Set-systems with signed solutions

AbstractQualitative properties of general collections of systems of m linear equations in n unknowns are studied. More precisely, a family Ax=B of systems of linear equations is determined by A=(A1,A2,…,An) where each Ai and B is a nonempty subset of Rm. The family consists of all systems Ax=b̃ where the ith column of A belongs to Ai, and b̃ belongs to B. These general collections include the purely qualitative systems (where each Ai and B is set of all vectors with a given sign pattern), cone-systems (where each Ai and B is a cone in Rm), and interval systems (where each Ai and B is an interval of vectors). We give necessary and sufficient conditions on A and B for the sign patterns of the solutions to Ax=b̃ to be independent of the choice of A and b̃