Two Families of Lattices

Abstract

Let \(X\) denote a set with \(q\) elements. Suppose \(\mathcal{L}(n, q)\) denotes the set \(X^n\) (resp. \(X^n \cup \{\Delta\}\)) whenever \(q = 2\) (resp. \(q \geq 3\)). For any two elements \(\alpha = (\alpha_1, \ldots, \alpha_n)\) and \(\beta = (\beta_1, \ldots, \beta_n) \in \mathcal{L}(n, q)\), define \(\alpha \leq \beta\) if and only if \(\beta = \Delta\) or \(\alpha_i = \beta_i\) whenever \(\alpha_i \neq 0\) for \(1 \leq i \leq n\). Then \(\mathcal{L}(n, q)\) is a lattice, denoted by \(\mathcal{L}_\bigcirc(n, q)\). Reversing the above partial order, we obtain the dual of \(\mathcal{L}_\bigcirc(n, q)\), denoted by \(\mathcal{L}_R(n, q)\). This paper discusses their geometricity, and computes their characteristic polynomials, determines their full automorphism groups. Moreover, we construct a family of quasi-strongly regular graphs from the lattice \(\mathcal{L}_\bigcirc(n, q)\).