No subject

[Jonathan Post]

As wikipedia comments:

It follows that a totally ordered set is a distributive
lattice<http://en.wikipedia.org/wiki/Distributive_lattice>
....

Totally ordered sets form a full
subcategory<http://en.wikipedia.org/wiki/Subcategory>of the
category <http://en.wikipedia.org/wiki/Category_%28mathematics%29> of partially
ordered sets <http://en.wikipedia.org/wiki/Partial_order>, with the
morphisms being maps which respect the orders....

A simple counting argument will verify that any finite total order (and
hence any subset thereof) has a least element. Thus every finite total order
is in fact a well order <http://en.wikipedia.org/wiki/Well_order>. Either by
direct proof or by observing that every well order is order
isomorphic<http://en.wikipedia.org/wiki/Order_isomorphic>to an
ordinal <http://en.wikipedia.org/wiki/Ordinal> one may show that every
finite total order is order
isomorphic<http://en.wikipedia.org/wiki/Order_isomorphic>to an initial
segment<http://en.wikipedia.org/w/index.php?title=Initial_segment&action=edit>of
the natural numbers ordered by <. In other words a total order with k
elements is induced by a bijection with the first k natural numbers. Hence
it is common to index finite total orders or countable well orders by
natural numbers in a fashion which respects the ordering.

Contrast with a partial order <http://en.wikipedia.org/wiki/Partial_order>,
which lacks the third condition. An example of a partial order is the
happened-before <http://en.wikipedia.org/wiki/Happened-before> relation.

Could the author of the sequence mean something like "the number of totally
ordered subset of all partial orders on n points, up to isomorphism"? Or
relate to your Category enumeration?

-- Jonathan Vos Post


On 12/6/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> http://www.research.att.com/~njas/sequences/A000669 is "Number of
> series-reduced planted trees with n leaves. ... also the number of
> total orderings of n unlabeled points."
>
> The number of total orderings of n unlabeled points is 1.
>
> What is this portion of the description intended to mean?
>
> Franklin T. Adams-Watters
>
> ________________________________________________________________________
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