Abstract
Our aim is to investigate groups and their weak congruence lattices in the abstract setting of lattices L with (local) closure operators C in the categorical sense, where L is regarded as a small category and C is a family of closure maps on the principal ideals of L. A useful tool for structural investigations of such “lattices with closure” is the so-called characteristic triangle, a certain substructure of the square L 2. For example, a purely order-theoretical investigation of the characteristic triangle shows that the Dedekind groups (alias Hamiltonian groups) are precisely those with modular weak congruence lattices; similar results are obtained for other classes of algebras.
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Presented by M. Ploščica.
The research of the first author was partially supported by the NFSR of Hungary (OTKA), Grant No. T 049433 and K 60148, and also by the grant for the project “Lattice methods and applications” detailed below, that covered his visit to Novi Sad. The research of the third and the fourth authors was partially supported by the Serbian Ministry of Science and Environment, Grant No. 144011, and by the Provincial Secretariat for Science and Tech. Development, Auton. Province of Vojvodina, Grant “Lattice methods and applications”.
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Czédli, G., Erné, M., Šešelja, B. et al. Characteristic triangles of closure operators with applications in general algebra. Algebra Univers. 62, 399–418 (2009). https://doi.org/10.1007/s00012-010-0059-2
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DOI: https://doi.org/10.1007/s00012-010-0059-2