Mark Kambites
We consider the preservation of the properties of automaticity and prefix-automaticity in Rees matrix semigroups over semigroupoids and small categories. Some of our results are new or improve upon existing results in the single-object case of Rees matrix semigroups over semigroups.
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We study the relationship between the loop problem of a semigroup, and that of a Rees matrix construction (with or without zero) over the semigroup. This allows us to characterize exactly those completely zero-simple semigroups for which the loop problem is context-free. We also establish some results concerning loop problems for subsemigroups and Rees quotients.
We introduce a class of finite semigroups obtained by considering Rees quotients of numerical semigroups. Several natural questions concerning this class, as well as particular subclasses obtained by considering some special ideals, are answered while others remain open. We exhibit nice presentations for these semigroups and prove that the Rees quotients by ideals of N, the positive integers under addition, constitute a set of generators for the pseudovariety of commutative and nilpotent semigroups.
Complexity problems associated with finite rings and finite semigroups, particularly semigroups of matrices over a field and the Rees matrix semigroups, are examined. Let M_nF be the ring of n x n matrices over the finite field F and let T_nF be the multiplicative semigroup of n x n matrices over the finite field F. It is proved that for any finite field F and positive integer n >= 2, the polynomial equivalence problem for the T_n F is co-NPcomplete, thus POL-EQ_Σ(M_n F) (POL-EQ_Σis a polynomial equivalence problem for which polynomials are presented as sums of monomials) is also co-NP-complet…
We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the problem well-defined, the uniform word problem for semigroups described by automatic structures is decidable. Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix to…
We describe how to construct all inverse semigroups Morita equivalent to a given inverse semigroup. This is done by taking the maximum inverse images of the regular Rees matrix semigroups over the inverse semigroup where the sandwich matrix satisfies what we call the McAlister conditions.
Commuting graphs of completely simple semigroups
本研究刻画群上Rees矩阵半群的交换图及其多项性质，推导完全简单半群交换图的性质与对应图类，还限定了群交换图部分性质的取值范围。
Primitive representations of finite groups as well as primitive finite groups were classified in the O'Nan-Scott Theorem. In this paper we classify faithful finite primitive semigroup representations. To each finite primitive representation, we associate an invariant, a finite dimensional matirix with entries in a primitive finite groups representation. To a large extent, this matrix determines the representation. In later papers in this series the invariant is further explored. Our primitivity resoullts rely on two main ideas, one of which is a small but important part of the theory of tame a…
Cross-connection theory developed by Nambooripad is the construction of a semigroup from its principal left (right) ideals using categories. We briefly describe the general cross-connection theory for regular semigroups and use it to study the {normal categories} arising from the semigroup $Sing(V)$ of singular linear transformations on an arbitrary vectorspace $V$ over a field $K$. There is an inbuilt notion of duality in the cross-connection theory, and we observe that it coincides with the conventional algebraic duality of vector spaces. We describe various cross-connections between these c…
Using a variant of Schreier's Theorem, and the theory of Green's relations, we show how to reduce the computation of an arbitrary subsemigroup of a finite regular semigroup to that of certain associated subgroups. Examples of semigroups to which these results apply include many important classes: transformation semigroups, partial permutation semigroups and inverse semigroups, partition monoids, matrix semigroups, and subsemigroups of finite regular Rees matrix and $0$-matrix semigroups over groups. For any subsemigroup of such a semigroup, it is possible to, among other things, efficiently co…
Ordered semigroups and ideal categories of principal ideal rings
本文从有序半群和范畴两个视角研究交换主理想环的理想集合，证明了I(R)是正则有序半群，I_R是具有子对象的范畴，并建立了二者之间的对应关系。
A Generalization of the Ehresmann-Schein-Nambooripad Theorem to Two-Sided Ehresmann Semigroupoids
本文提出双侧Ehresmann半群oids的概念，并将其与局部双序Ehresmann范畴相对应，从而推广了Ehresmann-Schein-Nambooripad定理。
Relating ample and biample topological categories with Boolean restriction and range semigroups
本文将Cockett–Garner等价性推广至布尔值域半群，证明其与充足拓扑范畴等价，并由此建立étale布尔值域半群与布尔双限制半群的等价及余支撑运算的刻画。
We fix a path model for the space of filters of the inverse semigroup $\mathcal{S}_Λ$ associated to a left cancellative small category $Λ$. Then, we compute its tight groupoid, thus giving a representation of its $C^*$-algebra as a (full) groupoid algebra. Using it, we characterize when these algebras are simple. Also, we determine amenability of the tight groupoid under mild, reasonable hypotheses.
We study simple and projective modules of a certain class of Ehresmann semigroups, a well-studied generalization of inverse semigroups. Let $S$ be a finite right (left) restriction Ehresmann semigroup whose corresponding Ehresmann category is an EI-category, that is, every endomorphism is an isomorphism. We show that the collection of finite right restriction Ehresmann semigroups whose categories are EI is a pseudovariety. We prove that the simple modules of the semigroup algebra $\Bbbk S$ (over any field $\Bbbk$) are formed by inducing the simple modules of the maximal subgroups of $S$ via th…
Étale categories, restriction semigroups, and their operator algebras
本文定义 étale 范畴与限制半群的算子代数及半叉积代数，解决 Kudryavtseva-Lawson 问题，并证明其推广了 étale 群胚与逆半群的 C*-代数。
This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb categories. If $\mathcal K$ denotes any of these categories, and if $σ\in\mathcal K_{nm}$ is a fixed morphism, then an associative operation $\star_σ$ may be defined on $\mathcal K_{mn}$ by $α\star_σβ=ασβ$. The resulting semigroup $\mathcal K_{mn}^σ=(\mathcal K_{mn},\star_σ)$ is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green's relati…
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 Categorical Realization of the (2-)Category of Monoids via Sch{ü}tzenberger Categories and Strict Factorization Systems
该研究通过舒岑贝格尔范畴与严格分解系统构造与么半群范畴等价的范畴，扩展至含幺半群范畴并转化为2-等价，所得2-范畴可用于研究么半群的Morita等价。
A note on Morita equivalence to graph inverse semigroups
该研究刻画了与图逆半群Morita等价的逆半群，并将相关结论推广到与左可消范畴关联的逆半群情形。
Fix (not necessarily distinct) objects $i$ and $j$ of a locally small category $S$, and write $S_{ij}$ for the set of all morphisms $i\to j$. Fix a morphism $a\in S_{ji}$, and define an operation $\star_a$ on $S_{ij}$ by $x\star_ay=xay$ for all $x,y\in S_{ij}$. Then $(S_{ij},\star_a)$ is a semigroup, known as a sandwich semigroup, and denoted by $S_{ij}^a$. This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green's relations and stability, focusing on the relationships between these properties on $S_{ij…
Fix sets $X$ and $Y$, and write $\mathcal{PT}_{XY}$ for the set of all partial functions $X\to Y$. Fix a partial function $a:Y\to X$, and define the operation $\star_a$ on $\mathcal{PT}_{XY}$ by $f\star_ag=fag$ for $f,g\in\mathcal{PT}_{XY}$. The sandwich semigroup $(\mathcal{PT}_{XY},\star_a)$ is denoted $\mathcal{PT}_{XY}^a$. We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $\mathcal{PT}_{XY}^a$, as well as its regular and idempotent-generated subsemigroups, Reg$(\mathcal{PT}_{XY}^a)$ and $\mathbb E(\mathcal{PT}_{XY}^a)$. After describ…
A perspective on non-commutative frame theory
本文以 restriction quantal frames 为桥梁，建立完全 restriction 幺半群与 étale localic 范畴的对偶，并推广至拓扑范畴及多种半群等价关系。
Global and local triviality for extensions of ample groupoids: an algebraic approach
通过逆半群与丰富群胚的对偶性，证明了扩张中心项在扭曲交叉积下的全局与局部平凡性。
A canonical Makanin-Razborov diagram and a pseudo topology for sets of tuples in free groups, semigroups, associative algebras and Lie algebras I
本文提出统一方法将Makanin-Razborov图与自由群中元组集合关联，证明单端MR图存在典范图，并据此定义伪闭包、秩及典范包络以研究各类代数簇的全局结构。
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