# A radical extension of the category of $S$-sets

Document Type : Research Paper

Authors

1 Faculty of Mathematics, Statistics and Computer Sciences, Department of Mathematics, Semnan University, Semnan, Iran.

2 Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran.

Abstract

Let S-Set be the category of $S$-sets, sets together with the actions of a semigroup $S$ on them. And, let S-Pos be the category of $S$-posets, posets together with the actions compatible with the orders on them. In this paper we show that the category S-Pos is a radical extension of S-Set; that is there is a radical on the category S-Pos, the order desolator radical, whose torsion-free class is S-Set.

To do this, first we give a precise definition of a radical on the category S-Posand construct some functors between the above mentioned categories and finally we show that  S-Pos is a radical extension of S-Set.

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### History

• Receive Date: 04 February 2015
• Revise Date: 09 May 2016
• Accept Date: 10 May 2016