Previous |  Up |  Next

Article

Keywords:
ordered set; linear extension; natural representation; lexicographic sum; dense subset
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.
References:
[1] G. Birkhoff: Lattice Theory. Providence, Rhode Island, 1967. MR 0227053 | Zbl 0153.02501
[2] M. M. Day: Arithmetic of ordered systems. Trans. Amer. Math. Soc. 58 (1945), 1–43. DOI 10.1090/S0002-9947-1945-0012262-4 | MR 0012262 | Zbl 0060.05813
[3] T. Fofanova, I. Rival, A. Rutkowski: Dimension two, fixed points nad dismantable ordered sets. Order 13 (1996), 245–253. MR 1420398
[4] F. Hausdorff: Grundzüge der Mengenlehre. Leipzig, 1914.
[5] T. Hiraguchi: On the dimension of partially ordered sets. Sci. Rep. Kanazawa Univ. 1 (1951), 77–94. MR 0070681 | Zbl 0200.00013
[6] H. A. Kierstead, E. C. Milner: The dimension of the finite subsets of $K$. Order 13 (1996), 227–231. MR 1420396
[7] D. Kurepa: Partitive sets and ordered chains. Rad Jugosl. Akad. Znan. Umjet. Odjel Mat. Fiz. Tehn. Nauke 6 (302) (1957), 197–235. MR 0097328 | 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. MR 0048795
[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. MR 0056049
[11] V. Novák: On the well dimension of ordered sets. Czechoslovak Math. J. 19 (94) (1969), 1–16. MR 0241325
[12] V. Novák: Über Erweiterungen geordneter Mengen. Arch. Math. (Brno) 9 (1973), 141–146. MR 0354456
[13] V. Novák: Some cardinal characteristics of ordered sets. Czechoslovak Math. J. 48 (123) (1998), 135–144. DOI 10.1023/A:1022523830353 | MR 1614021
[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. MR 0082958
[16] Ordered sets. Proc. NATO Adv. Study Inst. Banff (1981). Zbl 0519.05017
[17] M. Pouzet, I. Rival: Which ordered sets have a complete linear extension?. Canad. J. Math. 33 (1981), 1245–1254. DOI 10.4153/CJM-1981-093-9 | MR 0638378
[18] A. Rutkowski: Which countable ordered sets have a dense linear extension?. Math. Slovaca 46 (1996), 445–455. MR 1451035 | Zbl 0890.06003
[19] J. Schmidt: Lexikographische Operationen. Z. Math. Logik Grundlagen Math. 1 (1955), 127–170. DOI 10.1002/malq.19550010207 | MR 0071488 | Zbl 0065.03703
[20] J. Schmidt: Zur Kennzeichnung der Dedekind-Mac Neilleschen Hülle einer geordneten Menge. Arch. Math. 7 (1956), 241–249. DOI 10.1007/BF01900297 | MR 0084484
[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
[22] W. Sierpiński: Cardinal and Ordinal Numbers. Warszawa, 1958. MR 0095787
[23] W. Sierpiński: Sur une propriété des ensembles ordonnés. Fund. Math. 36 (1949), 56–67.
[24] G. Szász: Einführung in die Verbandstheorie. Leipzig, 1962. MR 0138567
[25] E. Szpilrajn: Sur l’extension de l’ordre partiel. Fund. Math. 16 (1930), 386–389.
Partner of
EuDML logo