Locally Moving Groups and Reconstruction Problems

Abstract

Let B be a complete atomless Boolean algebra. For g ∈ Aut(B) (***Math Type****). Let G ≤ Aut(B). We say that G is a locally moving subgroup of Aut(B) if {var(g) ∣ g ∈ G} is dense in B. We show (Corollary 1.4) that for every complete atomless Boolean algebra B and a locally moving subgroup G of Aut(B), the group G determines B up to isomorphism. We then use this theorem to prove that certain mathematical structures are determined by their automorphism group. In Section 3 we show such a reconstruction theorem for a certain class of locally compact topological spaces. Then we prove a reconstruction theorem for subspaces of the foliated torus. In Section 4 we present a reconstruction theorem for 2-homogeneous linear orders and circular orders.

This work was prepared in 1994–95 while the author was a visitor in Bowling Green State University, Bowling Green, Ohio and in the University of Colorado, Boulder, Colorado. The author is most grateful to both of these institutions.