arXiv subjects

A. R. Rajan

Publications and source records attributed to A. R. Rajan.

4 recordsLinked to original sources

arXiv2024

Semiring arising as Lattice of Groupsemirings

Much study has been done on semigroups which are unions of groups. There are several ways in which a union of groups can be made into a semigroup in which each of the component groups arises as subgroups of the constructed semigroup. An important class of such unions is a semilattice of groups. Group semirings are semirings $(G,+,\cdot )$ where $(G,\cdot )$ is a group and $(G,+)$ is a left zero semigroup. We consider construction of semirings from classes of group semirings $\{G_α:α\in D \}$ indexed by a distributive lattice $D$. It is shown that if $S=\cup\{G_α\}$ is a strong distributive lattice of group semirings $G_α$ then the multiplicative semigroup $(S,\cdot)$ of the semiring $(S,+,\cdot)$ is a Clifford semigroup and the additive semigroup $(S,+)$ is a left normal band. Further in this case all the groups $G_α$ are mutually isomorphic.

math.GR↗

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↗

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↗

arXiv2017

Normal category of partitions of a set

Let $T_X$ be the semigroup of all non-invertible transformations on an arbitrary set $X$. It is known that $T_X$ is a regular semigroup. The principal right(left) ideals of a regular semigroup $S$ with partial left(right) translations as morphisms form a normal category $\mathcal{R}( S )$($\mathcal{L}( S )$). Here we consider the category $Π(X)$ of partitions of a set $X$ and show that it admits a normal category structure and that $Π(X)$ is isomorphic to the category $\mathcal{R}( T_X )$. We also consider the normal dual $N^\ast \mathscr{P}(X)$ of the power-set category $\mathscr{P}(X)$ associated with $X$ and show that $N^\ast \mathscr{P}(X)$ is isomorphic to the partition category - $Π(X)$ of the set $X$.

math.GR↗