The Krull dimension of a finite lattice (X;6) is equal to the height of the poset of join prime elements of X minus 1. To every partially ordered set we assign an order-matrix, and we use these ordermatrices to characterize the join prime elements of finite lattices. In addition, we present a reduction algorithm for the computation of the height of a finite poset. The algorithm is based on the concept of the incidence matrix. Our main objective, ultimately, is to use these processes to calculate the Krull dimension of any given finite lattice.