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*To*: categories@mta.ca*Subject*: categories: Re: David Benson's questions on terminology*From*: "Robert J. MacG. Dawson" <Robert.Dawson@STMARYS.CA>*Date*: Tue, 28 Nov 2000 14:57:46 -0400*References*: <200011281143.LAA22264@koi-pc.dcs.qmw.ac.uk>*Sender*: cat-dist@mta.ca

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

**Follow-Ups**:**categories: Re: David Benson's questions on terminology***From:*Mamuka Jibladze <jib@rmi.acnet.ge>

**References**:**categories: David Benson's questions on terminology***From:*Paul Taylor <pt@dcs.qmw.ac.uk>

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