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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

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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