Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 19 1 2023 08 01 On free acts over semigroups and their lattices of radical subacts 81 102 103736 10.48308/cgasa.19.1.81 EN Mohammad Ali Naghipoor Department of Mathematics, Jahrom University, Jahrom, Iran. 0000-0002-2453-9604 Journal Article 2023 05 12 This study aims to investigate free objects in the category of acts over an arbitrary semigroup S. We consider two generalizations of free acts over arbitrary semigroups, namely acts with conditions (F1) and (F2), and give some new results about (minimal) prime subacts and radical subacts of any S-act with condition (F1). Furthermore, some lattice structures for some collections of radical subacts of free S-acts are introduced. We also obtain some results about the relationship between radical subacts of free S-acts and radical ideals of S. Moreover, for any prime ideal P of a semigroup S with a zero, we find a one-to-one correspondence between the collections of P-prime subacts of any two free S-acts. Also, it is shown that all free S-acts have isomorphic lattices of radical multiplication subacts. https://cgasa.sbu.ac.ir/article_103736_9697e63277a2b9f12f17acac93b69599.pdf