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arXiv · 1701.06098

Cross-connections of linear transformation semigroup

P. A. Azeef Muhammed
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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.

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P. A. Azeef Muhammed. 2017-01-21. Cross-connections of linear transformation semigroup. https://doi.org/10.1007/s00233-018-9942-5

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