The Structure of Regular Right Uniform Semigroups
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The Structure of Regular Right Uniform Semigroups Samira Hosseinzadeh Alikhalaji1 · Mojtaba Sedaghatjoo2 Mohammad Roueentan3
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Received: 23 May 2020 / Revised: 14 October 2020 / Accepted: 20 October 2020 © Iranian Mathematical Society 2020
Abstract In this paper, we investigate the right uniform notion on some classes of semigroups. The main objective of this paper is identifying the structure of regular right uniform semigroups which can be applied as a cornerstone of characterizing regular right subdirectly irreducible semigroups. Keywords Uniform semigroup · Regular uniform semigroup · Strongly right Noetherian uniform semigroup · Subdirectly irreducible Mathematics Subject Classification 20M30 · 20M17 · 08B26
1 Introduction and Preliminaries Taking inspiration from uniform modules, investigation on uniform acts over semigroups was initiated by Feller and Gantos in the category of Act0 -S ([5]). In a recent work on uniform acts [10], an introductory account on uniform acts as a generalization of (subdirectly) irreducible acts, in the category Act-S is presented. A semigroup S is called right uniform if it is uniform as a right S-act over itself. In this paper, structures of some classes of right uniform semigroups are investigated. The content
Communicated by Mohammad B. Asadi.
B
Mojtaba Sedaghatjoo [email protected] Samira Hosseinzadeh Alikhalaji [email protected] Mohammad Roueentan [email protected]
1
Science and Research Branch, Islamic Azad University, Tehran, Iran
2
Department of Mathematics, Faculty of Intelligent Systems, Engineering and Data Science, Persian Gulf University, Bushehr, Iran
3
College of Engineering, Lamerd Higher Education Center, Lamerd, Iran
123
Bulletin of the Iranian Mathematical Society
of the paper is organized in three sections. The first section is allocated to preliminaries and terminology needed in the sequel. Section 2 contains general results on right uniform semigroups, mainly, we identify conditions under which right uniformity is transferred from a semigroup S to S 1 (S 0 ) and vice versa. Moreover, we prove that the set of idempotents of a right uniform semigroup S is a left zero or a right zero subsemigroup, and hence any right uniform semigroup is an E-semigroup. Section 3 is devoted to investigating right uniformity on some classes of semigroups, in particular, we identify the structure of regular right uniform semigroups which can be used in the characterization of regular right subdirectly irreducible semigroups. Besides, we identify the structure of right uniform semigroups belonging to some subclasses of regular semigroups. Ultimately, we summarize all characterized classes of the right uniform semigroups in a table. Throughout this paper, S will denote an arbitrary semigroup which is not a singleton. To every semigroup S we can associate the monoid S 1 with identity element 1 adjoined if necessary: S = 1
S if S has an identity element, S ∪ {1} otherwise.
In a similar fashion to every semigroup S we can associate
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