Schröder–Bernstein property
A Schröder–Bernstein property is any mathematical property that matches the following pattern 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 In the classical (Cantor–)Schröder–Bernstein theorem, Not all statements of this form are true. For example, assume that 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 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. The same in the language of category theory: 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. Schröder–Bernstein problems and Schröder–Bernstein theorems. The problem of deciding whether a Schröder–Bernstein property (for a given class and two relations) holds or not, is called a Schröder–Bernstein problem. A theorem that states a Schröder–Bernstein property (for a given class and two relations), thus solving  the Schröder–Bernstein problem in the affirmative, is called a Schröder–Bernstein theorem (for the given class and two relations), not to be confused with the classical (Cantor–)Schröder–Bernstein theorem mentioned above. The Schröder–Bernstein theorem for measurable spaces states the Schröder–Bernstein property for the following case: In the Schröder–Bernstein theorem for operator algebras, Taking into account that commutative von Neumann algebras are closely related to measurable spaces, one may say that the Schröder–Bernstein theorem for operator algebras is in some sense a noncommutative counterpart of the Schröder–Bernstein theorem for measurable spaces. The Myhill isomorphism theorem  can be viewed as a Schröder–Bernstein theorem in computability theory. Banach spaces violate the Schröder–Bernstein property; here Many other Schröder–Bernstein problems related to various spaces and algebraic structures (groups, rings, fields etc.) are discussed by informal groups of mathematicians (see External Links below).