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On principal congruences and the number of congruences of a lattice with more ideals than filters

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Abstract

Let λ and κ be cardinal numbers such that κ is infinite and either 2 ≤ λ ≤ κ, or λ = 2κ. We prove that there exists a lattice L with exactly λ many congruences, 2κ many ideals, but only κ many filters. Furthermore, if λ ≥ 2 is an integer of the form 2m · 3n, then we can choose L to be a modular lattice generating one of the minimal modular nondistributive congruence varieties described by Ralph Freese in 1976, and this L is even relatively complemented for λ = 2. Related to some earlier results of George Grätzer and the first author, we also prove that if P is a bounded ordered set (in other words, a bounded poset) with at least two elements, G is a group, and κ is an infinite cardinal such that κ ≤ |P| and κ ≤ |G|, then there exists a lattice L of cardinality κ such that (i) the principal congruences of L form an ordered set isomorphic to P, (ii) the automorphism group of L is isomorphic to G, (iii) L has 2κ many ideals, but (iv) L has only κ many filters.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, 6720, Hungary

    Gábor Czédli

  2. Faculty of Mathematics and Computer Science of the University of Bucharest, RO 010014, Romania

    Claudia Mureşan

  3. Università degli Studi di Cagliari, Romania

    Claudia Mureşan

Authors
  1. Gábor Czédli
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  2. Claudia Mureşan
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Corresponding author

Correspondence to Gábor Czédli.

Additional information

Dedicated to the memory of George Allen Hutchinson

Communicated by Á. Szendrei

This research was supported by NFSR of Hungary (OTKA), grant number K 115518, and by the research grant Proprietà d’Ordine Nella Semantica Algebrica delle Logiche Non Classiche of Università degli Studi di Cagliari, Regione Autonoma della Sardegna, L. R. 7/2007, n. 7, 2015, CUP: F72F16002920002.

Acknowledgment.

The referee’s careful work and his hints to simplify the proof of Lemma 3.2 and the second proof of part (i) of Theorem 1.1 are appreciated.

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Czédli, G., Mureşan, C. On principal congruences and the number of congruences of a lattice with more ideals than filters. ActaSci.Math. 85, 363–380 (2019). https://doi.org/10.14232/actasm-018-538-y

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  • DOI: https://doi.org/10.14232/actasm-018-538-y

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