on c-injective generalized hyper s-acts

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.