# Cross-connections of linear transformation semigroup

## TL;DR

This paper applies cross-connection theory to the semigroup of singular linear transformations on a vector space, revealing that despite many categorical cross-connections, only one semigroup up to isomorphism emerges, with potential for interesting subsemigroups.

## Contribution

It extends cross-connection theory to the semigroup of singular linear transformations and identifies the unique semigroup structure arising from these categories.

## Key findings

- Only one semigroup up to isomorphism from the categories
- Duality in cross-connection coincides with algebraic duality
- Construction of interesting subsemigroups of the variant of the linear transformation semigroup

## Abstract

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.