Lattices Associated with Partial Linear Maps of Finite Vector Spaces

Abstract

Let \(\mathbb{F}_q^n\) denote the \(n\)-dimensional row vector space over the finite field \(\mathbb{F}_q\), where \(n \geq 2\). An \(l\)-partial linear map of \(\mathbb{F}_q^n\) is a pair \((V, f)\), where \(V\) is an \(l\)-dimensional subspace of \(\mathbb{F}_q^n\) and \(f: V \to \mathbb{F}_q^n\) is a linear map. Let \(\mathcal{L}\) be the set of all partial linear maps of \(\mathbb{F}_q^n\) containing \(1\). Ordered \(\mathcal{L}\) by ordinary and reverse inclusion, two families of finite posets are obtained. This paper proves that these posets are lattices, discusses their geometricity, and computes their characteristic polynomials.