Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T17:32:51.469Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural Partial Orders

Published online by Cambridge University Press:  20 November 2018

R. A. Dean  and
Gordon Keller
R. A. Dean
Affiliation:
California Institute of Technology, Pasadena, California
Gordon Keller
Affiliation:
California Institute of Technology, Pasadena, California
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let n be an ordinal. A partial ordering P of the ordinals T = T(n) = {w: w < n} is called natural if x P y implies x ⩽ y.

A natural partial ordering, hereafter abbreviated NPO, of T(n) is thus a coarsening of the natural total ordering of the ordinals. Every partial ordering of a finite set 5 is isomorphic to a natural partial ordering. This is a consequence of the theorem of Szpielrajn (5) which states that every partial ordering of a set may be refined to a total ordering. In this paper we consider only natural partial orderings. In the first section we obtain theorems about the lattice of all NPO's of T(n).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 535 - 554
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Birkhoft, G., Lattice theory, rev. éd., Amer. Math. Soc. Colloq. Publ., Vol. 25 (Amer. Math. Soc, New York, 1948).Google Scholar
2. Dilworth, R. P. and Crawley, P., Decomposition theory for lattices without chain conditions, Trans. Amer. Math. Soc, 96 (1960), 1–22.Google Scholar
3. Feller, W., An introduction to probability theory and its applications, Vol. 1, 2nd ed. (Wiley, New York, 1957).Google Scholar
4. Hall, M., Jr., Combinatorial theory (Blaisdell, Waltham, 1967).Google Scholar
5. Szpielrajn, E., Sur Vextension de Vordre partiel, Fund. Math., 16 (1930), 386–389.Google Scholar