A characterization of bisubmodular functions

Given a nonempty finite set E, let 3(E) be the set of all the ordered pairs of disjoint subsets of E. We call a function on 3(E) a biset function. A biset function f: 3(E) --> R is called bisubmodular if we have For All(X(1),Y-1),(X(2),Y-2)epsilon 3(E): f(X(1),Y-1)+f(X(2),Y-2)greater than or equal to f((X(1) boolean OR X(2))-(Y-1 boolean OR Y-2),(Y-1 boolean OR Y-2)-(X(1) boolean OR X(2))) +f(X(1) boolean AND X(2),Y-1 boolean AND Y-2). We give a simple necessary and sufficient condition for a biset function to be bisubmodular.