Re: David Benson's questions on terminology

Re: David Benson's questions on terminology

[Vaughan Pratt]

Universal algebraists (but not category theorists??) call b the _cover_ of
a when a<b with nothing in between.  See e.g. the index of such lattice
theory texts as Davey and Priestley or McKenzie, McNulty and Taylor
(Walter).  (MM&T distinguish upper cover and lower cover but obviously
an unqualified cover has to mean the upper kind to all but us Aussies.)
In a well-ordered set, "cover" and "successor" are synonymous: an ordinal
is a cover just when it is a successor ordinal.

Managed not to mention reflexivity---oops.

Vaughan Pratt

Re: David Benson's questions on terminology

[Robert J. MacG. Dawson]

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

Re: David Benson's questions on terminology

[Mamuka Jibladze]

> 	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