Maedeh Ghasempour - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Hamid Rasouli - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran
Ali Iranmanesh - Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran
Hasan Barzegar - Department of Mathematics, Tafresh University, Tafresh ۳۹۵۱۸- ۷۹۶۱۱, Iran
Abolfazl Tehranian - Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

This 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 allinclusions from cyclic GHS-subacts into indecomposable ones. Then we introducenew 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 beclassified as semi-injective.

hypermonoid, generalized hyper S-act, Injectivity, C-injectivity, semi-injectivity