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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
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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

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3. Dilworth, R. P., The normal completion of the lattice of continuous functions. Trans. Amer. Math. Soc, 68 (1950), 427–438.Google Scholar
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5. Stone, M. H., Boundedness properties of function lattices, Can. J. Math., 1 (1949), 176–186.Google Scholar