Operations with continuous extensions to compactifications
by
Reinhard Winkler
TU Wien, Institut für Diskrete Mathematik und Geometrie
Coauthors: Gabriel Maresch

Compactifications play an important role in several areas of mathematics. Examples: Group and semigroup compactifications in functional and abstract harmonic analysis; Odometers in the theory of dynamical systems; the Stone-Cech compactification of the natural numbers in combinatoric number theory. For General Algebra the following approach is natural.

Let (C, \iota) be a compactification of the set X, i.e.  \iota: X --> C with \iota(X) dense in the compact space C, and f: Xⁿ --> X an n-ary operation on X. The natural question is: Can f be continuously extended to C or, more explicitly: Is there some continuous F: Cⁿ --> C such that F(\iota(x₁), ... , \iota(x_n)) = \iota(f(x₁, ... , x_n)) for all x_i in X? This induces a Galois correspondence between operations and compactifications. It is easy to see that Galois closed sets of operations are clones. But many natural questions in this context seem to be non trivial but nevertheless worth being investigated.

Date received: May 17, 2004