Quasi Regular Semilattice and Association Schemes in Singular Linear Space

Abstract: Let \mathbb{F}_q^{n+l} denote the (n+l)-dimensional singular linear space over a finite field \mathbb{F}_q. For a fixed integer m\leq\min{n,l}, denote by \mathcal{L}^m_o(\mathbb{F}_q^{n+l}) the set of all subspaces of type (t,t_1), where t_1\leq t\leq m. Partially ordered by ordinary inclusion, one family of quasi regular semilattices is obtained. Moreover, we obtain a association schemes and discuss the bound of a M-clique.