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In mathematics, a median algebra is a set with a ternary operation satisfying a set of axioms which generalise the notion of median or majority function, as a Boolean function. The axioms are # # # # 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 * * also suffice. In a Boolean algebra, or more generally a distributive lattice, the median function 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
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