Wang, Li-Min Ideal nil-extensions of semigroups with semimodular congruence lattices. (English) Zbl 0789.20068 Semigroup Forum 47, No. 3, 353-358 (1993). 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) Cited in 1 Document 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 Keywords:nil semigroup; ideal nil-extensions; congruence lattice; semimodular PDFBibTeX XMLCite \textit{L.-M. Wang}, Semigroup Forum 47, No. 3, 353--358 (1993; Zbl 0789.20068) Full Text: DOI EuDML References: [1] Mitsch, H.,Semigroups and their lattices of congruences, Semigroup Forum26 (1983), 1–63. · Zbl 0513.20047 · doi:10.1007/BF02572819 [2] Trueman, D. C.,The lattices of congruences on direct products of cyclic semigroups and certain other semigroups, Proc. Roy. Soc. Edinburgh95A (1983), 203–214. · Zbl 0524.20038 [3] Jones, P. R.,Congruence semimodular varieties of semigroups, Lecture Notes in Math., No. 1320, Springer, Berlin-New York (1988), 162–171. · Zbl 0643.20038 [4] Jones, P. R.,Congruences contained in an equivalence on a semigroup, J. Austral. Math. Soc.A29 (1980), 162–176. · Zbl 0428.20035 · doi:10.1017/S1446788700021170 [5] Zhu, Pingyu,On the congruence lattice of the nilpotent extension of a semigroup, Acta Math. Sinica33 (1990), 679–683 (in Chinese). · Zbl 0716.20041 [6] Demlova, M. and V. Koubek,Minimal congruences and coextensions in semigroups, Lecture Notes in Math., No. 1320, Springer, Berlin-New York (1988), 28–83. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.