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The 3-Irreducible Partially Ordered Sets

Published online by Cambridge University Press:  20 November 2018

David Kelly
David Kelly*
Affiliation:
University of Manitoba, Winnipeg, Manitoba
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Abstract

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The dimension [4] of a partially ordered set (poset) is the minimum number of linear orders whose intersection is the partial ordering of the poset. For a positive integer m, a poset is m-irreducible[10] if it has dimension m and removal of any element lowers its dimension. By the compactness property of finite dimension, every m-irreducible poset is finite and every poset of dimension ≧ m contains an m-irreducible subposet.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 2 , 01 April 1977 , pp. 367 - 383
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Baker, K. A., Dimension, join-independence, and breadth in partially ordered sets, 1961 (unpublished).Google Scholar
2. Banaschewski, B., Hilllensysteme und Erweiterungen von Quasi-Ordnungen, Z. Math. Logik Grundlagen Math. 2 (1956), 117–130.Google Scholar
3. Birkhoff, G., Lattice theory, 3rd. ed. (American Mathematical Society, Providence, R.I., 1967).Google Scholar
4. Dushnik, B. and Miller, E. W., Partially ordered sets, Amer. J. Math. 63 (1941), 600–610.Google Scholar
5. Hiraguchi, T., On the dimension of partially ordered sets, Sci. Reports Kanazawa Univ. 1 (1951), 77–94.Google Scholar
6. Kelly, D. and Rival, I., Crowns, fences and dismantlable lattices, Can. J. Math. 26 (1974), 1257–1271.Google Scholar
7. Kelly, D. and Rival, I. Planar lattices, Can. J. Math. 27 (1975), 636–665.Google Scholar
8. Kelly, D. and Rival, I. Certain partially ordered sets of dimension three, J. Combinatorial Theory Ser. A 18 (1975), 239–242.Google Scholar
9. Schmidt, J., Zur Kennzeichnung der Dedekind-MacNeilleschen Hiille einer geordneten Menge, Arch. Math. 7 (1956), 241–249.Google Scholar
10. Trotter, W. T., Jr., Irreducible posets with large height exist, J. Combinatorial Theory Ser. A 17 (1974), 337–344.Google Scholar
11. Trotter, W. T., Jr., and Moore, J. I., Jr., Characterization problems for graphs, partially ordered sets, lattices, and families of sets, Discrete Math, (to appear).Google Scholar