Determining properties of a multipartite tournament from its lattice of convex subsets

The collection of convex subsets of a multipartite tournament T forms a lattice C(T ). Given a lattice structure for C(T ), we deduce properties of T . In particular, we find conditions under which we can detect clones in T . We also determine conditions on the lattice which will imply that T is bipartite, except for a few cases. We classify the ambiguous cases. Finally, we study a property of C(T ) we call the anti-bipartite condition. We prove a result on directed cycles in multipartite tournaments satisfying the anti-bipartite condition, and determine the minimum number of partitions in such multipartite tournaments. Australasian Journal of Combinatorics, Vol. 31, (2005), 217-230