Let \(F_q\) be a finite field with \(q\) elements and let \(F_q^n\) be the \(n\)-dimensional vector space over \(F_q\). An \(\ell\)-partial linear map of \(F_q^n\) is \((V, f)\) where \(V\) is an \(\ell\)-dimensional subspace of \(F_q^n\) and \(f: V \to F_q^n\) is a linear map. The authors consider the partially ordered sets of partial linear maps, one ordered by inclusion and the other ordered by reverse inclusion. They prove these posets are in fact finite lattices and for \(n \geq 2\) neither is a geometric lattice. The Möbius functions are given and characteristic polynomials are computed.