| Title:
|
Linear extensions of orderings (English) |
| Author:
|
Novák, Vítězslav |
| Author:
|
Novotný, Miroslav |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
50 |
| Issue:
|
4 |
| Year:
|
2000 |
| Pages:
|
853-864 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A construction is given which makes it possible to find all linear extensions of a given ordered set and, conversely, to find all orderings on a given set with a prescribed linear extension. Further, dense subsets of ordered sets are studied and a procedure is presented which extends a linear extension constructed on a dense subset to the whole set. (English) |
| Keyword:
|
ordered set |
| Keyword:
|
linear extension |
| Keyword:
|
natural representation |
| Keyword:
|
lexicographic sum |
| Keyword:
|
dense subset |
| MSC:
|
06A06 |
| idZBL:
|
Zbl 1079.06500 |
| idMR:
|
MR1792975 |
| . |
| Date available:
|
2009-09-24T10:38:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127615 |
| . |
| Reference:
|
[1] G. Birkhoff: Lattice Theory.Providence, Rhode Island, 1967. Zbl 0153.02501, MR 0227053 |
| Reference:
|
[2] M. M. Day: Arithmetic of ordered systems.Trans. Amer. Math. Soc. 58 (1945), 1–43. Zbl 0060.05813, MR 0012262, 10.1090/S0002-9947-1945-0012262-4 |
| Reference:
|
[3] T. Fofanova, I. Rival, A. Rutkowski: Dimension two, fixed points nad dismantable ordered sets.Order 13 (1996), 245–253. MR 1420398 |
| Reference:
|
[4] F. Hausdorff: Grundzüge der Mengenlehre.Leipzig, 1914. |
| Reference:
|
[5] T. Hiraguchi: On the dimension of partially ordered sets.Sci. Rep. Kanazawa Univ. 1 (1951), 77–94. Zbl 0200.00013, MR 0070681 |
| Reference:
|
[6] H. A. Kierstead, E. C. Milner: The dimension of the finite subsets of $K$.Order 13 (1996), 227–231. MR 1420396 |
| Reference:
|
[7] D. Kurepa: Partitive sets and ordered chains.Rad Jugosl. Akad. Znan. Umjet. Odjel Mat. Fiz. Tehn. Nauke 6 (302) (1957), 197–235. Zbl 0147.26301, MR 0097328 |
| Reference:
|
[8] J. Loś, C. Ryll-Nardzewski: On the application of Tychonoff’s theorem in mathematical proofs.Fund. Math. 38 (1951), 233–237. MR 0048795, 10.4064/fm-38-1-233-237 |
| Reference:
|
[9] E. Mendelson: Appendix.W. Sierpiński: Cardinal and Ordinal Numbers, Warszawa, 1958. |
| Reference:
|
[10] J. Novák: On partition of an ordered continuum.Fund. Math. 39 (1952), 53–64. MR 0056049, 10.4064/fm-39-1-53-64 |
| Reference:
|
[11] V. Novák: On the well dimension of ordered sets.Czechoslovak Math. J. 19 (94) (1969), 1–16. MR 0241325 |
| Reference:
|
[12] V. Novák: Über Erweiterungen geordneter Mengen.Arch. Math. (Brno) 9 (1973), 141–146. MR 0354456 |
| Reference:
|
[13] V. Novák: Some cardinal characteristics of ordered sets.Czechoslovak Math. J. 48 (123) (1998), 135–144. MR 1614021, 10.1023/A:1022523830353 |
| Reference:
|
[14] M. Novotný: O representaci částečně uspořádaných množin posloupnostmi nul a jedniček (On representation of partially ordered sets by means of sequences of 0’s and 1’s).Čas. pěst. mat. 78 (1953), 61–64. |
| Reference:
|
[15] M. Novotný: Bemerkung über die Darstellung teilweise geordneter Mengen.Spisy přír. fak. MU Brno 369 (1955), 451–458. MR 0082958 |
| Reference:
|
[16] : Ordered sets.Proc. NATO Adv. Study Inst. Banff (1981). Zbl 0519.05017 |
| Reference:
|
[17] M. Pouzet, I. Rival: Which ordered sets have a complete linear extension?.Canad. J. Math. 33 (1981), 1245–1254. MR 0638378, 10.4153/CJM-1981-093-9 |
| Reference:
|
[18] A. Rutkowski: Which countable ordered sets have a dense linear extension?.Math. Slovaca 46 (1996), 445–455. Zbl 0890.06003, MR 1451035 |
| Reference:
|
[19] J. Schmidt: Lexikographische Operationen.Z. Math. Logik Grundlagen Math. 1 (1955), 127–170. Zbl 0065.03703, MR 0071488, 10.1002/malq.19550010207 |
| Reference:
|
[20] J. Schmidt: Zur Kennzeichnung der Dedekind-Mac Neilleschen Hülle einer geordneten Menge.Arch. Math. 7 (1956), 241–249. MR 0084484, 10.1007/BF01900297 |
| Reference:
|
[21] V. Sedmak: Dimenzija djelomično uredenih skupova pridruženih poligonima i poliedrima (Dimension of partially ordered sets connected with polygons and polyhedra).Period. Math.-Phys. Astron. 7 (1952), 169–182. MR 0053495 |
| Reference:
|
[22] W. Sierpiński: Cardinal and Ordinal Numbers.Warszawa, 1958. MR 0095787 |
| Reference:
|
[23] W. Sierpiński: Sur une propriété des ensembles ordonnés.Fund. Math. 36 (1949), 56–67. 10.4064/fm-36-1-56-67 |
| Reference:
|
[24] G. Szász: Einführung in die Verbandstheorie.Leipzig, 1962. MR 0138567 |
| Reference:
|
[25] E. Szpilrajn: Sur l’extension de l’ordre partiel.Fund. Math. 16 (1930), 386–389. 10.4064/fm-16-1-386-389 |
| . |
