Shahid Beheshti UniversityCategories and General Algebraic Structures with Applications2345-585319120230801On C-injective generalized hyper S-acts12714110388110.48308/cgasa.19.1.127ENMaedeh GhasempourDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranHamid RasouliDepartment of Mathematics, Science and Research Branch, Islamic
Azad University, Tehran, IranAli IranmaneshDepartment of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, IranHasan BarzegarDepartment of Mathematics, Tafresh University, Tafresh 39518-
79611, IranAbolfazl TehranianDepartment of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, IranJournal Article20230825This paper explores generalized hyper S-acts (GHS-acts) over a hypermonoid S as generalizations of monoid acts within the context of algebraic hyperstructures. Specifically, we extend the definition of C-injectivity to GHS-acts and investigate their internal and homological properties. It is established that for being GHS-injectivity of GHS-acts with a fixed element, it suffices to consider all<br />inclusions from cyclic GHS-subacts into indecomposable ones. Then we introduce<br />new concepts known as semi-injectivity and semi-C-injectivity. By providing examples, we demonstrate that injectivity and semi-injectivity (C-injectivity and semiC-injectivity) are different concepts for GHS-acts, whereas they are the same in the context of acts over monoids. It is also shown that all pure GHS-acts are injective if and only if all pure cyclic GHS-acts are C-injective. Furthermore, we establish an equivalent condition on a hypermonoid S such that all quotients of SS exhibit semi-injectivity. Finally, we derive an equivalent condition for a hypermonoid to be<br />classified as semi-injective.https://cgasa.sbu.ac.ir/article_103881_92769525788a374436c796f5305133ba.pdf