Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-20T07:14:12.111Z Has data issue: false hasContentIssue false
  • English
  • Français

The k-Normal Completion of Function Lattices

Published online by Cambridge University Press:  20 November 2018

Henry B. Cohen
Henry B. Cohen*
Affiliation:
University of Pittsburgh
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.

A subset G of a non-empty partially ordered set C is called normal if it coincides with the set of all upper bounds of the set of lower bounds of G. This is equivalent to stipulating that G be the set of all upper bounds of some subset of C called a set of generators for G. When ordered by inclusion, the family of all normal subsets of C forms a complete lattice with maximum C and minimum empty or singleton. The meet operation is simply point set intersection; whence, the meet of a family Gi of normal subsets is the set of upper bounds of ∪ Fi where Fi generates Gi for each i. A normal subset is called proper if it is neither void nor C, and the proper normal subsets. of C form a boundedly complete lattice.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 669 - 675
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Bade, W. G., The space of all continuous functions on a compact Hausdorff space, Univ. of California, Berkeley, 1957, unpublished seminar notes.Google Scholar
2. Cohen, H. B., The k-extremally disconnected spaces as projectives, Can. J. Math., 16 (1964), 253–260.Google Scholar
3. Dilworth, R. P., The normal completion of the lattice of continuous functions. Trans. Amer. Math. Soc, 68 (1950), 427–438.Google Scholar
4. Gillman, L. and Jerison, M., Rings of continuous functions (New York, 1960).Google Scholar
5. Stone, M. H., Boundedness properties of function lattices, Can. J. Math., 1 (1949), 176–186.Google Scholar