Any 2-asummable bipartite function is weighted threshold

Possible characterizations of which positive boolean functions are weighted threshold were studied in the 60s and 70s. It is known that a boolean function is weighted threshold if and only if it is kk-asummable for every value of kk. Furthermore, for some particular subfamilies of functions (those with up to eight variables, and graph functions), it is known that a function is weighted threshold if and only if it is 2-asummable.In this work we prove that bipartite functions also satisfy this property: a bipartite function is weighted threshold if and only if it is 2-asummable. In a bipartite function the set of variables can be partitioned in two classes, such that all the variables in the same class play exactly the same role in the function.