Cross-connections of 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.

References in corpus (4)

• Variants of finite full transformation semigroups
• Cross-connections of completely simple semigroups
• Cross-connections of the singular transformation semigroup
• Cross-connections and variants of the full transformation semigroup

Cited by in corpus (2)

• Inductive groupoids and cross-connections of regular semigroups
• Cross-connections of the singular transformation semigroup