We study a certain class of greedy algorithms for combinatorial optimization, and present a generalized version of this algorithmic pattern encompassing several previously published methods. We analyze the properties of the solutions produced by such algorithms, and provide proofs of their optimality through the concept of lexicographic max-ordering. By presenting a unified formulation of this class of greedy algorithms, we hope to facilitate the development of new such algorithms, and to provide a deeper understanding on the properties of existing algorithms.