Abstract
For finite dimensional hermitean inner product spaces V, over \(*\)-fields F, and in the presence of orthogonal bases providing form elements in the prime subfield of F, we show that quantifier-free definable relations in the subspace lattice \(\mathsf{L}(V)\), endowed with the involution induced by orthogonality, admit quantifier-free descriptions within F, also in terms of Grassmann–Plücker coordinates. In the latter setting, homogeneous descriptions are obtained if one allows quantification type \(\Sigma _1\). In absence of involution, these results remain valid.
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Martin Ziegler acknowledges funding by the National Research Foundation of Korea (Grant NRF-2017R1E1A1A03071032) and co-funding to EU H2020 MSCA IRSES project 731143 by the International Research & Development Program of the Korean Ministry of Science and ICT (Grant NRF-2016K1A3A7A03950702).
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Herrmann, C., Ziegler, M. Definable relations in finite dimensional subspace lattices with involution. Part II: Quantifier-free and homogeneous descriptions. Algebra Univers. 80, 28 (2019). https://doi.org/10.1007/s00012-019-0603-7
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DOI: https://doi.org/10.1007/s00012-019-0603-7