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This study aims at giving some new results about the prime and radical subacts of any act over a semigroup S. The notion of prime radical is used to define a closure operator in the category of S-acts, some categorical properties of which are studied. Moreover, a lattice structure is introduced for the collection of radical subacts of an S-act. Also, two different types of Zariski spaces of prime varieties in the category of acts over a semigroup are presented. It is shown that a certain type of Zariski space of acts is isomorphic to a lattice of radical subacts. [ABSTRACT FROM AUTHOR]