Polyboxes, cube tilings and rigidity

A non-empty subset A of X = X-1 x ... x X (d) is a (proper) box if A = A(1) x ... x A(d) and A(i) subset of X (i) for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A(i) = B-i , A(i) = X-i \B-i , A(i) is not an element of{B-i ,X-i\B-i }. Let F and G be two systems of disjoint boxes. Can one decide whether boolean OR F = boolean OR G? In general, the answer is 'no', but as is shown in the paper, it is 'yes' if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A(i) = X-i\B-i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that boolean OR F = boolean OR G implies F = G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.