Abstract
A pre-orthogonality on a projective geometry is a symmetric binary relation, ⊥, such that for each point \({p, p^{\perp} = \{q | p \perp q \}}\) is a subspace. An orthogonality is a pre-orthogonality such that each p ⊥ is a hyperplane. Such ⊥ is called anisotropic iff it is irreflexive. For projective geometries with an anisotropic pre-orthogonality, we show how to find a (large) projective subgeometry with a natural embedding for the lattices of subspaces and with an orthogonality induced by the given pre-orthogonality. We also discuss (faithful) representations of modular ortholattices within this context and derive a condition which allows us to transform a representation by means of an anisotropic pre-orthogonality into an anisotropic orthogeometry by means of an anisotropic orthogonality.
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Herrmann, C., Roddy, M.S. On geometric representations of modular ortholattices. Algebra Univers. 71, 285–297 (2014). https://doi.org/10.1007/s00012-014-0278-z
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DOI: https://doi.org/10.1007/s00012-014-0278-z