A (0, 1) matrix A is said to be ideal if all the vertices of the polytope Q(A) = {x | Ax ⩾ 1, 0 ⩽ x ⩽ 1} are integral. The issue of finding a satisfactory characterization of those matrices which are minimally non-ideal is a well known open problem. An outstanding result toward the solution of this problem, due to Alfred Lehman, is the description of crucial properties of minimally non-ideal matrices. In this paper we consider the extension of the notion of ideality to (0,±1) matrices. By means of a standard transformation, we associate with any (0,±1) matrix A a suitable (0,1) matrix D(A). Then we introduce the concept of disjoint completion A+ of a (0,±1) matrix A and we show that A is ideal if and only if D(A+) is ideal. Moreover, we introduce a suitable concept of a minimally non-ideal (0,±1) matrix and we prove a Lehman-type characterization of minimally non-ideal (0,±1) matrices.
