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A Schröder–Bernstein property is any mathematical property that matches the following pattern : If, for some mathematical objects ''X'' and ''Y'', both ''X'' is similar to a part of ''Y'' and ''Y'' is similar to a part of ''X'' then ''X'' and ''Y'' are similar (to each other). The name Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) property is in analogy to the theorem of the same name (from set theory). ==Schröder–Bernstein properties== In order to define a specific Schröder–Bernstein property one should decide * what kind of mathematical objects are ''X'' and ''Y'', * what is meant by "a part", * what is meant by "similar". In the classical (Cantor–)Schröder–Bernstein theorem, * objects are sets (maybe infinite), * "a part" is interpreted as a subset, * "similar" is interpreted as equinumerous. Not all statements of this form are true. For example, assume that * objects are triangles, * "a part" means a triangle inside the given triangle, * "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles"). Then the statement fails badly: every triangle ''X'' evidently is similar to some triangle inside ''Y'', and the other way round; however, ''X'' and ''Y'' need not be similar. A Schröder–Bernstein property is a joint property of * a class of objects, * a binary relation "be a part of", * a binary relation "be similar to" (similarity). Instead of the relation "be a part of" one may use a binary relation "be embeddable into" (embeddability) interpreted as "be similar to some part of". Then a Schröder–Bernstein property takes the following form. :If ''X'' is embeddable into ''Y'' and ''Y'' is embeddable into ''X'' then ''X'' and ''Y'' are similar. The same in the language of category theory: :If objects ''X'', ''Y'' are such that ''X'' injects into ''Y'' (more formally, there exists a monomorphism from ''X'' to ''Y'') and also ''Y'' injects into ''X'' then ''X'' and ''Y'' are isomorphic (more formally, there exists an isomorphism from ''X'' to ''Y''). The relation "injects into" is a preorder (that is, a reflexive and transitive relation), and "be isomorphic" is an equivalence relation. Also embeddability is usually a preorder, and similarity is usually an equivalence relation (which is natural, but not provable in the absence of formal definitions). Generally, a preorder leads to an equivalence relation and a partial order between the corresponding equivalence classes. The Schröder–Bernstein property claims that the embeddability preorder (assuming that it is a preorder) leads to the similarity equivalence relation, and a partial order (not just preorder) between classes of similar objects. 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「Schröder–Bernstein property」の詳細全文を読む スポンサード リンク
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