Min-max systems, or min-max-plus systems are algebraic models of discrete-event dynamic systems. They are nonlinear extensions of the well known linear max-plus system models. The structural property "inseparability" of min-max systems was proposed by us in a recent paper. It turns out that inseparability is equivalent to the property of "irreducibility" proposed recently by van der Woude and Subiono, although the two properties appear in very different forms. We will prove this fact here.