The lattice of subspaces of a vector space over a finite field
Ivan Chajda, Helmut Länger · Soft Computing · 2019
For finite m and q we study the lattice $$\mathbf {L}(\mathbf {V})=(L(\mathbf {V}),+,\cap ,\{\vec {0}\},V)$$ of subspaces of an m-dimensional vector space $$\mathbf {V}$$ over a field $$\mathbf {K}$$ of cardinality q. We present formulas for the number of d-dimensional subspaces of $$\mathbf {V}$$ , for the number of complements of a subspace and for the number of e-dimensional subspaces including a given d-dimensional subspace. It was shown in Eckmann and Zabey (Helv Phys Acta 42:420–424, 1969) that $$\mathbf {L}(\mathbf {V})$$ possesses an orthocomplementation only in case $$m=2$$ and $${{\,\mathrm{char}\,}}\mathbf {K} e 2$$ . Hence, only in this case $$\mathbf {L}(\mathbf {V})$$ can be considered as an orthomodular lattice. On the contrary, we show that a complementation $$'$$ on $$\mathbf {L}(\mathbf {V})$$ can be chosen in such a way that $$(L(\mathbf {V}),+,\cap ,{}')$$ is both weakly orthomodular and dually weakly orthomodular. Moreover, we show that $$(L(\mathbf {V}),+,\cap ,{}^\perp ,\{\vec {0}\},V)$$ is paraorthomodular in the sense of Giuntini et al. (Stud Log 104:1145–1177, 2016).