From: "Robert J. MacG. Dawson" <Robert.Dawson@STMARYS.CA>
To: categories@mta.ca
Subject: Re: David Benson's questions on terminology
Date: Tue, 28 Nov 2000 14:57:46 -0400 [thread overview]
Message-ID: <3A24002A.2762B24B@stmarys.ca> (raw)
In-Reply-To: <200011281143.LAA22264@koi-pc.dcs.qmw.ac.uk>
Paul Taylor wrote:
>
> (1) I would say (rather strongly) that it is ill-conceived to
> try to generalise the successor relation from the natural numbers
> to arbitrary partial orders. The successor relation is an aspect
> of the inductive/recursive/well founded structure on N, and it
> is wrong to confuse well founded relations (which are necessarily
> IRreflexive) with partial arders (which are Reflexive).
>
> See Sections 2.7, 3.1 and elsewhere in "Practical Foundations".
I don't think David was trying to generalize the successor
relation in the sense of finding a "moral equivalent" in a poset for
the natural numbers' successor _function_. All he wants - I think -
is a notation for "a > b and there is no a>c>b". I would suggest
using an indefinite article with a noun formation:
" a is _a_ successor of b"
or a prepositional formation that does not connote uniqueness or
necessary existence:
"a is immediately above b"
Bob Pare and I used "<!" for this in our 1993 paper on tileorders.
It may be - is this what you're getting at, Paul? - that if one
finds a successor relation is natural or useful for what one's looking at,
then one should wonder hard about whether it would be better thought of as
a well-founded structure rather than as a poset, so as to avoid the
repetition of "and not equal to". But there are certainly cases where
after the wondering one would conclude "no it isn't."
-Robert Dawson
next parent reply other threads:[~2000-11-28 18:57 UTC|newest]
Thread overview: 3+ messages / expand[flat|nested] mbox.gz Atom feed top
[not found] <200011281143.LAA22264@koi-pc.dcs.qmw.ac.uk>
2000-11-28 18:57 ` Robert J. MacG. Dawson [this message]
2000-11-29 13:12 ` Mamuka Jibladze
2000-11-29 9:25 Vaughan Pratt
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