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

Czédli Gábor and Mureşan Claudia: On principal congruences and the number of congruences of a lattice with more ideals than filters. In: Acta scientiarum mathematicarum, (85) 3-4. pp. 363-380. (2019)

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Abstract

Let λ and κ be cardinal numbers such that κ is infinite and either2 ≤ λ ≤ κ, or λ = 2κ. We prove that there exists a lattice L exactly λ many congruences ,2κ many ideals, but only κ many filters. Furthermore, if λ ≤ 2isan 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 ifPis a bounded ordered set (in other words, a boundedposet) 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 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.

Item Type: Article
Journal or Publication Title: Acta scientiarum mathematicarum
Date: 2019
Volume: 85
Number: 3-4
ISSN: 2064-8316
Page Range: pp. 363-380
Related URLs: http://acta.bibl.u-szeged.hu/66425/
DOI: 10.14232/actasm-018-538-y
Uncontrolled Keywords: Rácselmélet - rács szűrő - egyezések
Additional Information: Ábrákkal ill. Bibliogr.: p. 378-380.
Date Deposited: 2020. Apr. 23. 11:02
Last Modified: 2021. Mar. 25. 15:34
URI: http://acta.bibl.u-szeged.hu/id/eprint/66321

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