Order Preserving Maps of Posets

TL;DR

This paper investigates the structure of order-preserving maps between finite posets, revealing their lattice properties, dualities, and applications to counting maps and characteristic polynomials.

Contribution

It establishes that the hom-poset from a poset to the collection of order ideals forms a distributive lattice and characterizes it via a product of posets, introducing new structural insights.

Findings

Hom(P,J(Q)) is a distributive lattice.

Hom(P,J(Q)) is isomorphic to J(P*×Q).

Calculated the number of order-preserving maps to Boolean algebras.

Abstract

For any two finite posets $P$ and $Q$, let $\Hom(P,Q)$ be the hom-poset consisting of all order preserving maps from $P$ to $Q$, and $J(Q)$ the collection of all order ideals of $Q$. In this paper, we study some basic properties of the hom-poset $\Hom(P,Q)$ and prove that $\Hom\big(P,J(Q)\big)$ is a distributive lattice and characterized by \[ \Hom\big(P,J(Q)\big)\cong J(P^*\times Q), \] where $P^*$ is the dual of $P$. Consequently, we obtain that $\Hom\big(P,J(Q)\big)$ and $\Hom\big(Q,J(P)\big)$ are dual isomorphic, i.e., \[ \Hom\big(P,J(Q)\big)\cong \Hom^{*}\big(Q,J(P)\big). \] As applications, we calculate the number of order preserving maps from any poset to the boolean algebra, and the characteristic polynomial of $\Hom\big(P,J(Q)\big)$.