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Structure des demi-groupes dont le treillis des sous-demi-groupes est distributif. (French) Zbl 0112.25601

Summary: The author determines those semigroups \(D\) for which the lattice \(T(D)\) of subsemigroups is distributive. The problem can be reduced to the corresponding problem for groups and subgroups, which was solved by the reviewer [Duke Math. J. 4, 247–269 (1938; Zbl 0020.34801; JFM 64.0055.01)]. Among the most interesting results one may mention: \(T(D)\) is distributive if and only if, for any pair of elements \(a\) and \(b\), either \(ab\) is a power of \(a\) or \(b\), or \(ab\) generates a finite cyclic group such that a power of \(ab\) is a power of \(a\) and another power a power of \(b\). A cyclic semigroup \(D\) has distributive \(T(D)\) if and only if \(D\) is finite and the period begins at a rank \(\le 5\).

MSC:

20M10 General structure theory for semigroups
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