On the \(le\)-semigroups whose semigroup of bi-ideal elements is a normal band
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. Here we show that \(\mathcal{B}(S)\) is a band if and only if it is a normal band and give a complete characterization of the...
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| Date: | 2016 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2016
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/141 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | 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. Here we show that \(\mathcal{B}(S)\) is a band if and only if it is a normal band and give a complete characterization of the \(le\)-semigroups \(S\) for which the associated semigroup \(\mathcal{B}(S)\) is in each of the seven nontrivial subvarieties of normal bands. We also show that the set \(\mathcal{B}_{m}(S)\) of all minimal bi-ideal elements of \(S\) forms a rectangular band and that \(\mathcal{B}_{m}(S)\) is a bi-ideal of the semigroup~\(\mathcal{B(S)}\). |
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