: It is known that for convex TU games the vertices of the core are the marginal vectors, and this result remains true for games where the set of feasible coalitions is a distributive lattice. We propose a larger class of vertices for games on distributive lattices, called min-max vertices, obtained by minimizing or maximizing in a given order the coordinates of a core element. We show that, for connected hierarchies (and for the general case under some restrictions), there is a simple recursive formula generating some of them. Moreover, we find under which conditions two different orders induce the same vertex for every game, and show that there exist balanced games whose core has vertices which are not min-max vertices if and only if n > 4.