  In this paper, we considered impartial games on a simplicial complex. Each
vertex of a given simplicial complex acts as a position of an impartial game.
Each player in turn chooses a face of the simplicial complex and, for each
position on each vertex of that face, the player can make an arbitrary number
of moves. Moreover, the player can make only a single move for each position on
each vertex, not on that face. We show how the P-positions of this game can be
characterized using the P-position length. This result can be considered an
extension of the emperor sum theory. While the emperor sum only allowed
multiple moves for a single component, this study examines the case where
multiple moves can be made for multiple components, and clarifies areas that
the emperor sum theory did not cover.
