翻訳と辞書 Words near each other ・ Medialink ・ Medialuna ・ Medialuna (genus) ・ MediaMan ・ MediaMath ・ MediaMax CD-3 ・ Mediamax news agency ・ Mediamerx ・ MediaMiser ・ MediaMonkey ・ MediaMonks ・ Median ・ Median (disambiguation) ・ Median (geometry) ・ Median absolute deviation ・ Median algebra ・ Median alveolar cyst ・ Median antebrachial vein ・ Median aperture ・ Median arcuate ligament ・ Median arcuate ligament syndrome ・ Median artery ・ Median center of United States population ・ Median cubital vein ・ Median cut ・ Median eminence ・ Median filter ・ Median follow-up ・ Median graph ・ Median household income Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Median algebra ： ウィキペディア英語版 Median algebra In mathematics, a median algebra is a set with a ternary operation $\backslash langle\; x,y,z\; \backslash rangle$ satisfying a set of axioms which generalise the notion of median or majority function, as a Boolean function. The axioms are # $\backslash langle\; x,y,y\; \backslash rangle\; =\; y$ # $\backslash langle\; x,y,z\; \backslash rangle\; =\; \backslash langle\; z,x,y\; \backslash rangle$ # $\backslash langle\; x,y,z\; \backslash rangle\; =\; \backslash langle\; x,z,y\; \backslash rangle$ # $\backslash langle\; \backslash langle\; x,w,y\backslash rangle\; ,w,z\; \backslash rangle\; =\; \backslash langle\; x,w,\; \backslash langle\; y,w,z\; \backslash rangle\backslash rangle$ The second and third axioms imply commutativity: it is possible (but not easy) to show that in the presence of the other three, axiom (3) is redundant. The fourth axiom implies associativity. There are other possible axiom systems: for example the two * $\backslash langle\; x,y,y\; \backslash rangle\; =\; y$ * $\backslash langle\; u,v,\; \backslash langle\; u,w,x\; \backslash rangle\backslash rangle\; =\; \backslash langle\; u,x,\; \backslash langle\; w,u,v\; \backslash rangle\backslash rangle$ also suffice. In a Boolean algebra, or more generally a distributive lattice, the median function $\backslash langle\; x,y,z\; \backslash rangle\; =\; (x\; \backslash vee\; y)\; \backslash wedge\; (y\; \backslash vee\; z)\; \backslash wedge\; (z\; \backslash vee\; x)$ satisfies these axioms, so that every Boolean algebra and every distributive lattice forms a median algebra. Birkhoff and Kiss showed that a median algebra with elements 0 and 1 satisfying < 0,x,1 > = x is a distributive lattice. ==Relation to median graphs== A median graph is an undirected graph in which for every three vertices ''x'', ''y'', and ''z'' there is a unique vertex < x,y,z > that belongs to shortest paths between any two of ''x'', ''y'', and ''z''. If this is the case, then the operation < x,y,z > defines a median algebra having the vertices of the graph as its elements. Conversely, in any median algebra, one may define an ''interval'' (''z'' ) to be the set of elements ''y'' such that < x,y,z > = ''y''. One may define a graph from a median algebra by creating a vertex for each algebra element and an edge for each pair (''x'', ''z'') such that the interval (''z'' ) contains no other elements. If the algebra has the property that every interval is finite, then this graph is a median graph, and it accurately represents the algebra in that the median operation defined by shortest paths on the graph coincides with the algebra's original median operation. 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「Median algebra」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.