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Cross-connections of the singular transformation semigroup

School of Mathematics, Indian Institute of Science Education and Research, Thiruvananthapuram, Kerala-695016, India

Corresponding author.

Present address: Institute of Natural Sciences and Mathematics, Ural Federal University, Lenina 51, 620000 Ekaterinburg, Russia.

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and

A. R. Rajan

Department of Mathematics, University of Kerala, Thiruvananthapuram, Kerala-695581, India

E-mail Address: arrunivker@yahoo.com

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https://doi.org/10.1142/S0219498818500470Cited by:6 (Source: Crossref)

Abstract

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 $Sing (X)$ $Sing (X)$ of all non-invertible transformations on a set $X$ . The categories involved are characterized as the powerset category $𝒫 (X)$ 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 $𝒫 (X)$ . This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of $Sing (X)$ .

Communicated by A. Facchini

Keywords:

AMSC: 20M10, 20M17, 20M50

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