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Geometric Hyperplanes of Lie Incidence Geometries
Geometriae Dedicata, 1997Let \(\Gamma=({\mathcal P},{\mathcal L})\) be a geometry of points and lines. A subspace of \(\Gamma\) is a set of points which contains every line that meets it in at least two points. An embedding \(\mu\) of \(\Gamma\) in a finite-dimensional vector space \(V\) consists of a map \(\mu_1\) of \({\mathcal P}\) into the set of 1-subspaces of \(V\) and a
Cooperstein, Bruce N., Shult, Ernest E.
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On finite models of Hilbert's incidence geometry
Discrete MathematicsIn this paper, the authors give a lower bound on the number of such models with \({n}\) points using finite models of the first group of Hilbert's axioms of Euclidean geometry (denote with \(A\)). By \(\mathrm{HilbInc}(n)\), the authors denote the number of nonisomorphic models of \(A\) with the point set \({1, 2,\dots,n}\) and calculate the exact ...
Bojan Basic, Kristina Ago
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Incidence loops and their geometry
1992Publisher Summary This chapter discusses the concept of incidence loops and their geometry. An incidence group (P, L,·) is a group (P,·) together with a structure (P, L) of an incidence space such that both structures are compatible. The notion of incidence group can be generalized by weakening the assumptions concerning the algebraic structure of P;
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Preliminaries and Incidence Geometry (I)
2015This chapter contains a brief summary of several types of mathematical knowledge needed to read this book, including the elements of logic, set theory, mapping theory, and algebraic structures such as number systems and vector spaces. Definitions of basis, dimension, linear mappings, isomorphism, matrices and determinants are given; there is also ...
Edward John Specht +3 more
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On apartments in incidence geometry
2009info:eu-repo/semantics ...
Buekenhout, Francis, Leemans, Dimitri
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On incidence matrices of finite affine geometries
Ars Comb., 1997The author proves that for each two levels of the lattice \(\text{AG}(n-1,q)\) of flats of a finite affine geometry the incidence \(0-1\) matrix (\(a_{XY}=1\) iff \(X\leq Y\)) has full rank.
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