Linear extensions of orderings. (English) Zbl 1079.06500
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.
MSC:
| 06A06 | Partial orders, general |
References:
| [1] | G. Birkhoff: Lattice Theory. Providence, Rhode Island, 1967. · Zbl 0153.02501 |
| [2] | M. M. Day: Arithmetic of ordered systems. Trans. Amer. Math. Soc. 58 (1945), 1-43. · Zbl 0060.05813 · doi:10.2307/1990233 |
| [3] | T. Fofanova, I. Rival, A. Rutkowski: Dimension two, fixed points nad dismantable ordered sets. Order 13 (1996), 245-253. · Zbl 0867.06003 |
| [4] | F. Hausdorff: Grundzüge der Mengenlehre. Leipzig, 1914. · JFM 45.0123.01 |
| [5] | T. Hiraguchi: On the dimension of partially ordered sets. Sci. Rep. Kanazawa Univ. 1 (1951), 77-94. · Zbl 0200.00013 |
| [6] | H. A. Kierstead, E. C. Milner: The dimension of the finite subsets of \(K\). Order 13 (1996), 227-231. · Zbl 0866.06001 |
| [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 |
| [8] | J. Loś, C. Ryll-Nardzewski: On the application of Tychonoff’s theorem in mathematical proofs. Fund. Math. 38 (1951), 233-237. · Zbl 0044.27403 |
| [9] | E. Mendelson: Appendix. W. Sierpiński: Cardinal and Ordinal Numbers, Warszawa, 1958. |
| [10] | J. Novák: On partition of an ordered continuum. Fund. Math. 39 (1952), 53-64. · Zbl 0050.05303 |
| [11] | V. Novák: On the well dimension of ordered sets. Czechoslovak Math. J. 19 (94) (1969), 1-16. · Zbl 0175.01203 |
| [12] | V. Novák: Über Erweiterungen geordneter Mengen. Arch. Math. (Brno) 9 (1973), 141-146. |
| [13] | V. Novák: Some cardinal characteristics of ordered sets. Czechoslovak Math. J. 48 (123) (1998), 135-144. · Zbl 0927.06001 · doi:10.1023/A:1022523830353 |
| [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. |
| [15] | M. Novotný: Bemerkung über die Darstellung teilweise geordneter Mengen. Spisy přír. fak. MU Brno 369 (1955), 451-458. · Zbl 0068.04301 |
| [16] | Ordered sets. Proc. NATO Adv. Study Inst. Banff (1981). · Zbl 0519.05017 · doi:10.1016/0095-8956(81)90048-4 |
| [17] | M. Pouzet, I. Rival: Which ordered sets have a complete linear extension?. Canad. J. Math. 33 (1981), 1245-1254. · Zbl 0479.06001 · doi:10.4153/CJM-1981-093-9 |
| [18] | A. Rutkowski: Which countable ordered sets have a dense linear extension?. Math. Slovaca 46 (1996), 445-455. · Zbl 0890.06003 |
| [19] | J. Schmidt: Lexikographische Operationen. Z. Math. Logik Grundlagen Math. 1 (1955), 127-170. · Zbl 0065.03703 · doi:10.1002/malq.19550010207 |
| [20] | J. Schmidt: Zur Kennzeichnung der Dedekind-Mac Neilleschen Hülle einer geordneten Menge. Arch. Math. 7 (1956), 241-249. · Zbl 0073.03801 · doi:10.1007/BF01900297 |
| [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. |
| [22] | W. Sierpiński: Cardinal and Ordinal Numbers. Warszawa, 1958. · Zbl 0083.26803 |
| [23] | W. Sierpiński: Sur une propriété des ensembles ordonnés. Fund. Math. 36 (1949), 56-67. · Zbl 0039.27801 |
| [24] | G. Szász: Einführung in die Verbandstheorie. Leipzig, 1962. · Zbl 0101.26901 |
| [25] | E. Szpilrajn: Sur l’extension de l’ordre partiel. Fund. Math. 16 (1930), 386-389. · JFM 56.0843.02 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
