Congruences and Trajectories in Planar Semimodular Lattices
A 1955 result of J. Jakubík states that for the prime intervals p and q of a finite lattice, con(p) ≥ con(q) iff p is congruence-projective to q (via intervals of arbitrary size). The problem is how to determine whether con(p) ≥ con(q) involving only prime intervals. Two recent papers approached thi...
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| Format: | Article |
| Language: | English |
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University of Zielona Góra
2018-06-01
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| Series: | Discussiones Mathematicae - General Algebra and Applications |
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| Online Access: | https://doi.org/10.7151/dmgaa.1280 |
| _version_ | 1827839946505846784 |
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| author | Grätzer G. |
| author_facet | Grätzer G. |
| author_sort | Grätzer G. |
| collection | DOAJ |
| description | A 1955 result of J. Jakubík states that for the prime intervals p and q of a finite lattice, con(p) ≥ con(q) iff p is congruence-projective to q (via intervals of arbitrary size). The problem is how to determine whether con(p) ≥ con(q) involving only prime intervals. Two recent papers approached this problem in different ways. G. Czédli’s used trajectories for slim rectangular lattices-a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how my approach can be used to reprove Czédli’s result and generalize it to arbitrary slim, planar, semimodular lattices. |
| first_indexed | 2024-03-12T07:27:26Z |
| format | Article |
| id | doaj.art-a1be4efe9cde42a78a6c503279ceb88d |
| institution | Directory Open Access Journal |
| issn | 2084-0373 |
| language | English |
| last_indexed | 2024-03-12T07:27:26Z |
| publishDate | 2018-06-01 |
| publisher | University of Zielona Góra |
| record_format | Article |
| series | Discussiones Mathematicae - General Algebra and Applications |
| spelling | doaj.art-a1be4efe9cde42a78a6c503279ceb88d2023-09-02T22:05:48ZengUniversity of Zielona GóraDiscussiones Mathematicae - General Algebra and Applications2084-03732018-06-0138113114210.7151/dmgaa.1280dmgaa.1280Congruences and Trajectories in Planar Semimodular LatticesGrätzer G.0Department of Mathematics University of Manitoba Winnipeg, MB R3T 2N2, CanadaA 1955 result of J. Jakubík states that for the prime intervals p and q of a finite lattice, con(p) ≥ con(q) iff p is congruence-projective to q (via intervals of arbitrary size). The problem is how to determine whether con(p) ≥ con(q) involving only prime intervals. Two recent papers approached this problem in different ways. G. Czédli’s used trajectories for slim rectangular lattices-a special subclass of slim, planar, semimodular lattices. I used the concept of prime-projectivity for arbitrary finite lattices. In this note I show how my approach can be used to reprove Czédli’s result and generalize it to arbitrary slim, planar, semimodular lattices.https://doi.org/10.7151/dmgaa.1280semimodular latticeplanar latticeslim latticerectangular latticecongruencetrajectoryprime intervalprimary: 06c10secondary: 06b10 |
| spellingShingle | Grätzer G. Congruences and Trajectories in Planar Semimodular Lattices Discussiones Mathematicae - General Algebra and Applications semimodular lattice planar lattice slim lattice rectangular lattice congruence trajectory prime interval primary: 06c10 secondary: 06b10 |
| title | Congruences and Trajectories in Planar Semimodular Lattices |
| title_full | Congruences and Trajectories in Planar Semimodular Lattices |
| title_fullStr | Congruences and Trajectories in Planar Semimodular Lattices |
| title_full_unstemmed | Congruences and Trajectories in Planar Semimodular Lattices |
| title_short | Congruences and Trajectories in Planar Semimodular Lattices |
| title_sort | congruences and trajectories in planar semimodular lattices |
| topic | semimodular lattice planar lattice slim lattice rectangular lattice congruence trajectory prime interval primary: 06c10 secondary: 06b10 |
| url | https://doi.org/10.7151/dmgaa.1280 |
| work_keys_str_mv | AT gratzerg congruencesandtrajectoriesinplanarsemimodularlattices |