Lattices Generated by Partial Injective Maps of Finite Sets

Abstract

Let \(n\) be a positive integer with \(n\geq 2\) and \([n] := \{1, 2, \ldots, n\}\). An \(m\)-partial injective map of \([n]\) is a pair \((A, f)\), where \(A\) is an \(m\)-subset of \([n]\) and \(f: A \rightarrow [n]\) 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.