Semigroup Forum

, Volume 97, Issue 3, pp 457–470 | Cite as

Cross-connections of linear transformation semigroups

  • P. A. A. MuhammedEmail author
Research Article


Cross-connection theory developed by Nambooripad is the construction of a regular semigroup from its principal left (right) ideals using categories. We use the cross-connection theory to study the structure of the semigroup \(\textit{Sing}(V)\) of singular linear transformations on an arbitrary vector space 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 variants of the linear transformation semigroup.


Regular semigroup Cross-connections Normal category Linear transformation semigroup Dual Variant 



The author is grateful to A. R. Rajan, University of Kerala, Thiruvananthapuram, India for several fruitful discussions during the preparation of this article. I am very grateful to M. V. Volkov, Ural Federal University, Russia for his guidance and comments which helped improve the manuscript considerably. I also thank the referee for his/her careful reading of the paper and encouraging suggestions.


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Authors and Affiliations

  1. 1.Institute of Natural Sciences and MathematicsUral Federal UniversityEkaterinburgRussia
  2. 2.School of MathematicsIndian Institute of Science Education and ResearchThiruvananthapuramIndia

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