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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Generalized polygon ： ウィキペディア英語版 Generalized polygon In combinatorial theory, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized ''n''-gons encompass as special cases projective planes (generalized triangles, ''n'' = 3) and generalized quadrangles (''n'' = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the ''Moufang property'' have been completely classified by Tits and Weiss. Every generalized ''n''-gon with ''n'' even is also a near polygon. ==Definition== A generalized ''2''-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line. For ''$n\; \backslash geq\; 3$'' a generalized ''n''-gon is an incidence structure ($P,L,I$), where $P$ is the set of points, $L$ is the set of lines and $I\backslash subseteq\; P\backslash times\; L$ is the incidence relation, such that: * It is a partial linear space. * It has no ordinary ''m''-gons as subgeometry for ''''. * It has an ordinary ''n''-gon as a subgeometry. * For any $\backslash \; \backslash subseteq\; P\; \backslash cup\; L$ there exists a subgeometry ($P\text{'},\; L\text{'},\; I\text{'}$) isomorphic to an ordinary ''n''-gon such that $\backslash \; \backslash subseteq\; P\text{'}\; \backslash cup\; L\text{'}$. An equivalent but sometimes simpler way to express these conditions is: consider the bipartite ''incidence graph'' with the vertex set $P\; \backslash cup\; L$ and the edges connecting the incident pairs of points and lines. * The girth of the incidence graph is twice the diameter ''n'' of the incidence graph. From this it should be clear that the incidence graphs of generalized polygons are Moore graphs. A generalized polygon is of order ''(s,t)'' if: * all vertices of the incidence graph corresponding to the elements of $L$ have the same degree ''s'' + 1 for some natural number ''s''; in other words, every line contains exactly ''s'' + 1 points, * all vertices of the incidence graph corresponding to the elements of $P$ have the same degree ''t'' + 1 for some natural number ''t''; in other words, every point lies on exactly ''t'' + 1 lines. We say a generalized polygon is thick if every point (line) is incident with at least three lines (points). All thick generalized polygons have an order. The dual of a generalized ''n''-gon ($P,L,I$), is the incidence structure with notion of points and lines reversed and the incidence relation taken to be the inverse relation of $I$. It can easily be shown that this is again a generalized ''n''-gon. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Generalized polygon」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.