On the algorithmic construction of the 1960 sectional complement
In 1960, G. Gratzer and E. T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of a sectionally complemented finite lattice L. For u < v in L, they constructed a sectional complement, which is now called the 1960 sectional complement. In 1999, G....
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| Dokumentumtípus: | Cikk |
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Bolyai Institute, University of Szeged
Szeged
2011
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| Sorozat: | Acta scientiarum mathematicarum
77 No. 1-2 |
| Kulcsszavak: | Matematika |
| Tárgyszavak: | |
| Online Access: | http://acta.bibl.u-szeged.hu/16377 |
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| 040 | |a SZTE Egyetemi Kiadványok Repozitórium |b hun | ||
| 041 | |a eng | ||
| 100 | 1 | |a Grätzer George A. | |
| 245 | 1 | 3 | |a On the algorithmic construction of the 1960 sectional complement |h [elektronikus dokumentum] / |c Grätzer George A. |
| 260 | |a Bolyai Institute, University of Szeged |b Szeged |c 2011 | ||
| 300 | |a 35-45 | ||
| 490 | 0 | |a Acta scientiarum mathematicarum |v 77 No. 1-2 | |
| 520 | 3 | |a In 1960, G. Gratzer and E. T. Schmidt proved that every finite distributive lattice can be represented as the congruence lattice of a sectionally complemented finite lattice L. For u < v in L, they constructed a sectional complement, which is now called the 1960 sectional complement. In 1999, G. Gratzer and E. T. Schmidt discovered a very simple way of constructing a sectional complement in the ideal lattice of a chopped lattice made up of two sectionally complemented finite lattices overlapping in only two elements—the Atom Lemma. The question was raised whether this simple process can be generalized to an algorithm that finds the 1960 sectional complement. In 2006, G. Gratzer and M. Roddy discovered such an algorithm— allowing a wide latitude how it is carried out. In this paper we prove that the wide latitude apparent in the algorithm is deceptive: whichever way the algorithm is carried out, it produces the same sectional complement. This solves, in fact, Problems 2 and 3 of the GratzerRoddy paper. Surprisingly, the unique sectional complement provided by the algorithm is the 1960 sectional complement, solving Problem 1 of the same paper. | |
| 650 | 4 | |a Természettudományok | |
| 650 | 4 | |a Matematika | |
| 695 | |a Matematika | ||
| 700 | 0 | 1 | |a Klus G. |e aut |
| 700 | 0 | 1 | |a Nguyen Athena |e aut |
| 856 | 4 | 0 | |u http://acta.bibl.u-szeged.hu/16377/1/math_077_numb_001_002_035-045.pdf |z Dokumentum-elérés |