On prime and radical subacts of acts over semigroups

Abstract

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.

Keywords:

• Prime subact
• radical subact
• S-act
• Zariski lattice

2020 Mathematics Subject Classification:

• Primary: 20M30; 20M50

Acknowledgment

The author would like to express his appreciation to the anonymous referees for their helpful comments and constructive suggestions.