Multi-topological Algebras by Matt Insall and Jeff Dalton University of Missouri – Rolla

Topological algebras are algebraic structures with a topology that is compatible, in the sense that the operations are continuous in the appropriate product topologies on the underlying sets. For example, a topological group is a group G on which there is a topology t such that the group operation is continuous in the product topology on GxG and the inversion operation is continuous in the topology on G. The theories of topological semigroups, of topological groups, and of topological fields and vector spaces are all quite well developed. But for applications, one may not be fortunate enough to have the operations continuous in the product topologies. On the other hand, powers of algebraic structures can frequently host a variety of useful topologies, and the operations may be continuous in some of these.

A multi-topological algebra is an algebra A, together with a system of topologies on the appropriate powers of the underlying set of A, such that on the corresponding topological spaces, the operations of the algebra in question are continuous. The authors have never seen this definition in any other works, so it is reasonable for us to assume that the theory of such structures is entirely undeveloped. We propose to develop it. We will begin with multi-topological dioids, which are of interest to the second author due to their applicability in the design of scheduling algorithms in manufacturing processes, and with multi-topological semigroups, multi-topological groups, multi-topological fields, etc., by investigating which results in the classical theories of topological algebras of those types carry over naturally to our setting. Along the way we will devise examples which indicate which results do not carry over at all or which do not carry over easily. We will develop a web site to disseminate our simpler results, and we will publish in archival journals those results that we obtain which are more complicated. We will hire students and post-doctoral fellows to learn from us the fundamentals and carry on the work beyond what we will have time to do ourselves. We will hold a conference for the study of such structures, and invite also classical topological algebraists to the conference so they may contribute to the discussions and literature of this new area. We will invite nonstandard analysts to join in determining how to use nonstandard analysis effectively in this new setting also.