It is known when we call a poset P, a P-chain permutational poset, given a subset of permutations P of the symmetric group S₍. In this work, we use the same idea to study subsets of words of length n, that are not necessarily permutations, for example: especially when they are certain classes of restricted growth functions induced by set partitions in standard form over n=\1, 2 n\. Varying n only, and also varying n and k (the number of blocks of the set partitions) simultaneously, we can show that those posets form a projective system of trees and lattices (after giving a lattice structure in a natural way). These poset structures can be extended over signed restricted growth functions for standard type B set partitions over n=\-1, -2, n, 0, 1, 2 n\ as well. We investigate properties of the tree and lattice structures of these projective systems. In this scenario we further bring up some other posets like P-Partition posets of snake graph of continued fractions, Ascent lattices on Dyck Paths, certain type of lattice induced by generalisec fibonnaci number and Stanley order, lattices induced by non-crossing set partitions.
Amrita Acharyya (Fri,) studied this question.