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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 19th 2014

    New page: plump ordinal. Is there a general name for a pair of inequalities <\lt and \le which satisfy the expected laws? They’re kind of weird from a categorical perspective; \le is a poset (or preorder), of course, and <\lt is an endo-profunctor of it, but <\lt has a multiplication and a counit, so it’s neither a monad nor a comonad in ProfProf.

    • CommentRowNumber2.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 20th 2014
    • (edited Feb 20th 2014)

    I don't know a name, but there should be one! Another place that this pair comes up is in non-located (one-sided or MacNeille) real numbers.

    We can leave transitivity of <\lt out of the list of axioms (since it follows in two ways), but we also should add its irreflexivity: If x<xx \lt x, then False.

    An unbiased list of axioms is (in addition to <\lt \Rightarrow \leq) this:

    • If x 0x nx_0 \leq \cdots \leq x_n, then x 0x nx_0 \leq x_n;
    • The conclusion can be strengthened to x 0<x nx_0 \lt x_n iff at least one of the premises can be strengthened to x i<x i+1x_i \lt x_{i+1}.

    The entire forward direction of the second item here is missing from your list; it may be too strong.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 20th 2014

    I think it is too strong. Consider the ordinal x={0|P}x=\{0|P\}. Then 0x0\le x and x1x\le 1 and 0<10\lt 1, but “0<x0\lt x or x<1x\lt 1” makes P decidable.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 20th 2014

    (And all those ordinals are plump.)

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 20th 2014

    Another way to say what we’re talking about is that the same set is equipped with both a partial order and a quasiorder in a “compatible” way.

    • CommentRowNumber6.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 20th 2014

    Yes, this would restrict attention to some sort of super-plump ordinal (and so not be for the page plump ordinal).

    But I don't exactly understand Taylor's motivation for restricting to plump ordinals in the first place.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 24th 2014

    I don’t entirely understand his motivation myself, but it seems to be at least partly that he wants a successor satisfying xyx \le y iff x<syx \lt s y. Joyal-Moerdijk also arrived at an equivalent notion of plump ordinal from algebraic-set-theory considerations.

    My recent interest in plump ordinals comes from the fact that they embed constructively in the surreals.

    • CommentRowNumber8.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 24th 2014

    My recent interest in plump ordinals comes from the fact that they embed constructively in the surreals.

    Yes, I noticed that! I also see that you have a higher inductive definition of plump ordinals themselves. This makes them more interesting to me too.

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeSep 21st 2023

    Examples

    diff, v5, current