Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-12-03T20:36:36.293Z Has data issue: false hasContentIssue false
  • English
  • Français

A Logarithmic Property for Exponents of Partially Ordered Sets

Published online by Cambridge University Press:  20 November 2018

Dwight Duffus  and
Ivan Rival
Dwight Duffus
Affiliation:
The University of Calgary Calgary, Alberta
Ivan Rival
Affiliation:
The University of Calgary Calgary, Alberta
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.

In an effort to unify the arithmetic of cardinal and ordinal numbers, Garrett Birkhoff [2; 3; 4; 5] (cf. [6]) defined several operations on partially ordered sets of which at least one, (cardinal) exponentiation, is of considerable independent interest: for partially ordered sets P and Q let PQ denote the set of all order-preserving maps of Q to P partially ordered by f ≦ g if and only if f(x) ≦ g(x) for each x ∈ Q.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 4 , 01 August 1978 , pp. 797 - 807
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Banaschewski, B., Hullensysteme und Erweiterungen von Quasi-Ordnungen, Z. Math. Logik Grundlagen Math. 2 (1956), 117–130.Google Scholar
2. Birkhoff, G., Extended arithmetic, Duke Math. J. 3 (1937), 311–316.Google Scholar
3. Birkhoff, G., Rings of sets, Duke Math. J. 3 (1937), 443–454.Google Scholar
4. Birkhoff, G., Generalized arithmetic, Duke Math. J. 9 (1942), 283–302.Google Scholar
5. Birkhoff, G., Lattice theory (American Mathematical Society, Providence, R.I., second edition, 1948).Google Scholar
6. Day, M. M., Arithmetic of ordered systems, Trans. Amer. Math. Soc. 58 (1945), 1–43.Google Scholar
7. Fuchs, E., Isomorphismus der Kardinalpotenzen, Arch. Math. (Brno) 1 (1965), 83–93.Google Scholar
8. Hashimoto, J., On the product decomposition of partially ordered sets, Math. Japonicae 1 (1948), 120–123.Google Scholar
9. Hashimoto, J., On direct product decomposition of partially ordered sets, Ann. of Math. (2) 54 (1951), 315–318.Google Scholar
10. L., Lovâsz, Operations with structures, Acta. Math. Acad. Sci. Hung. 18 (1967), 321–328.Google Scholar
11. On the cancellation law among finite relational structures, Period. Math. Hungar. 1 (1971), 145–156.Google Scholar
12. Novotny, M., Tjber gewisse Eigenschaften von Kardinaloperationen, Spisy Prirod. Fak. Univ. Brno (1960), 465–484.Google Scholar
13. Schmidt, J., Zur Kennzeichnung der Dedekind-MacNeilleschen Hillle einer geordneten Menge, Arch. Math. 7 (1956), 241–249.Google Scholar