Accumulation points of congruence densities of finite lattices: Dedicated to George Grätzer on the occasion of his ninetieth birthday, with appreciation and admiration, and in friendship

Gábor Czédli

doi:10.56754/0719-0646.2803.455

Authors

Gábor Czédli University of Szeged, Bolyai Institute, Szeged, Aradi vértanúk tere 1, Hungary https://orcid.org/0000-0001-9990-3573

MSC:

06B10, 06C10

DOI:

https://doi.org/10.56754/0719-0646.2803.455

Keywords:

Congruence density, accumulation point, lattice congruence, dually well-ordered set, semimodular lattice, modular lattice

Abstract

Let \(\mathcal{W}\) be a non-trivial variety of lattices and let \(L\) be a finite lattice in \(\mathcal{W}\). The congruence density of \(L\) with respect to \(\mathcal{W}\) is the number of congruences of \(L\) divided by the maximum number of congruences of \(|L|\)-element lattices belonging to \(\mathcal{W}\). We prove that, with respect to the order and multiplication of the real numbers, the set \(\mathrm{SCD}(\mathcal{W})\) of congruence densities of finite members of \(\mathcal{W}\) as well as its topological closure are countably infinite dually well-ordered monoids. We also prove that the set of accumulation points of \(\mathrm{SCD}(\mathcal{W})\) is either a singleton or it is countably infinite; furthermore, it is a singleton if and only if \(\mathcal{W}\) is a subvariety of the variety of modular lattices. This gives a complicated characterization of modularity: a non-singleton lattice \(K\) is modular if and only if \(\mathrm{SCD}(\mathcal{V}(K))\), where \(\mathcal{V}(K)\) denotes the variety generated by \(K\), has only one accumulation point. The class \(\mathcal{S}\) of semimodular lattices is not a variety, but \(\mathrm{SCD}(\mathcal{S})\) is still meaningful. We prove that \(\mathrm{SCD}(\mathcal{S})\) has exactly one accumulation point.

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