Lattices Generated by Partial Injective Maps of Finite Sets

Abstract

Let $n$ be a positive integer with $n\ge 2$ and $[n]:=\{1,2,\dots ,n\}$. An $m$-partial injective map of $[n]$ is a pair $(A,f)$, where $A$ is an $m$-subset of $[n]$ and $f:A\to [n]$ is an injective map. Let $P=L\cup \{I\}$, where $L$ is the set of all partial injective maps of $[n]$. Partially ordering $P$ by ordinary or reverse inclusion yields two families of finite posets. This article proves that these posets are atomic lattices, discusses their geometricity, and computes their characteristic polynomials.