Construction of Quasi-Regular Semilattices with Singular Linear Spaces

Abstract

Let ${\mathbb{F}}_q^{(}n+1)$ denote the $(n+l)$-dimensional projective space over a finite field ${\mathbb{F}}_q$. For a fixed integer $m\le min\{n,l\}$, denote by ${\mathcal{L}}_o^m({\mathbb{F}}_q^{n+1})$ the set of all subspaces of type $(t,t_1)$, where $t_1\le t\le m$. Partially ordered by ordinary inclusion, one family of quasi-regular semilattices is obtained. Moreover, we compute all its parameters.