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On normal semigroups. (English) Zbl 0358.20074
A subsemigroup \(A\) of a semigroup \(S\) is called a bi-ideal of \(S\) if \(ASA \subseteq A\). The set of all bi-ideals of a semigroup \(S\) is also a semigroup under the multiplication of subsets, which we denote by \(B(S)\). A semigroup \(S\) is called normal if \(xS=Sx\) for all \(x \in S\), and is called viable if \(ab = ba\) whenever \(ab\) and \(ba\) are idempotents. The author gives characterizations of semigroups that are semilattices of groups: For a semigroup \(S\) the following conditions are equivalent:
(1) \(S\) is a semilattice of groups.
(2) \(B(S)\) is a semilattice of groups.
(3) \(B(S)\) is regular and normal.
(4) \(B(S)\) is regular and viable.
Reviewer: Nuboaki Kuroki

MSC:
20M10 General structure theory for semigroups
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