Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43520171031A radical extension of the category of $S$-sets115311631013ENM.HaddadiFaculty of Mathematics, Statistics and Computer Sciences, Department of Mathematics, Semnan University, Semnan, Iran.M.M.EbrahimiDepartment of Mathematics, Shahid Beheshti University, Tehran 19839, Iran.Journal Article20150204Let 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.<br /><br /> 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.http://bims.iranjournals.ir/article_1013_e39fbaec4f209d34b2b2b5b8fbbd286c.pdf