Extension structures and compactifications
by
Werner Gähler
Scheibenbergstr. 37, D-12685 Berlin, Germany
Coauthors: Patrik Eklund (Department of Computer Science, Umeå University, SE-90187, Umeå, Sweden)

The notion of extension structure [2] depends on a fixed set functor j: \sf SET --> \sf SET. We therefore also speak on a j-extension structure. This notion contains important structural properties of Cauchy structures and is basically for a general completion theory. Let X be a set. By a j-extension structure on X we mean a triple (S, T, ~ ) consisting of two sets S subset or equal jX and T subset or equal jX ×X and an equivalence relation ~ on S such that, writing M \usebox\ko x instead of (M, x) in T, we have that (1)  M \usebox\ko x implies M in S, (2) M \usebox\ko x and M ~ N imply N \usebox\ko x, and (3) from M \usebox\ko x and N \usebox\ko x it follows M ~ N.

A j-extension structure (S, T, ~ ) and also the related j-extension space (X, (S, T, ~ )) is said to be separated provided that M \usebox\ko x and M\usebox\ko y imply x = y, and they are said to be complete provided that for each M in S there is an element x of X such that M \usebox\ko x holds.

We have that the category of all separated and complete j-extension spaces is an epireflective subcategory of the category of all separated j-extension spaces. As is shown in [3], the related completion construction can be applied for generating completion theorems in algebra, lattice theory and general topology, in particular they lead to a universal completion for Cauchy-spaces in the fuzzy filter case. Since compactifications can be identified with special Cauchy-completions, even different types of compactifications can be generated.

Among others, we present in this paper new results on the Richardson compactification for the fuzzy filter case applying new results on fuzzy filters. Note that this type of compactification was already treated in [1]. For a fixed non-degenerate infinitely distributive complete lattice L, here we mean by a fuzzy filter a mapping M: L^X --> L such that M([`0]) = 0 and M([`1]) = 1 for the constant fuzzy sets [`0], [`1] in L^X and M(f /\ g) = M(f) /\ M(g) for all fuzzy sets f, g in L^X.

A fuzzy filter M is said to be bounded in case M(f) <= supf holds for all f in L^X. A fuzzy filter M will be called distinguished provided that M(f \/ g) = M(f) \/ M(g) holds for all f, g in L^X. Moreover, M will be called balanced provided that a distinguished fuzzy filter finer than M exists.

In the general fuzzy filter case under the zero-condition of L (\alpha > 0,  \beta > 0   ===>  \alpha /\ \beta > 0) all fuzzy filters are balanced. In this case the Richardson compactification exists.

In the case of bounded fuzzy filters there may be fuzzy filters which are not balanced. If in this case we restrict us to balanced fuzzy filters, then the Richardson compactification also exists.

[1] P. Eklund, W. Gähler, Completions and compactifications by means of monads, in: Fuzzy Logic, State of Art, Kluwer, Dortrecht/Boston/London 1993, 39-56

[2] W. Gähler, Completion theory, in: Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 48/1991, 8

[3] W. Gähler, Extension structures and completions in topology and algebra, 2000

Date received: June 5, 2000

Copyright © 2000 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 # caeq-34.