Schröder–Bernstein property

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

Mirror-in-mirror images as counterexample: The left image can be embedded into the right one and vice versa (below, left/mid); yet, both aren't similar. The Schröder-Bernstein theorem applied to the unstructured pixel sets obtains a noncontinuous bijection (right).

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.

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.

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[1] states the Schröder–Bernstein property for the following case:

In the Schröder–Bernstein theorem for operator algebras,[2]

Taking into account that commutative von Neumann algebras are closely related to measurable spaces,[3] 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;[4][5] 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).

Notes

  1. ↑ Srivastava 1998, see Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).
  2. ↑ Kadison & Ringrose 1986, see Proposition 6.2.4 (on page 406).
  3. ↑ Kadison & Ringrose 1986, see Theorem 9.4.1 (on page 666).
  4. 1 2 Casazza 1989
  5. 1 2 Gowers 1996

See also

References

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