Subject: categories: Re: David Benson's questions on terminology

From: "Robert J. MacG. Dawson" <[email-masked]>
Date: Tue, 28 Nov 2000 14:57:46 -0400

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

---

From: Mamuka Jibladze <[email-masked]>
Date: Wed, 29 Nov 2000 15:12:53 +0200 (EET)

> Bob Pare and I used "<!" for this in our 1993 paper on tileorders.

In that case I believe there is established terminology/notation in
lattice theory, they say "a covers b", denoted b -< a, and corresponding
intervals (i.e. intervals with [a,b]={a,b}) are called gaps.

Mamuka