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International Conference on Applicable General Topology
August 12-18, 2001
Hacettepe University
Ankara, Turkey |
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Organizers
L. M. Brown (Ankara), G. Brümmer (Cape Town), M. Diker (Ankara), M. Henriksen (Claremont), R. D. Kopperman (New York), G. M. Reed (Oxford), I. L. Reilly (Auckland), S. Salbany (Pretoria), D. Spreen (Siegen)
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Textural Categories, a Preliminary Report.
by
Senol Dost
Hacettepe University
Coauthors: Lawrence M. Brown (Hacettepe University)
By a texturing [2-4] of a set S we mean a subset S of the
power set P(S) which is a point separating complete, completely
distributive lattice with respect to inclusion which contains S and
\emptyset, and for which arbitrary meets coincide with intersections
and finite joins coincide with unions. The pair (S, S) is then
called a texture space, or simply texture. A texture
(S, S) is complemented if we have an idempotent mapping
\sigma:S --> S which reverses inclusion. It is simple
if the only molecules (join irreducible elements) of S are the
sets Ps, s in S, where Ps is the smallest element of S
containing s.
By a ditopology (\tau, \kappa) on (S, S) we mean a family
\tau subset or equal S of open sets containing S, \emptyset and closed
under finite meets and arbitrary joins, and a generally unrelated family
\kappa subset or equal S
of closed sets containing S, \emptyset and closed under finite
joins and arbitrary meets. If \sigma is a complementation and \kappa = \sigma(\tau) then (\tau, \kappa) is called complemented.
Textures, perhaps restricted to be simple or complemented, and on which may be
defined a possibly complemented ditopology, can be taken as the objects for a
variety of categories. Natural candidates for the morphisms between textures
(S, S) and (T, T) include
- Functions from S and T preserving the texturings S and
T in some suitable way and where appropriate restricted to
preserve complementation or have continuity properties.
- Structure preserving mappings between S and T.
- Difunctions [2] from (S, S) to (T, T), or more generally
relations, corelations or direlations from (S, S) to (T, T),
again restricted where appropriate to preserve complementation or have suitable
continuity properties.
Of natural interest are interrelations between the various categories obtained
by choosing the objects and morphisms as outlined above, and it is also of
interest to consider connections between these categories and certain categories
which occur in the literature. In this preliminary report we consider in
particular relations between certain textural categories and (subcategories of)
the category of sober topological spaces, categories of fuzzy lattices
and of Hutton Spaces in the sense of [1].
References
- D. Adnadjevic and A. P. Sostak, On inductive dimensions for
fuzzy topological spaces, in Fuzzy Topology, Fuzzy Sets and Systems 73 (1),
1995, 5-12.
- L. M. Brown, Relations and functions on texture spaces,
Preprint.
- L. M. Brown and M. Diker, Ditopological texture spaces and
intuitionistic sets, Fuzzy Sets and Systems 98, 1998, 217-224.
- L. M. Brown and R. Ertürk, Fuzzy Sets as Texture Spaces, I.
Representation Theorems, Fuzzy Sets and Systems, 110 (2), 2000, 227-236.
Date received: July 26, 2001
Copyright © 2001 by the author(s).
The author(s) of this document and the organizers of the conference
have granted their consent to include this abstract in
Atlas Conferences Inc.
Document # cagx-58.