Abstract
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.
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References
G. Birkhoff: Lattice Theory. Providence, Rhode Island, 1967.
M. M. Day: Arithmetic of ordered systems. Trans. Amer. Math. Soc. 58 (1945), 1–43.
T. Fofanova, I. Rival, A. Rutkowski: Dimension two, fixed points nad dismantable ordered sets. Order 13 (1996), 245–253.
F. Hausdorff: Grundzüge der Mengenlehre. Leipzig, 1914.
T. Hiraguchi: On the dimension of partially ordered sets. Sci. Rep. Kanazawa Univ. 1 (1951), 77–94.
H. A. Kierstead, E. C. Milner: The dimension of the finite subsets of K. Order 13 (1996), 227–231.
D. Kurepa: Partitive sets and ordered chains. Rad Jugosl. Akad. Znan. Umjet. Odjel Mat. Fiz. Tehn. Nauke 6(302) (1957), 197–235.
J. Loś, C. Ryll-Nardzewski: On the application of Tychonoff's theorem in mathematical proofs. Fund. Math. 38 (1951), 233–237.
E. Mendelson: Appendix. W. Sierpiński: Cardinal and Ordinal Numbers. Warszawa, 1958.
J. Novák: On partition of an ordered continuum. Fund. Math. 39 (1952), 53–64.
V. Novák: On the well dimension of ordered sets. Czechoslovak Math. J. 19(94) (1969), 1–16.
V. Novák: Über Erweiterungen geordneter Mengen. Arch. Math. (Brno) 9 (1973), 141–146.
V. Novák: Some cardinal characteristics of ordered sets. Czechoslovak Math. J. 48(123) (1998), 135–144.
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.
M. Novotný: Bemerkung über die Darstellung teilweise geordneter Mengen. Spisy přír. fak. MU Brno 369 (1955), 451–458.
Ordered sets. Proc. NATO Adv. Study Inst. Banff (1981)(I. Rival, ed.).
M. Pouzet, I. Rival: Which ordered sets have a complete linear extension?. Canad. J. Math. 33 (1981), 1245–1254.
A. Rutkowski: Which countable ordered sets have a dense linear extension?. Math. Slovaca 46 (1996), 445–455.
J. Schmidt: Lexikographische Operationen. Z. Math. Logik Grundlagen Math. 1 (1955), 127–170.
J. Schmidt: Zur Kennzeichnung der Dedekind-Mac Neilleschen Hülle einer geordneten Menge. Arch. Math. 7 (1956), 241–249.
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.
W. Sierpiński: Cardinal and Ordinal Numbers. Warszawa, 1958.
W. Sierpiński: Sur une propriété des ensembles ordonnés. Fund. Math. 36 (1949), 56–67.
G. Szász: Einführung in die Verbandstheorie. Leipzig, 1962.
E. Szpilrajn: Sur l'extension de l'ordre partiel. Fund. Math. 16 (1930), 386–389.
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Novák, V., Novotný, M. Linear extensions of orderings. Czechoslovak Mathematical Journal 50, 853–864 (2000). https://doi.org/10.1023/A:1022424931030
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DOI: https://doi.org/10.1023/A:1022424931030