On generalized bi-\(\Gamma\)-ideals in \(\Gamma\)-semigroups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1347.20074
author: Abul Basar
author: M. Y. Abbasi
publication date: 2016-04-13T00:00:00Z
Mathematics Subject Classification ID: 20M99
zbMATH DE Number: 6568360
zbMATH Keywords: \(\Gamma\)-semigroups
zbMATH Keywords: prime bi-ideals
zbMATH Keywords: semiprime bi-ideals
MaRDI profile type: Publication
title: On generalized bi-\(\Gamma\)-ideals in \(\Gamma\)-semigroups.
published in: Quasigroups and Related Systems
review text: This paper is based on wrong definitions, in spite of that there is a serious problem with the proofs of the results.NEWLINENEWLINE Look, for example, at the proofs of Propositions 3.1, 3.4, at Proposition 3.6 and its proof. This is, for example, the definition of the semiprime generalized bi-\(\Gamma\)-ideal given in the paper: ``A generalized bi-\(\Gamma\)-ideal of a \(\Gamma\)-semigroup is said to be semiprime if for any bi-\(\Gamma\)-ideal \(B_1\) of \(S\), if \(B_1\Gamma B_1\subseteq B\), then \(B_1\subseteq B\)''. The correct is: ``if for any subset \(A\) of \(S\), \(A\Gamma A\subseteq B\) implies \(A\subseteq B\)'' (and not ``for any generalized bi-ideal \(B_1\)'' of \(S\)). And this is equivalent to saying that ``for any \(a\in S\), \(a\Gamma a\subseteq B\) implies \(a\in B\) (see, for example [\textit{M. Petrich}, Introduction to semigroups. Columbus, Ohio: Charles E. Merrill Publishing Company (1973; Zbl 0321.20037)]).NEWLINENEWLINE Some of my remarks: It is much better to use the term ``bi-ideal'' instead of ``generalized bi-ideal'' and this is because in most of the cases a bi-ideal needs not be a subsemigroup. In addition, there are not two kinds of bi-ideals (left, right ideals, subsemigroups etc.) in a \(\Gamma\)-semigroup to distinguish them as bi-ideal and bi-\(\Gamma\)-ideal, left ideal and left \(\Gamma\)-ideal, subsemigroup and sub-\(\Gamma\)-semigroup, and use the terms bi-\(\Gamma\)-ideal, sub-\(\Gamma\)-semigroup, etc. Finally, it is well known that most of the results on semigroups can be transferred to \(\Gamma\)-semigroups just putting a ``Gamma'' in the appropriate place. So for the \(\Gamma\)-semigroups the authors should indicate inside the paper, the papers on which their paper is based and not just to cite them in the References. They just have to be careful to check the results on semigroups before they transfer them.
reviewed by: Niovi Kehayopulu
description: scientific article; zbMATH DE number 6568360