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