Separating OR, SUM, and XOR circuits

Given a boolean n×n matrix A we consider arithmetic circuits for computing the transformation x↦Ax over different semirings. Namely, we study three circuit models: monotone OR-circuits, monotone SUM-circuits (addition of non-negative integers), and non-monotone XOR-circuits (addition modulo 2). Our focus is on separating OR-circuits from the two other models in terms of circuit complexity: