arXiv subjects

P. A. Azeef Muhammed

Publications and source records attributed to P. A. Azeef Muhammed.

16 recordsLinked to original sources

arXiv2026

Maximal subgroups of free projection- and idempotent-generated semigroups with applications to partition monoids

This paper investigates the maximal subgroups of a free projection-generated regular $*$-semigroup $PG(P)$ over a projection algebra $P$, and their relationship to the maximal subgroups of the free idempotent-generated semigroup $IG(E)$ over the corresponding biordered set $E = E(P)$. In the first part of the paper we obtain a number of general presentations by generators and defining relations, in each case reflecting salient combinatorial/topological properties of the groups. In the second part we apply these to explicitly compute the groups when $P = P(P_n)$ and $E = E(P_n)$ arise from the partition monoid $P_n$. Specifically, we show that the maximal subgroup of $PG(P(P_n))$ corresponding to a projection of rank $r\leq n-2$ is (isomorphic to) the symmetric group $S_r$. In $IG(E(P_n))$, the corresponding subgroup is the direct product $Z \times S_r$. The appearance of the infinite cyclic group $Z$ is explained by a connection to a certain twisted partition monoid $P_n^Φ$, which has the same biordered set as $P_n$.

math.GR↗

arXiv2026

Involutions of (twisted) diagram monoids

We classify the involutions of all of the most well-studied diagram monoids -- namely the partition, planar partition, partial Brauer, Motzkin, Brauer and Temperley--Lieb monoids -- and characterise those that give rise to star-regular or regular star-monoid structures. We then complete the same program for the associated twisted diagram monoids, with respect to both the canonical float-counting twisting, and the recently-discovered rank-based twisting. This necessitates developing a general theory of involutions of twisted products. Some of our results were quite unexpected. For example, a Brauer monoid is star-regular with respect to many of its involutions, but only a regular star-monoid for one of them. We will also see that twisted diagram monoids over the integers are always star-regular, thereby providing new and very natural examples of star-regular monoids. Along the way, we also obtain (by necessity) a number of results of independent interest; specifically, we classify the automorphisms of the Motzkin and Temperley--Lieb monoids (and hence also of the planar partition monoids), and we show that all of our diagram monoids generically have trivial centre.

math.RA↗

arXiv2025

Left reductive regular semigroups

In this paper we develop an ideal structure theory for the class of left reductive regular semigroups and apply it to several subclasses of popular interest. In these classes we observe that the right ideal structure of the semigroup is `embedded' inside the left ideal one, and so we can construct these semigroups starting with only one object (unlike in other more general cases). To this end, we introduce an upgraded version of Nambooripad's normal category as our building block, which we call a connected category. The main theorem of the paper describes a category equivalence between the category of left (and right) reductive regular semigroups and the category of connected categories. Then, we specialise our result to describe constructions of L- (and R-) unipotent semigroups, right (and left) regular bands, inverse semigroups and arbitrary regular monoids. Finally, we provide concrete (and rather simple) descriptions to the connected categories that arise from finite transformation semigroups, linear transformation semigroups (over a finite dimensional vector space) and symmetric inverse monoids.

math.GR↗

arXiv2025

Twisted products of monoids

A twisting of a monoid $S$ is a map $Φ:S\times S\to\mathbb{N}$ satisfying the identity $Φ(a,b) + Φ(ab,c) = Φ(a,bc) + Φ(b,c)$. Together with an additive commutative monoid $M$, and a fixed $q\in M$, this gives rise a so-called twisted product $M\times_Φ^qS$, which has underlying set $M\times S$ and multiplication $(i,a)(j,b) = (i+j+Φ(a,b)q,ab)$. This construction has appeared in the special cases where $M$ is $\mathbb{N}$ or $\mathbb{Z}$ under addition, $S$ is a diagram monoid (e.g.~partition, Brauer or Temperley-Lieb), and $Φ$ counts floating components in concatenated diagrams. In this paper we identify a special kind of `tight' twisting, and give a thorough structural description of the resulting twisted products. This involves characterising Green's relations, (von Neumann) regular elements, idempotents, biordered sets, maximal subgroups, Schützenberger groups, and more. We also consider a number of examples, including several apparently new ones, which take as their starting point certain generalisations of Sylvester's rank inequality from linear algebra.

math.GR↗

arXiv2025

Projection algebras and free projection- and idempotent-generated regular $*$-semigroups

The purpose of this paper is to introduce a new family of semigroups - the free projection-generated regular $*$-semigroups - and initiate their systematic study. Such a semigroup $PG(P)$ is constructed from a projection algebra $P$, using the recent groupoid approach to regular $*$-semigroups. The assignment $P\mapsto PG(P)$ is a left adjoint to the forgetful functor that maps a regular $*$-semigroup $S$ to its projection algebra $P(S)$. In fact, the category of projection algebras is coreflective in the category of regular $*$-semigroups. The algebra $P(S)$ uniquely determines the biordered structure of the idempotents $E(S)$, up to isomorphism, and this leads to a category equivalence between projection algebras and regular $*$-biordered sets. As a consequence, $PG(P)$ can be viewed as a quotient of the classical free idempotent-generated (regular) semigroups $IG(E)$ and $RIG(E)$, where $E=E(PG(P))$; this is witnessed by a number of presentations in terms of generators and defining relations. The semigroup $PG(P)$ can also be interpreted topologically, through a natural link to the fundamental groupoid of a simplicial complex explicitly constructed from $P$. The theory is then illustrated on a number of examples. In one direction, the free construction applied to the projection algebras of adjacency semigroups yields a new family of graph-based path semigroups. In another, it turns out that, remarkably, the Temperley-Lieb monoid $TL_n$ is the free regular $*$-semigroup over its own projection algebra $P(TL_n)$.

math.RA↗

arXiv2024

Categorical representation of DRC-semigroups

DRC-semigroups model associative systems with domain and range operations, and contain many important classes, such as inverse, restriction, Ehresmann, regular $*$-, and $*$-regular semigroups. In this paper we show that the category of DRC-semigroups is isomorphic to a category of certain biordered categories whose object sets are projection algebras in the sense of Jones. This extends the recent groupoid approach to regular $*$-semigroups of the first and third authors. We also establish the existence of free DRC-semigroups by constructing a left adjoint to the forgetful functor into the category of projection algebras.

math.RA↗

arXiv2023

A groupoid approach to regular $*$-semigroups

In this paper we develop a new groupoid-based structure theory for the class of regular $*$-semigroups. This class occupies something of a `sweet spot' between the important classes of inverse and regular semigroups, and contains many natural examples. Some of the most significant families include the partition, Brauer and Temperley-Lieb monoids, among other diagram monoids. Our main result is that the category of regular $*$-semigroups is isomorphic to the category of so-called `chained projection groupoids'. Such a groupoid is in fact a triple $(P,\mathcal G,\varepsilon)$, where: $\bullet$ $P$ is a projection algebra (in the sense of Imaoka and Jones), $\bullet$ $\mathcal G$ is an ordered groupoid with object set $P$, and $\bullet$ $\varepsilon:\mathscr C\to\mathcal G$ is a special functor, where $\mathscr C$ is a certain natural `chain groupoid' constructed from $P$. Roughly speaking: the groupoid $\mathcal G=\mathcal G(S)$ remembers only the `easy' products in a regular $*$-semigroup $S$; the projection algebra $P=P(S)$ remembers only the `conjugation action' of the projections of $S$; and the functor $\varepsilon=\varepsilon(S)$ tells us how $\mathcal G$ and $P$ `fit together' in order to recover the entire structure of $S$. In this way, we obtain the first completely general structure theorem for regular $*$-semigroups. As a consequence of our main result, we give a new proof of the celebrated Ehresmann--Schein--Nambooripad Theorem, which establishes an isomorphism between the categories of inverse semigroups and inductive groupoids. Other applications will be given in future works. We consider several examples along the way, and pose a number of problems that we believe are worthy of further attention.

math.RA↗

arXiv2022

Cross-connections in Clifford semigroups

An inverse Clifford semigroup (often referred to as just a Clifford semigroup) is a semilattice of groups. It is an inverse semigroup and in fact, one of the earliest studied classes of semigroups. In this short note, we discuss various structural aspects of a Clifford semigroup from a cross-connection perspective. In particular, given a Clifford semigroup, we show that the semigroup of normal cones is isomorphic to the original semigroup, even when it is not a monoid. Hence, we see that cross-connection description degenerates in Clifford semigroups. Further, we specialise the discussion to provide the description of the cross-connection structure in an arbitrary semilattice, also.

math.GR↗

arXiv2021

Cross-connection structure of locally inverse semigroups

Locally inverse semigroups are regular semigroups whose idempotents form pseudo-semilattices. We characterise the categories that correspond to locally inverse semigroups in the realm of Nambooripad's cross-connection theory. Further, we specialise our cross-connection description of locally inverse semigroups to inverse semigroups and completely 0-simple semigroups, obtaining structure theorems for these classes. In particular, we show that the structure theorem for inverse semigroups can be obtained using only one category, quite analogous to the Ehresmann-Schein-Nambooripad Theorem; for completely 0-simple semigroups, we show that cross-connections coincide with structure matrices, thus recovering the Rees Theorem by categorical tools.

math.GR↗

arXiv2021

A Tale of Two Categories: Inductive groupoids and Cross-connections

A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category whose object set is a strict preorder and the morphisms admit a factorisation property. A pair of `related' normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad \cite{mem,cross} as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.

math.CT↗

arXiv2020

The mathematical work of K.S.S. Nambooripad

We provide an overview of the mathematical work of K.S.S. Nambooripad, with a focus on his contributions to the theory of regular semigroups. In particular, we outline Nambooripad's seminal contributions to the structure theory of regular semigroups via his theory of {\em inductive groupoids}, and also via his theory of {\em cross connections}. We also provide information about outgrowths of his work in the algebraic theory of semigroups and its connections with several other fields of mathematics, in particular with the theory of operator algebras.

math.GR↗

arXiv2019

Cross-connection structure of concordant semigroups

Cross-connection theory provides the construction of a semigroup from its ideal structure using small categories. A concordant semigroup is an idempotent-connected abundant semigroup whose idempotents generate a regular subsemigroup. We characterize the categories arising from the generalised Green relations in the concordant semigroup as consistent categories and describe their interrelationship using cross-connections. Conversely, given a pair of cross-connected consistent categories, we build a concordant semigroup. We use this correspondence to prove a category equivalence between the category of concordant semigroups and the category of cross-connected consistent categories. In the process, we illustrate how our construction is a generalisation of Nambooripad's cross-connection analysis of regular semigroups. We also identify the inductive cancellative category associated with a pair of cross-connected consistent categories.

math.GR↗

arXiv2018

Inductive groupoids and cross-connections of regular semigroups

There are two major structure theorems for an arbitrary regular semigroup using categories, both due to Nambooripad. The first construction using inductive groupoids departs from the biordered set structure of a given regular semigroup. This approach belongs to the realm of the celebrated Ehresmann--Schein--Nambooripad Theorem and its subsequent generalisations. The second construction is a generalisation of Grillet's work on cross-connected partially ordered sets, arising from the principal ideals of the given semigroup. In this article, we establish a direct equivalence between these two seemingly different constructions. We show how the cross-connection representation of a regular semigroup may be constructed directly from the inductive groupoid of the semigroup, and vice versa.

math.GR↗

arXiv2017

Cross-connections of linear transformation semigroup

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 categories and show that although there are many cross-connections, upto isomorphism, we have only one semigroup arising from these categories. But if we restrict the categories suitably, we can construct some interesting subsemigroups of the {variant} of the linear transformation semigroup.

math.RA↗

arXiv2017

Cross-connections and variants of the full transformation semigroup

Cross-connection theory propounded by K. S. S. Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant $\mathscr{T}_X^θ$ of the full transformation semigroup $(\mathscr{T}_X,\cdot)$ for an arbitrary $θ\in \mathscr{T}_X$ is the semigroup $\mathscr{T}_X^θ= (\mathscr{T}_X,\ast)$ with the binary operation $α\ast β= α\cdotθ\cdotβ$ where $α, β\in \mathscr{T}_X$. In this article, we describe the ideal structure of the regular part $Reg(\mathscr{T}_X^θ)$ of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of $Reg(\mathscr{T}_X^θ)$ and describe how they are \emph{cross-connected} by a functor induced by the sandwich transformation $θ$. This lead us to a structure theorem for the semigroup and give the representation of $Reg(\mathscr{T}_X^θ)$ as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup.

math.GR↗

arXiv2017

Cross-connections of the singular transformation semigroup

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 is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to $Sing(X)$. We also describe the right reductive subsemigroups of $Sing(X)$ with the category of principal left ideals isomorphic to $\mathscr{P}(X)$. This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of $Sing(X)$.

math.GR↗