Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-08T07:19:19.346Z Has data issue: false hasContentIssue false
  • English
  • Français

Structure Results for Function Lattices

Published online by Cambridge University Press:  20 November 2018

D. Duffus ,
B. Jónsson  and
I. Rival
D. Duffus
Affiliation:
University of Calgary Calgary, Alberta
B. Jónsson
Affiliation:
University of Calgary Calgary, Alberta
I. Rival
Affiliation:
Vanderbilt University Nashville, Tennessee
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.

For partially ordered sets X and F let Yx denote the set of all order-preserving maps of X to Y partially ordered by f ≦ g if and only if f(x)≦ g (x) for each x ∈ X [1; 4; 6]. If X is unordered then Yx is the usual direct product of partially ordered sets, while if both X and Y are finite unordered sets then Yx is the commonplace exponent of cardinal numbers. This generalized exponentiation has an important vindication especially for those partially ordered sets that are lattices.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 2 , 01 April 1978 , pp. 392 - 400
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Birkhoff, G., Extended arithmetic, Duke Math. J. 3 (1937), 311–316.Google Scholar
2. Birkhoff, G., C-wgs of sets, Duke Math. J. 3 (1937), 443–454.Google Scholar
3. Birkhoff, G. Lattice theory (American Mathematical Society, Providence, R.I., second edition, 1948, third edition, 1967).Google Scholar
4. Birkhoff, G. Generalized arithmetic, Duke Math. J. 9 (1942), 283–302.Google Scholar
5. Davey, B. A., Duffus, D., Quackenbush, R. W. and Rival, I., Exponents of finite simple lattices, J. London Math. Soc, (to appear).Google Scholar
6. Day, M. M., Arithmetic of ordered systems, Trans. Amer. Math. Soc. 58 (1945), 1–43.Google Scholar
7. Priestley, H. A., Representation of distributive lattices by means of ordered Stone spaces, Bull. London Math. Soc. 2 (1970), 186–190.Google Scholar
8. Priestley, H. A., Ordered topological spaces and the representation of distributive lattices, Proc. London Math. Soc. (3) 24 (1972), 507–530.Google Scholar