Some Properties of Grey Injective S-Acts

In this paper, we extend the concept of grey injective S-acts within the framework of grey systems theory. Grey systems, as introduced by Deng, are a significant tool for handling uncertain information, and this study extends the concept to the domain of S-acts in algebra. We begin by defining grey injective S-acts and show that every grey S-act can be embedded in a grey injective S-act. Also, we investigate the Baer’s lemma for grey injectivity of grey S-acts and apply it in certain implications. Further, the notion of grey divisible S-acts is introduced, and we establish that every grey injective S-act is also a grey divisible S-act. This paper also provides a detailed examination of the injectivity property in the category of grey S-acts, leveraging Skornjakov’s Theorem to demonstrate that the product of a family of grey S-acts is grey injective if and only if each member of the family is grey injective. These findings contribute to a broader understanding of the algebraic structures underpinning grey systems and their applicability in uncertain environments. The theoretical results presented herein are grounded in rigorous proofs and are expected to have implications for future studies in both pure and applied mathematics, particularly in areas requiring the management of imprecise data.