morphic s-acts over commutative monoids with unique zero

In this paper, we introduce and investigate the notion of morphic s-acts over commutative monoids, focusing on the category of right s-acts that possess a unique zero element. inspired by the concept of morphic modules introduced by el mehdi bouba, we define an s-act to be morphic if, for each element, its orbit corresponds to the annihilator of some element of the monoid, and the annihilator in the monoid satisfies a specific structural form. we examine how this property behaves under standard constructions, such as subacts, homomorphic images, preimages, direct sums, pullbacks, and rees quotients. several preservation results are established, and conditions ensuring the morphic property are identified. throughout, we provide illustrative examples to clarify the theory and demonstrate its scope within the framework of commutative monoids.