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In order theory, a branch of mathematics, a semiorder is a type of ordering that may be determined for a set of items with numerical scores by declaring two items to be incomparable when their scores are within a given margin of error of each other, and by using the numerical comparison of their scores when those scores are sufficiently far apart. Semiorders were introduced and applied in mathematical psychology by as a model of human preference without the assumption that indifference is transitive. They generalize strict weak orderings, form a special case of partial orders and interval orders, and can be characterized among the partial orders by two forbidden four-item suborders. ==Definition== Let ''X'' be a set of items, and let < be a binary relation on ''X''. Items ''x'' and ''y'' are said to be ''incomparable'', written here as ''x'' ~ ''y'', if neither ''x'' < ''y'' nor ''y'' < ''x'' is true. Then the pair (''X'',<) is a semiorder if it satisfies the following three axioms:〔 describes an equivalent set of four axioms, the first two of which combine the definition of incomparability and the first axiom listed here.〕 *For all ''x'' and ''y'', it is not possible for both ''x'' < ''y'' and ''y'' < ''x'' to be true. That is, < must be an irreflexive, antisymmetric relation *For all ''x'', ''y'', ''z'', and ''w'', if it is true that ''x'' < ''y'', ''y'' ~ ''z'', and ''z'' < ''w'', then it must also be true that ''x'' < ''w''. *For all ''x'', ''y'', ''z'', and ''w'', if it is true that ''x'' < ''y'', ''y'' < ''z'', and ''y'' ~ ''w'', then it cannot also be true that ''x'' ~ ''w'' and ''z'' ~ ''w'' simultaneously. It follows from the first axiom that ''x'' ~ ''x'', and therefore the second axiom (with ''y'' = ''z'') implies that < is a transitive relation. One may define a partial order (''X'',≤) from a semiorder (''X'',<) by declaring that whenever either or . Of the axioms that a partial order is required to obey, reflexivity follows automatically from this definition, antisymmetry follows from the first semiorder axiom, and transitivity follows from the second semiorder axiom. Conversely, from a partial order defined in this way, the semiorder may be recovered by declaring that whenever and . The first of the semiorder axioms listed above follows automatically from the axioms defining a partial order, but the others do not. The second and third semiorder axioms forbid partial orders of four items forming two disjoint chains: the second axiom forbids two chains of two items each, while the third item forbids a three-item chain with one unrelated item. 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「semiorder」の詳細全文を読む スポンサード リンク
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