A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space. The notion is equivalent to that of a linear hypergraph.
Let an incidence structure, for which the elements of are called points and the elements of are called lines. S is a partial linear space, if the following axioms hold:
any line is incident with at least two points
any pair of distinct points is incident with at most one line
If there is a unique line incident with every pair of distinct points, then we get a linear space.
Propertiesedit
The De Bruijn–Erdős theorem shows that in any finite linear space which is not a single point or a single line, we have .
partial, linear, space, partial, linear, space, also, semilinear, near, linear, space, basic, incidence, structure, field, incidence, geometry, that, carries, slightly, less, structure, than, linear, space, notion, equivalent, that, linear, hypergraph, content. A partial linear space also semilinear or near linear space is a basic incidence structure in the field of incidence geometry that carries slightly less structure than a linear space The notion is equivalent to that of a linear hypergraph Contents 1 Definition 2 Properties 3 Examples 4 References 5 External linksDefinition editLet S P L I displaystyle S mathcal P mathcal L textbf I nbsp an incidence structure for which the elements of P displaystyle mathcal P nbsp are called points and the elements of L displaystyle mathcal L nbsp are called lines S is a partial linear space if the following axioms hold any line is incident with at least two points any pair of distinct points is incident with at most one lineIf there is a unique line incident with every pair of distinct points then we get a linear space Properties editThe De Bruijn Erdos theorem shows that in any finite linear space S P L I displaystyle S mathcal P mathcal L textbf I nbsp which is not a single point or a single line we have P L displaystyle mathcal P leq mathcal L nbsp Examples editProjective space Affine space Polar space Generalized quadrangle Generalized polygon Near polygonReferences editShult Ernest E 2011 Points and Lines Universitext Springer doi 10 1007 978 3 642 15627 4 ISBN 978 3 642 15626 7 Lynn Batten Combinatorics of Finite Geometries Cambridge University Press 1986 ISBN 0 521 31857 2 p 1 22 Lynn Batten and Albrecht Beutelspacher The Theory of Finite Linear Spaces Cambridge University Press Cambridge 1992 Eric Moorhouse Incidence Geometry Lecture notes archived External links editpartial linear space at the University of Kiel partial linear space at PlanetMath Retrieved from https en wikipedia org w index php title Partial linear space amp oldid 1162587288, wikipedia, wiki, book, books, library,
