A partially ordered set  is ‘top r separable’ if its ground set can be partitioned as  so that  for all  and all  with  For a given n-set, the minimum number of k-subsets whose transitive closure of chains on each subsets is top r-separable, denoted by  is called the separation number of  and r. In this paper, we first give some basic properties of  and obtain that  and  Next, we prove that  and   if  In addition, this result can be seen as  in terms of triangular number  of order a natural number r.

finite ordered set, transitive closure, separability