On the \(le\)-semigroups whose semigroup of bi-ideal elements is a normal band
type: Item
instance of: scholarly article
zbMATH Open document ID: 1400.06007
author: M. Kumbhakar
author: Anjan Kumar Bhuniya
publication date: 2016-04-13T00:00:00Z
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 6568275
zbMATH Keywords: \(le\)-semigroup
zbMATH Keywords: regular
zbMATH Keywords: intra-regular
zbMATH Keywords: bi-ideal element
zbMATH Keywords: normal band
zbMATH Keywords: rectangular band
zbMATH Keywords: left zero band
MaRDI profile type: Publication
title: On the \(le\)-semigroups whose semigroup of bi-ideal elements is a normal band
published in: Algebra and Discrete Mathematics
review text: The definitions of the different subvarieties of normal bands on p. 174 should be given in a clear way. A band is called left (right) normal if it satisfies the identity \(abc=acb\) (\(abc=bac\)); it is called normal if it satisfies the identity \(abca=acba\). But a semigroup is called rectangular band if it satisfies the identity \(a=aba\). As one can easily prove, every semigroup satisfying the relation \(a=aba\) is a band. Every left (right) zero semigroup, that is every semigroup satisfying the identity \(ab=a\) \((ab=b)\) is a band.NEWLINENEWLINENEWLINEThis being so, the two lines between the Theorem 3.9 and Theorem 3.10 should be formulated as follows: A semilattice is a left and right normal band. ``Conversely'', every left and right normal band is a semilattice.NEWLINENEWLINE NEWLINEThere is confusion in the paper. In Proposition 3.1, for example, for an \(le\)-semigroup \(S\), the authors should clarify what ``the semigroup \({\mathcal B}(S)\) of all bi-ideal elements of \(S\) is regular'' means. Besides, the \(\Rightarrow\)-part of the proposition holds for \(poe\)-semigroups and the \(\Leftarrow\)-part for \(\vee e\)-semigroups in general.NEWLINENEWLINE On p. 175, lines 8 to 10, the authors write: ``Hence the bi-ideal element \(\beta (a)\) generated by \(a\) reduces to the form \(\beta (a)=aea\). Thus in a regular \(le\)-semigroup the notions of bi-ideal elements as we have defined and that defined by N. Kehayopulu are the same''. This cannot be a conclusion from the \(\beta (a)=aea\). In fact, if \(S\) is a regular \(poe\)-semigroup and \(beb\leq b\), then \(b=beb\), so \(b^2=(beb)b\leq beb=b\) which shows that in case of regular \(poe\)-semigroups the two definitions are equivalent.NEWLINENEWLINEIn Theorem 3.2: Let \(S\) be an \(le\)-semigroup. Then \({\mathcal R}(S){\mathcal L}(S)\subseteq {\mathcal B} (S)\). If moreover, \(S\) is a regular \(le\)-semigroup, then \({\mathcal R}(S){\mathcal L}(S)={\mathcal B}(S)\). For the proof of the theorem, the auth…
reviewed by: Niovi Kehayopulu
description: scientific article; zbMATH DE number 6568275