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Ideal nil-extensions of semigroups with semimodular congruence lattices. (English) Zbl 0789.20068

Let \(S\) be a semigroup and \(K\) be an ideal of \(S\), then \(S\) is an ideal nil-extension of \(K\) if \(S/K \cong Q\) where \(Q\) is a nil semigroup. For ideal nil-extensions \(S\) the question is investigated when the congruence lattice \(C(S)\) of \(S\) is semimodular. It holds: For an ideal nil- extension \(S\) of \(K\) the lattice \(C(S)\) is semimodular whenever \(C(K)\) is modular. Further, it is proved that if \(S\) is an ideal nil-extension of \(K\) such that the congruences on \(K\) can be extended to \(S\) preserving inclusion, then \(C(S)\) is semimodular iff \(C(K)\) is semimodular. This is used to show that ideal nil-extensions \(S\) of \(K\), where \(K\) is either an inverse semigroup or a monoid, satisfy: \(C(S)\) is semimodular iff \(C(K)\) is semimodular.
Reviewer: M.Demlová (Praha)

MSC:

20M10 General structure theory for semigroups
08A30 Subalgebras, congruence relations
06C10 Semimodular lattices, geometric lattices
20M12 Ideal theory for semigroups
20M15 Mappings of semigroups
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References:

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