# categories: More about names and notation

Dear Colleagues,

Thank you for the responses!

(1) Several pointed out that my <<followers>> relations are
said to be <<covering>> relations in combinatorics.
These relations are defined as relations which are
1. acyclic
2. without interpolants, as Robert Dawson mentioned.
A followers relation for discrete partial order R is
the least relation whose reflexive and transitive closure
is R.
This agrees entirely with the definition of a
successor relation being the least relation whose
transitive closure is a discrete strict total order and whose
reflexive and transitive closure is a discrete total order.
In the case of successor, there is a distinct
next element, as in the succession of the Kings (and Queens)
of England.  In the case of followers, there is in
general a set of next elements, such as the followers
of Cromwell.

(2) While several alternatives were offered, I will try
Paul Taylor's <<instance>> of a relation to describe
an ordered pair (x,y) \in R.

(3) The problem of a good name for the sets Nat_k remains.
Suggestions included
finite ordinals
numerals
order ideals
and by far the most appropriate for my purposes,
index sets
[This problem of a good name is worthy of some further attention.
Computer science second year students will think
of <<finite ordinals>> as a number, not a set,
of <<numerals>> as representations of the digits in a number system,
and have not be exposed to order ideals.]

Cheers,
David
--
Professor David B. Benson                                (509) 335-2706
School of EE and Computer Science (EME 102A)             (509) 335-3818 fax
PO Box 642752, Washington State University               dbenson@eecs.wsu.edu
Pullman WA 99164-2752   U.S.A.