 1987.
$E$-unitary and $F$-inverse monoids, and closure operators on group Cayley graphs
本文建立了$E$-酉逆幺半群与$F$-逆幺半群分别等价于群Cayley图上特定闭包算子范畴，并引入$F$-Schützenberger图与$P$-展开以研究其表示。
Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial $n$ may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial monomorphisms of a free group. This gives a solution of the word problem. Another solution is obtained by an approach similar to that of Garside. We give also the analogues of Artin presentation with two generators and Sergiescu graph-presentations.
The word problem of finitely presented special inverse monoids via their groups of units
本文利用关系字因式分解给出有限表示群前缀成员资格可判定的充分条件，据此证明特殊逆幺半群、极大群像与单位群字问题的算法等价性，并应用于多类 E-酉特殊逆幺半群。
An immersion $f : {\mathcal D} \rightarrow \mathcal C$ between $Δ$-complexes is a $Δ$-map that induces injections from star sets of $\mathcal D$ to star sets of $\mathcal C$. We study immersions between finite-dimensional connected $Δ$-complexes by replacing the fundamental group of the base space by an appropriate inverse monoid. We show how conjugacy classes of the closed inverse submonoids of this inverse monoid may be used to classify connected immersions into the complex. This extends earlier results of Margolis and Meakin for immersions between graphs and of Meakin and Szakács on immersi…
We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an example of which now arises as the fundamental groupoid of our version of the Squier complex. A further key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. The relation module of a presentation is then defined as the abelianised kernel of this idempo…
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。
The compressed word problem for a finitely generated monoid M asks whether two given compressed words over the generators of M represent the same element of M. For string compression, straight-line programs, i.e., context-free grammars that generate a single string, are used in this paper. It is shown that the compressed word problem for a free inverse monoid of finite rank at least two is complete for Pi^p_2 (second universal level of the polynomial time hierarchy). Moreover, it is shown that there exists a fixed finite idempotent presentation (i.e., a finite set of relations involving idempo…
A new approach to universal $F$-inverse monoids in enriched signature
本文利用扩展生成集上的 Margolis-Meakin 扩张及对偶闭包算子，证明泛 $F$-逆幺半群 $F(G)$ 是其商逆幺半群，并给出泛性质的简化证明。
We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.
We study a class of inverse monoids of the form M = Inv< X | w=1 >, where the single relator w has a combinatorial property that we call sparse. For a sparse word w, we prove that the word problem for M is decidable. We also show that the set of words in (X \cup X^{-1})^* that represent the identity in M is a deterministic context free language, and that the set of geodesics in the Schutzenberger graph of the identity of M is a regular language.
Forbidden configurations for coherency
本文针对 $E$-Ehresmann 幺半群提出阻止相干性的元素构型，判定多类自由 Ehresmann、逆及 ample 幺半群的非相干性，并证明自由左 Ehresmann 幺半群弱相干且可嵌入 $E$-酉逆幺半群。
Adjoining universal inverses to families of elements of free monoids
本文研究自由幺半群附加泛双侧逆元后的正规形，给出有限集情形的算法及无限集情形的存在性，并提出相关问题。
A slight modification of the definition of the Thompson-Higman groups G_k1 and F_k1 leads to inverse monoids that map onto G_k1 (respectively F_k1), and that have interesting properties: they are finitely generated, and residually finite. These inverse monoids are closely related to the suffix expansion of G_k1 (respectively F_k1).
Maximal subgroups of finitely presented special inverse monoids
本文研究了有限表示特殊逆幺半群的最大子群，证明其可能的最大子群恰好是递归表示群，并指出单位群恰好是有限生成递归表示群，首次揭示了超越单位群的子群结构复杂性。
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 …
We develop the theory of fragile words by introducing the concept of eraser morphism and extending the concept to more general contexts such as (free) inverse monoids. We characterize the image of the eraser morphism in the free group case, and show that it has decidable membership problem. We establish several algorithmic properties of the class of finite-${\cal{J}}$-above (inverse) monoids. We prove that the image of the eraser morphism in the free inverse monoid case (and more generally, in the finite-${\cal{J}}$-above case) has decidable membership problem, and relate its kernel to the fre…
Subgroups of $E$-unitary and $R_1$-injective special inverse monoids
本文研究了$E$-酉和$\mathcal{R}_1$-内射特殊逆幺半群的极大子群结构，证明其极大子群受单位群严格控制，并构造证明了任意有限群均可作为极大子群出现。
Topological finiteness properties of monoids. Part 2: special monoids, one-relator monoids, amalgamated free products, and HNN extensions
本文应用拓扑方法证明特殊幺半群从其单位群继承有限性性质，并据此证明单关系幺半群的 Lyndon 恒等式定理，同时发展 Bass-Serre 理论研究融合自由积与 HNN 扩张的闭包性质。
Geometric Inverse Semigroup Theory: a note on the Milnor-Schwarz Lemma for inverse monoids
本文将Milnor-Schwarz引理推广到作用于路度度空间预层的逆单oid，提供了两种证明方法，一种基于群作用的初等技术，另一种涉及Vietoris-Rips复形。
If $f : \tildeΓ \rightarrow Γ$ is a covering map between connected graphs, and $H$ is the subgroup of $π_1(Γ,v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $ N(H)/H$, where $N(H)$ is the normalizer of $H$ in $π_1(Γ,v)$. We show that an entirely analogous result holds for immersions between connected graphs, where the subgroup $H$ is replaced by the closed inverse submonoid of the inverse monoid $L(Γ,v)$ used to construct the immersion. We observe a relationship between group actions on graphs and deck transformations …
A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question of Brough, Cain & Pfeiffer on whether the group of units of a context-free monoid is always finitely generated, and highlights some of the contrasts between context-free monoids and context-free groups.
Nilpotent groups, solvable groups and factorizable inverse monoids
本文引入逆半群中的次中心与中心幂等元序列，证明了可因子化逆幺半群中此类序列的同构性（类比Jordan-Hölder定理），并据此定义了G-幂零与G-可解逆幺半群，为研究幂零群和可解群提供了新途径。
The word problem for two-generator one-relator inverse monoids
本文研究两生成元单关系逆幺半群的字问题，证明了其可判定与不可判定结果，给出首个不可判定字问题的E-酉两生成元单关系逆幺半群等例子，还给出其可判定字问题的充分条件。
We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (l-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive l-monoids). This contrasts with the fact that, as proved by Repnitskii, there exist inverse-free equations that are valid in all Abelian l-groups but not in all commutative distributive l-monoids, and, as we prove here, there exist inverse-free equations that hold in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a dec…
Short presentations for transformation monoids
本文给出了有限全变换幺半群、对称逆幺半群和部分变换幺半群在对称群表示基础上所需最少附加关系的表示，分别需要4、3和8条附加关系，并针对n≥7、n≥3和n≥7给出了具体表示。
On the growth and integral (co)homology of free regular star-monoids
本文研究自由正则星幺半群的增长率与整系数同调，证明其增长为中间型、非有理非自动，并确定同调群在三维及以上为零，从而推出其不具有同调有限性FP₂。
Commutative decomposition of infinite symmetric groups and transformation monoids
研究无限对称群的可交换子群宽度，证明其最大值为9，改进下界至4，并分析变换单群的可交换子单群宽度，显示其为3，同时指出无限对称逆单群的可交换逆子单群宽度为无限。
Growth and identities of monogenic free adequate monoids
本文研究单生成自由适当幺半群的增长与恒等式，发现左适当情形呈中间增长且富化恒等式与Sylvester幺半群一致，而双侧情形呈指数增长且无非平凡幺半群恒等式。
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.
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