Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801On free acts over semigroups and their lattices of radical subacts8110210373610.48308/cgasa.19.1.81ENMohammad Ali NaghipoorDepartment of Mathematics, Jahrom University, Jahrom, Iran.0000-0002-2453-9604Journal Article20230512This 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_312f4bc60914a9c197921bed98249f59.pdf