On generalized bi-\(\Gamma\)-ideals in \(\Gamma\)-semigroups.
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.
scientific article; zbMATH DE number 6568360