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I need a reference that discusses the difference between a weakly ordered set and a partially ordered set.

My understanding is that a weakly ordered set $<X, \succsim>$ is one where the partial ordering induces a total ordering on the equivalence classes of $X$. This is correct, yes?

S.J.
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  • Yes, that’s correct; equivalently, it’s a total preorder, i.e., a binary relation that is total, transitive, and reflexive. (Reflexivity actually follows from the first two conditions, but it’s part of the definition of a preorder, so I include it.) – Brian M. Scott May 23 '15 at 17:16
  • Are you interested in the reference as a reference, or do you have some specific questions about the difference(s) between the two concepts? – Brian M. Scott May 24 '15 at 07:02
  • I would like to have something I can cite. – S.J. May 24 '15 at 17:07

1 Answers1

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After a long search, I found a clear source I can site.

Louis Narens, Probabilistic Lattices (2015) in chapter 2 on page 29.

According to the text:

A partial ordering is one that is

  • reflexive: $x \succsim x$
  • transitive: $x \succsim y$ and $y \succsim z$ implies $x \succsim z$
  • and anti-symmetric: if $x \succsim y$ and $y \succsim x$, then $x = y$

A weak ordering is one that is

  • transitive (stated above)
  • and connected: either $x \succsim y$ or $y \succsim x$.

A total ordering is one that is

  • a weak ordering,
  • and anti-symmetric (stated above).

"Note that a weak ordering is a partial ordering."

S.J.
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