The Failure of Cancellation Laws for Equidecomposability Types

Abstract

AbstractLet B be a Boolean algebra and G a group of automorphisms of B. Define an equivalence relation ∼ on B by letting x ∼ y if there are x1, x2,…,xn, y1, y2, …yn in B such that x is the disjoint union of the xi, y is the disjoint union of the yi, and for each i there is a member of G taking xi to yi. The equivalence classes under ∼ are called equidecomposability types. Addition of equidecomposability types is given by (x) + (y) = (x V y) provided x ∧ y = 0. An example is given of a complete Boolean algebra B and a group G of automorphisms of B with X, Y ∊ B such that (X) + (X) = (Y) + (Y) but (X) ≠ (Y), answering a question of Wagon (see [5 p. 231 problem 14]). Moreover B may be taken to be the algebra of Borel subsets of Cantor space modulo sets of the first category. It is also remarked that in this case equidecomposability types do not form a weak cardinal algebra.