From: "Ulrich Bodenhofer" <ulrich.bodenhofer@scch.at>
Subject: References on fundamental relationships between linear and partial orderings
Date: Wed, 14 Feb 2001 09:31:19 +0100
Newsgroups: sci.math,sci.math.research
Summary: Szpilrajn's Theorem (partial orders embed in linear orders) etc.

Hi,

I am looking for standard references in which I can find the following
fundamental properties (possibly with proofs):

1. For each partial ordering, there exists an extension which is
    a linear ordering (Szpilrajn's Theorem).
2. Every partial ordering can be represented as an intersection of
    linear orderings.
3. An ordering is linear if and only if there does not exists a larger
    ordering (one-to-one correspondence of linearity and maximality).

Assuming the Axiom of Choice, the proofs of all three propositions are
rather simple by using Zorn's Lemma. However, I would like to know
where they appeared first or where the community usually refers to
in this context. I only know

@article{Szpilrajn:30,
  author =  {E. Szpilrajn},
  title =  {Sur l'extension de l'ordre partiel},
  journal =  {Fund. Math.},
  volume =  {16},
  pages =  {386--389},
  year =  {1930}
}

and some textbooks on algebra.

Any help is gratefully appreciated!

Ulrich
==============================================================================

From: "Dave L. Renfro" <renfrod@central.edu>
Subject: Re: References on fundamental relationships between linear and partial orderings
Date: 15 Feb 01 13:42:45 -0500 (EST)
Newsgroups: sci.math.research

[quote of portion of previous post deteled --dhr]

The most complete introductory (i.e. no forcing) treatment I
know is ---->>>

Joseph G. Rosenstein, "Linear Orderings", Pure and Applied
Mathematics 98, Academic Press, 1982, xvii + 487 pages.
MR 84m:06001; Zbl 488.04002

You might also want to look through Sierpinski's book
"Cardinal and Ordinal Numbers".

Dave L. Renfro
==============================================================================

From: "Ulrich Bodenhofer" <ulrich.bodenhofer@scch.at>
Subject: Re: References on fundamental relationships between linear and partial orderings
Date: Mon, 19 Feb 2001 15:01:06 +0100
Newsgroups: sci.math.research

Thanks a lot! Our university library has the book and I am eager
to see it.

In the meantime, I found another reference which seems to be the
first paper in which the "intersection representation" occurs:

@article{DushnikMiller:41,
  author =  {B. Dushnik and E. W. Miller},
  title =  {Partially Ordered Sets},
  journal =  {Amer. J. Math.},
  volume =  {63},
  pages =  {600--610},
  year =  {1941}
}


Best regards,
Ulrich Bodenhofer

[quote of previous post deteled --djr]