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.
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.2010 Mathematics Subject Classification. 20M20, 20M30. Key words and phrases. completely 0-simple semigroup, rectangular band, right congruence, right subdirectly irreducible semigroup, uniform act.Definition 2.1. For a semigroup S, an S-act A is called unif orm if every nonzero subact is large in A. Also a semigroup S is called right (left) uniform if the right (left) S-act S S ( S S) is uniform.Definition 2.2. A subset B of a right S-act A is called separated, provided that for a = b ∈ B there exists s ∈ S\{1} such that as = bs.The next two results, stated below, are preliminary results of uniform acts needed in the next argument.Corollary 2.3. [9, Corollary 3.15] If S is a right uniform semigroup and xy = y for x, y ∈ S, then x is a left identity or y is a left zero.
Abstract. In this article we introduce the notion of fully idempotent acts and by this we give a characterization of commutative monoids over which all cyclic right acts are injective.
In this paper, we introduce (Rees) artinian S-acts as those acts that satisfy the descending chain condition on their (Rees) congruences. Then, we show that artinian S-acts are those acts whose all factor acts are finitely (Rees) cogenerated. In addition, we continue the study of (Rees) noetherian S-acts, namely, those acts whose (Rees) congruences are finitely generated. Moreover, as a useful tool for the investigation of chain conditions, we prove that the properties of being (Rees) noetherian and artinian are inherited in Rees short exact sequences. Next, we specifically consider the chain conditions on monoids. Finally, we prove that every right Rees artinian, commutative monoid is right Rees noetherian.
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