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In this paper we investigate right uniform notion on some classes of semigroups. The main objective of this paper is realizing the structure of regular right uniform semigroups which can be applied as a cornerstone of characterizing regular right subdirectly irreducible semigroups.

The aim of this paper is characterizing right subdirectly irreducible completely 0-simple semigroups. We prove that such semigroups are indeed groups with least nontrivial subgroups. On the other hand we prove that right irreducible completely 0-simple semigroups are groups for which nontrivial subgroups have nontrivial intersection. Ultimately, we characterize the class of uniform acts as an overclass of subdirectly irreducible acts over rectangular bands., Comment: 10 pages

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.

In this paper we prove that for a monoid $S$, products of indecomposable right $S$-acts are indecomposable if and only if $S$ contains a right zero. Besides, we prove that subacts of indecomposable right $S$-acts are indecomposable if and only if $S$ is left reversible. Ultimately, we prove that the one element right $S$-act $\Theta_S$ is product flat if and only if $S$ contains a left zero., Comment: 10 pages

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., Comment: 13 pages