Lexicographic Exponentiation of Chains.
by
Salma Kuhlmann
University of Saskatchewan
Coauthors: Charles Holland, Stephen McCleary

In his seminal 1908 Mathematische Annalen paper, Hausdorff introduces arithmetic operations on chains (lexicographic sums and products) generalizing Cantor's ordinal arithmetic. Roughly, the lexicographic product of a family of chains (indexed by a chain) is the Cartesian product, ordered lexicographically from the right. Several fascinating open problems concerning lexicographic chains arise naturally. In this talk, I will present an overview of some of these problems (that we have been studying in the last few years) and the progress made to date. I will focus on the following 3 Problems: I) The Isomorphism Problem: We investigate whether isomorphism of lexicographic powers implies isomorphism of the exponents. II) The Convex Embedding Problem: We investigate when it is possible to embed the exponent in the lexicographic power, such that the image is convex. (This problem was encountered while studying non-archimedean ordered exponential fields, and I will briefly indicate the connection). III) 2-Transitivity of the Automorphism Group: We study the automorphism group of some lexicographic powers and prove that they are 2-transitive. Some of our results generalize results of Hausdorff in this direction.

I will end the talk with a list of open problems and ideas for further research.

Date received: December 26, 2002