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Abstract. The paper is devoted to the investigation of uniform notion for acts over semigroups perceived as an overclass of subdirectly irreducible acts. We establish conditions to fill the gap between these classes of acts. Besides we prove that uniform acts with two zeros are subdirectly irreducible. Ultimately we investigate monoids which are uniform as right acts over themselves and we characterize regular ones.

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The main purpose of this paper is investigating classes of acts that are injective relative to all embeddings with indecomposable domains or codomains. We give some homological classifications of monoids in light of such kinds of injectivity. Our approach to indecomposable property provides a new characterization of right absolutely injective monoids as ones that all indecomposable acts are injective.