Re: David Benson's questions on terminology

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.

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