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In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections.〔; .〕 ==Definition== Formally, given two posets and , an order isomorphism from to is a bijective function from to with the property that, for every and in , if and only if . That is, it is a bijective order-embedding.〔This is the definition used by . For and it is a consequence of a different definition.〕 It is also possible to define an order isomorphism to be a surjective order-embedding. The two assumptions that cover all the elements of and that it preserve orderings, are enough to ensure that is also one-to-one, for if then (by the assumption that preserves the order) it would follow that and , implying by the definition of a partial order that . Yet another characterization of order isomorphisms is that they are exactly the monotone bijections that have a monotone inverse.〔This is the definition used by and .〕 An order isomorphism from a partially ordered set to itself is called an order automorphism.〔, p. 13.〕 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「order isomorphism」の詳細全文を読む スポンサード リンク
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