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Which Ordered Sets have a Complete Linear Extension?

Published online by Cambridge University Press:  20 November 2018

Maurice Pouzet  and
Ivan Rival
Maurice Pouzet
Affiliation:
Université de Lyon I, Villeurbanne, France
Ivan Rival
Affiliation:
University of Calgary, Calgary, Alberta
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It is a well known and useful fact [4] that every (partially) ordered set P has a linear extension L (that is, a totally ordered set (chain) on the same underlying set as P and satisfying a ≦ b in L whenever a ≦ b in P). It is just as well known that an ordered set P can be embedded in an ordered set P′ which, in turn, has a complete linear extension L′ (that is, a linear extension in which every subset has both a supremum and an infimum); just take L′ to be the “completion by cuts” of L. However, an arbitrary ordered set P need not, itself, have a complete linear extension (for example, if P is the chain of integers or, for that matter, if P is any noncomplete chain). It is natural to ask which ordered sets have a complete linear extension?

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 5 , 01 October 1981 , pp. 1245 - 1254
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Bonnet, R. and Pouzet, M., Extension et stratification d''ensembles dispersés, C.R. Acad. Sc. Paris 268 (1969), 1512–1515.Google Scholar
2. Dushnik, B. and Miller, E. W., Partially ordered sets, Amer. J. Math. 63 (1941), 600–610.Google Scholar
3. Raney, G. N., Completely distributive complete lattices, Proc. Amer. Math. Soc. 3 (1952), 677–680.Google Scholar
4. Szpilrajn, E., Sur Vextension de Vordre partiel, Fund. Math. 16 (1930), 386–389.Google Scholar