Let an injective function  where  is the vertex set of a graph G and  is the power set of a non-empty set X, be given. Consider the induced function  defined by  where  denotes the symmetric difference of the two sets. The function f is called a  k-uniform dcsl (and X a k-uniform dcsl-set) of the graph G, if there exists a constant k such that where  is the length of a shortest path between u and v in G. If a graph G admits a k-uniform dcsl, then G is called a k-uniform dcsl graph. The k-uniform dcsl index of a graph G, denoted by  is the minimum of the cardinalities of X, as X varies over all k-uniform dcsl-sets of G. A linear extension  of a partial order  is a linear order on the elements of  such that  in  implies  in  for all  A set  of linear extensions of  is a realizer of  if, for every incomparable pair  there are  with  in  and  in  The dimension of  denoted by  is the minimum of the cardinalities of realizers of  Let  be the range of a k-uniform dcsl of the path  on  vertices. The purpose of this paper is to prove that  whether or not  forms a lattice with respect to set inclusion

1-uniform distance-compatible set-labeling, k-uniform distance-compatible set-labeling, dcsl index, k-uniform dcsl index, dimension of the poset, partition of the poset, lattice.