Orthogonality and complementation in the lattice of subspaces of a finite-dimensional vector space over a finite field
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Publication:6333964
DOI10.21136/MB.2021.0042-20zbMATH Open1513.06018arXiv2002.00368MaRDI QIDQ6333964
Publication date: 2 February 2020
Abstract: We investigate the lattice L(V) of subspaces of an m-dimensional vector space V over a finite field GF(q) with q being the n-th power of a prime p. It is well-known that this lattice is modular and that orthogonality is an antitone involution. The lattice L(V) satisfies the Chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when L(V) is orthomodular. For m > 1 and m not divisible by p we show that L(V) contains a certain (non-Boolean) orthomodular lattice as a subposet. Finally, for q being a prime and m = 2 we characterize orthomodularity of L(V) by a simple condition.
Finite fields (field-theoretic aspects) (12E20) Complemented lattices, orthocomplemented lattices and posets (06C15) Modular lattices, Desarguesian lattices (06C05) Vector spaces, linear dependence, rank, lineability (15A03)
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