On the \(le\)-semigroups whose semigroup of bi-ideal elements is a normal band
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 authors use Lemma 2 in [the reviewer, Semigroup Forum 23, No. 1, 85--86 (1981; Zbl 0467.06010)]. This can be easily proved independently, \(S\) does not need to be an \(le\)-semigroup, the \(\Rightarrow\)-part holds in \(poe\)-semigroups and the \(\Leftarrow\)-part in \(\vee e\)-semigroups. NEWLINENEWLINEIn Proposition 3.3: Let \(S\) be a regular \(le\)-semigroup. Then \({\mathcal R}(S)\) and \({\mathcal L}(S)\) are bands. In this proposition, we have to delete the \(le\)-semigroup and write \(poe\)-semigroup instead. NEWLINENEWLINEAfter Proposition 3.3, the authors write: ``The following important result can be proved similarly to that in [the reviewer, Semigroup Forum 49, No. 2, 267--268 (1994; Zbl 0811.06014)]'' and give Theorem 3.4: An \(le\)-semigroup \(S\) is both regular and intra-regular if and only if \({\mathcal B}(S)\) is a band. This theorem can be proved independently, the \(\Rightarrow\)-part holds for \(poe\)-semigroups and the \(\Leftarrow\)-part for \(\vee e\)-semigroups. NEWLINENEWLINEIn the proof of Theorem 3.5, they say: ``Thus \({\mathcal B }(S)\) is a locally testable semigroup. Since a locally testable semigroup is a band if and only if it is a normal band [\textit{Y. Zalcstein}, Semigroup Forum 5, 216--227 (1973; Zbl 0273.20049), Theorem 4], so \({\mathcal B} (S)\) is a normal band''. In fact, they should say: ``Since \({\mathcal B} (S)\) is a band and \(b{\mathcal B}(S)b\) is a semilattice for every \(b\in {\mathcal B} (S)\), by [Zalcstein, loc. cit., Theorem 5], \({\mathcal B}(S)\) is a normal band''. The theorem can be proved independently without using the concept of a testable semigroup and \(S\) need not to be an \(le\)-semigroup in it, the \(\Rightarrow\)-part holds for a \(poe\)-semigroup and the \(\Leftarrow\)-part for a \(\vee e\)-semigroup. NEWLINENEWLINEIn Lemma 3.6: An \(le\)-semigroup \(S\) is left duo if and only if \(ae\leq ea\) for all \(a\in S\). The proof of the \(\Rightarrow\)-part of Lemma 3.6 is wrong. The set \(\{x\in S \mid x\leq sa \text{ for some } s\in S\}\) cannot be the left ideal of \(S\) generated by \(a\) (as it does not contain the element \(a\)). Proposition 3.7 and Theorems 3.8, 3.9 and 3.10 are based on Lemma 3.6. NEWLINENEWLINEIn Theorem 3.11: Let \(S\) be an \(le\)-semigroup. Then \({\mathcal B}(S)\) is a rectangular band if and only if \(S\) is regular and \(eae=ebe\) for all \(a,b\in S\). Again here, \(S\) needs not to be an \(le\)-semigroup. The \(\Rightarrow\)-part holds for \(\vee e\)-semigroups and the \(\Leftarrow\)-part for \(poe\)-semigroups. NEWLINENEWLINEIn Theorem 3.12: Let \(S\) be an \(le\)-semigroup. Then \({\mathcal B}(S)\) is a left zero band if and only if \(S\) is regular and \(ae\leq eb\) for all \(a,b\in S\). Here, the \(\Rightarrow\)-part holds for \(\vee e\)-semigroups and the \(\Leftarrow\)-part for \(poe\)-semigroups in general. In the rest of the paper, \({\mathcal R}_m (S)\) (\({\mathcal L}_m (S)\)) denotes the set of minimal right (left) ideal elements of \(S\) and \({\mathcal B}(S)\) denotes the set of bi-ideal elements of \(S\). According to Theorem 4.2, if \(S\) is an \(le\)-semigroup, then \({\mathcal B}_m (S)={\mathcal R}_m (S){\mathcal L}_m (S)\). This theorem holds for \(poe\)-semigroups. According to Theorem 4.3, if \(S\) is an \(le\)-semigroup such that the set \({\mathcal L}_m (S)\) is nonempty, then \({\mathcal L}_m (S)\) is a left ideal, moreover a left zero band of the semigroup \({\mathcal L}(S)\). This theorem also holds for \(poe\)-semigroups in general. According to Theorem 4.4, if \(S\) is an \(le\)-semigroup such that \({\mathcal B}(S)\) is nonempty, then \({\mathcal B}_m (S)\) is a bi-ideal of the \({\mathcal B}(S)\). Moreover \({\mathcal B}(S)\) is a rectangular band. This theorem also holds for \(poe\)-semigroups. We work in an ordered semigroup \((S,\cdot,\leq)\). So they should clarify, for example, what ``\({\mathcal L}_m (S)\) is a left ideal of the semigroup \({\mathcal L}(S)\)'' means; what ``\({\mathcal B}_m (S)\) is a bi-ideal of the semigroup \({\mathcal B}(S)\)'' means. NEWLINENEWLINEIn the first three lines of the abstract, the authors write ``It is well known that the semigroup \({\mathcal B}(S)\) of all bi-ideal elements of an \(le\)-semigroup \(S\) is a band if and only if \(S\) is both regular and intra-regular''. NEWLINENEWLINEIn Theorem 3.4 of the paper: An \(le\)-semigroup \(S\) is both regular and intra-regular if and only if \({\mathcal B}(S)\) is a band. In spite of its title, in the whole paper the set \(S\) need not to be an \(le\)-semigroup. But we must mention the following as well that is the most important. The definition of the \(le\)-semigroup in the paper is wrong. NEWLINENEWLINEIn the ``Preliminaries'', the authors write ``An \(le\)-semigroup is an algebra \((S,\cdot,\vee,\wedge,e)\) such that \((S,\cdot)\) is a semigroup, \((S,\vee,\wedge,e)\) is a lattice with a greatest element denoted by \(e\), and for all \(a,b,c\in S\), \(a(b\vee c)=ab\vee ac\) and \((a\vee b)c=ac\vee bc\)''. Which means that \(S\) is endowed with an order relation and, according to that relation, is a lattice. An after that, the authors say: ``The usual order relation \(\leq\) on the set \(S\) is defined by: for \(a,b\in S\) \(a\leq b\) is \(a\vee b=b\)''.
scientific article; zbMATH DE number 6568275