On principal congruences and the number of congruences of a lattice with more ideals than filters
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 gener...
Elmentve itt :
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| Dokumentumtípus: | Cikk |
| Megjelent: |
2019
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| Sorozat: | Acta scientiarum mathematicarum
85 No. 3-4 |
| Kulcsszavak: | Rácselmélet - rács szűrő - egyezések |
| doi: | 10.14232/actasm-018-538-y |
| Online Access: | http://acta.bibl.u-szeged.hu/66321 |
| Tartalmi kivonat: | 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. |
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| Terjedelem/Fizikai jellemzők: | 363-380 |
| ISSN: | 2064-8316 |