Lattices Associated with a Finite Vector Space

Let be a n-dimensional row vector space over a finite field For , let be a d- dimensional subspace of . denotes the set of all the spaces which are the subspaces of and not the subspaces of except . We define the partial order on by ordinary inclusion (resp. reverse inclusion), and then is a poset, denoted by (resp. ). In this paper we show that both and are finite atomic lattices. Further, we discuss the geometricity of and , and obtain their characteristic polynomials.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Yue, M. (2014) Lattices Associated with a Finite Vector Space. Applied Mathematics, 5, 672-676. doi: 10.4236/am.2014.54064.

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