Small Pseudo Principally Quasi-Injective Acts

This work aims to present a small pseudo principally quasi-injective acts as a novel generalization of pseudo principally quasi-injective S-acts over monoids. If each S-monomorphism from a small principal subact of an S-act MS into NS can be extended to S-homomorphism from MS into NS , an S-act NS is termed as a small pseudo principally M-injective. If an S-act MS is a small pseudo principally M-injective, it is called a small pseudo principally quasi-injective. This type of generalization has several properties. Additionally, the circumstances under which subacts inherit the Small Pseudo, principally quasi-injective acts, are studied. Examples are provided to demonstrate this concept. Small pseudo principally quasi-injective acts can coincide with small principally quasi-injective acts if certain criteria are met. We discuss new characterizations of small principally quasi-injective acts. The link between classes of small, principally quasi-injective acts with other classes of injectivity is shown.