A Characterization of Subdirectly Irreducible Acts

Kozhukhov, I. B.; Haliullina, A. R.

doi:10.17223/20710410/27/1

Cited by 3 publications

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“…In this paper we investigate uniform acts with a fundamental account on the basic notions and their interrelationships with injective and subdirectly irreducible acts. We prove that uniform acts possessing two zeros are subdirectly irreducible and characterize subdirectly irreducible acts over right zero semigroups with a different approach than what is done in [9] (in Russian). It is known that for a uniform module M , an endomorphism is monomorphism if and only if it is not nilpotent.…”

Section: Introductionmentioning

confidence: 93%

“…In [9], subdirectly irreducible acts over right zero semigroups are characterized. In what follows we do the same with another approach.…”

Section: Proofmentioning

confidence: 99%

“…Apparently, the first investigation on subdirectly irreducible acts goes back to 1974, by E.N. Roiz and was followed by some authors for instance Khozhukhov in [6,7,8,9]. Evidently, a pioneering work on uniform acts was initiated by Feller and Gantos in [2] which they characterized injective uniform acts as the ones with a local endomorphism monoid of their injective envelops.…”

Section: Introductionmentioning

confidence: 99%

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On uniform acts over semigroups

Roueentan¹,

Sedaghatjoo

2017

Semigroup Forum

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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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Section: Introductionmentioning

confidence: 93%

“…In [9], subdirectly irreducible acts over right zero semigroups are characterized. In what follows we do the same with another approach.…”

Section: Proofmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On uniform acts over semigroups

Roueentan¹,

Sedaghatjoo

2017

Semigroup Forum

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“…Although, we have characterized, monocyclic right congruences on completely 0-simple semigroups in the case that I is a singleton, it seems the problem of characterizing subdirectly irreducible acts over completely 0-simple semigroups needs more evidences to be dealt with. However, in [7], there is a characterization of subdirectly irreducible class of acts over rectangular bands, in what follows we clarify the structure of such acts explicitly. Thereafter, we characterize an overclass namely uniform acts over rectangular bands.…”

Section: Subdirectly Irreducible and Uniform Acts Over Rectangular Bandsmentioning

confidence: 99%

“…On the other hand, representations of semigroups by transformations of sets (acts over semigroups), have been played an undeniable role in semigroup theory which a great deal of works have been devoted to this area. Thereby, in light of determining any variety of algebras of the same type by its subdirectly irreducible members (Birkhoff's theorem), the class of subdirectly irreducible acts, was a matter of interest in literatures and was followed by some authors for instance Kozhukhov in [4,5,6,7]. Uniform acts that are generalizations of subdirectly irreducible acts, were investigated in a specific case in [1] by Feller and Gantos as a pioneering work whilst the recent work [9], presents a general account on this issue.…”

Section: Introductionmentioning

confidence: 99%

The structure of subdirectly irreducible and uniform acts over rectangular bands

Sedaghatjoo

Roueentan²

2020

Semigroup Forum

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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.

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Acts Over Semigroups

Kozhukhov

Михалев

2023

J Math Sci

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We present a review of results obtained mainly in the last two decades in a number of areas of the theory of acts over semigroups. The authors limited themselves to the structural theory of acts. The acts over completely (0-)simple semigroups, with conditions on the congruence lattice, diagonal acts, biacts and multiacts, and also partial acts are considered. Our work is an expanded version of the report made by the authors in October 2017 at the conference of Institute of Mathematics of Technical University of Berlin, dedicated to the 75th anniversary of Professor Ulrich Knauer, supplemented by results of later works.

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A Characterization of Subdirectly Irreducible Acts

Cited by 3 publications

References 0 publications

On uniform acts over semigroups

On uniform acts over semigroups

The structure of subdirectly irreducible and uniform acts over rectangular bands

Acts Over Semigroups

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