[xtm-wg] XTM-CMS: Set theory stuff

["Wrightson, Ann" <Ann.Wrightson@sweetandmaxwell.co.uk>]

Here's a copy of the stuff I did late in the week-end. Sure to have bugs -
if you see any, please tell me...

Ann W.


Some Basic Formal and Ontological Properties of Topic Maps

AMW 15/10/00

1. A Topic Map is part of the World

What follows pertains to a single Topic Map : "the Topic Map", which is 
an example exemplifying properties expected of all XTM Topic Maps.
  
All sets below are sets with a membership function which can be effectively
evaluated.

1a: The components of the topic map form a set

2. The World contains divers particulars which can be located;
some of these are Resources with respect to the Topic Map

2a  Those components of the Topic Map which refer directly to Resources form
a set.

2b At any stage in the evaluation/use of the Topic Map, the Resources in the
World for which there 
is a resolved reference in the topic map, form a set

2c At any stage in the evaluation/use of the Topic Map, the mapping from the
set of Topic Map components 
which refer directly to Resources as in 2a, and the set of Resources as in
2b, is a function.

2d There is nothing to prevent Topic Map components also being Resources 
(with respect to either "the Topic Map" or another Topic Map)

3. The World contains notions, concepts, entities etc which are Subjects of
the 
information in the Topic Map

3a The World contains Resources which, by virtue of phenomena outside the
Topic Map,
act as designators for Subjects with respect to the Topic Map, and these 
Resources form a set.

3b Those components of the Topic Map which refer directly to the Resources
which are 
acting as designators as in 3a, form a set, and this set is a subset of the
set in 2a.

3c The mapping from the set in 3b to the set in 3a, is a function.

4. Within the Topic Map, there are components which are intended to
represent Subjects, 
(these, at least, are called Topics), and these form a set.

4a. There is a function from the set in 4. to the set in 3b (topic identity)

5. Within the Topic Map, there are components which are intended to
represent relations
amongst Subjects, and these components form a set. (topic association). 

5a. Each member of the set as in 5 is a set which is a subset of the set in
4.

5b. Within the topic map, there are components which are intended to
represent
functions from the sets as in 5a. to the set in 4. (association-role)

5c. Within the Topic Map, there are components which are intended to
represent one-many 
relations between the members of the set in 5., and the members of the set
in 4. 
(association scope, as comprehension function of a set of types), and these
form a set.

6. Within the Topic Map, there are components which are intended to
represent 
relations between Subjects and sets of Resources, and these form a set
(topic occurrence). 

6a. Within the topic map, there are components which are intended to
represent
a function from the set as in 6. to the set in 4. (occurrence-role), and
these form a set

6c. Within the Topic Map, there are components which are intended to
represent one-many 
relations between the members of the set in 6., and the members of the set
in 4. 
(occurrence scope, as comprehension function of a set of types), and these
form a set.

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