Inverse limits of various posets

TL;DR

This paper explores inverse limits of various posets related to set partitions, restricted growth functions, and lattice structures, extending to signed functions and analyzing their projective systems and properties.

Contribution

It introduces a framework for studying inverse limits of posets derived from set partitions, restricted growth functions, and related structures, including extensions to type B partitions and other combinatorial lattices.

Findings

Posets form projective systems of trees and lattices.

Extensions to signed restricted growth functions are possible.

Properties of these systems are systematically analyzed.

Abstract

It is known when we call a poset P, a $\mathcal{P}$-chain permutational poset, given a subset of permutations $\mathcal{P}$ of the symmetric group $S_{n}$. 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 $[n]=\{1,2\cdots n\}$. 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 $\langle n\rangle=\{-1,-2,\cdots n,0,1,2\cdots n\}$ 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 $\mathcal{P}$-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.