A monotonic priority set is a new data structure which supports maximum-finding and deletions over a set of weighted points in the plane. Global updates to the weights can also be made, incrementing the weights of all points above a given threshold in one of the coordinates. The weights are assumed to be always monotonic in both coordinates. An efficient implementation of this structure is presented and two main applications are described. The first is to the problem of optimal assembly of code for computers with two kinds of jump instruction: long and short. The task in the second application is the implementation of a queuing discipline based on the ranks with respect to two different criteria.