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| An [[incidence structure]] <math>C=(P,L,I)</math> consists of points <math>P</math>, lines <math>L</math>, and flags <math>I \subseteq P \times L</math> where a point <math>p</math> is said to be incident with a line <math>l</math> if <math>(p,l) \in I</math>. It is a ([[Wikt:finite|finite]]) '''partial geometry''' if there are [[integer]]s <math>s,t,\alpha\geq 1</math> such that:
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| * For any pair of distinct points <math>p</math> and <math>q</math>, there is at most one line incident with both of them.
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| * Each line is incident with <math>s+1</math> points.
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| * Each point is incident with <math>t+1</math> lines.
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| * If a point <math>p</math> and a line <math>l</math> are not incident, there are exactly <math>\alpha</math> pairs <math>(q,m)\in I</math>, such that <math>p</math> is incident with <math>m</math> and <math>q</math> is incident with <math>l</math>.
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| A partial geometry with these parameters is denoted by <math>pg(s,t,\alpha)</math>.
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| ==Properties==
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| * The number of points is given by <math>\frac{(s+1)(s t+\alpha)}{\alpha}</math> and the number of lines by <math>\frac{(t+1)(s t+\alpha)}{\alpha}</math>.
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| * The point graph of a <math>pg(s,t,\alpha)</math> is a [[strongly regular graph]] : <math>srg((s+1)\frac{(s t+\alpha)}{\alpha},s(t+1),s-1+t(\alpha-1),\alpha(t+1))</math>.
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| * Partial geometries are dual structures : the dual of a <math>pg(s,t,\alpha)</math> is simply a <math>pg(t,s,\alpha)</math>.
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| ==Special case==
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| * The [[generalized quadrangle]]s are exactly those partial geometries <math>pg(s,t,\alpha)</math> with <math>\alpha=1</math>.
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| ==See also==
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| * [[Maximal arc]]
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| {{DEFAULTSORT:Partial Geometry}}
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| [[Category:Incidence geometry]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. For years he's been living in Mississippi and he doesn't plan on altering it. Office supervising is exactly where her primary income comes from. To perform domino is something I truly enjoy performing.
Stop by my blog post: real psychics (http://1.234.36.240/fxac/m001_2/7330)