On multipartite posets

A poset P=(X,≼)P=(X,≼) is m-partite if X   has a partition X=X1∪⋯∪XmX=X1∪⋯∪Xm such that (1) each XiXi forms an antichain in PP, and (2) x≺yx≺y implies x∈Xix∈Xi and y∈Xjy∈Xj where i