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A 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).
is often called a '''Schröder–Bernstein''' (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) '''property''' in analogy to the [[Schröder-Bernstein theorem|theorem]] of the same name (from set theory).


A mathematical property is said to be a '''Schröder–Bernstein''' (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) '''property''' if it is formulated in the following form.
==Schröder-Bernstein properties==
:If ''X'' is similar to a part of ''Y'' and also ''Y'' is similar to a part of ''X'' then ''X'' and ''Y'' are similar (to each other).
In order to define a specific Schröder-Bernstein property one should decide
In order to be specific one should decide
* what kind of mathematical objects are ''X'' and ''Y'',
* what kind of mathematical objects are ''X'' and ''Y'',
* what is meant by "a part",
* what is meant by "a part",
* what is meant by "similar".
* what is meant by "similar".


In the [[Schröder-Bernstein theorem|classical (Cantor-)Schröder–Bernstein theorem]],
In the classical (Cantor-)Schröder–Bernstein theorem,
* objects are [[Set (mathematics)|sets]] (maybe infinite),
* objects are [[Set (mathematics)|sets]] (maybe [[Finite and infinite|infinite]]),
* "a part" is interpreted as a [[subset]],
* "a part" is interpreted as a [[subset]],
* "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]].
* "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]].
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* "a part" means a triangle inside the given triangle,
* "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").
* "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 no be similar.
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 Schröder–Bernstein property is a joint property of
* a class of objects,
* a class of objects,
* a binary relation "be a part of",
* a [[Relation (mathematics)|binary relation]] "be a part of",
* a binary relation "be similar".
* a binary relation "be similar to" (similarity).
Instead of the relation "be a part of" one may use a binary relation "be embeddable into" interpreted as "be similar to some part of". Then a Schröder–Bernstein property takes the following form.
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.
: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]]:
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'').
: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 [[Order (relation)#Partial order|preorder]] (that is, reflexive and [[Transitive relation|transitive]] relation), and "isomorphic" is an [[equivalence relation]]. Also the relation "embeddable into" is usually a preorder, and similarity is usually an equivalence relation (which is natural, but not provable in the absence of formal definition). 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 an order between classes of similar objects.
The relation "injects into" is a [[Order (relation)#Partial order|preorder]] (that is, a reflexive and [[Transitive relation|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 [[Order (relation)#Partial order|partial order]] between the corresponding [[Equivalence relation|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.


A 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 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 space]]s<ref>{{harvnb|Srivastava|1998}}, see Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).</ref> states the Schröder–Bernstein property for the following case:
The Schröder–Bernstein theorem for [[measurable space]]s<ref>{{harvnb|Srivastava|1998}}, see Proposition 3.3.6 (on page 96), and the first paragraph of Section 3.3 (on page 94).</ref> states the Schröder–Bernstein property for the following case:
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* "a part" is interpreted as a measurable subset treated as a measurable space,
* "a part" is interpreted as a measurable subset treated as a measurable space,
* "similar" is interpreted as isomorphic.
* "similar" is interpreted as isomorphic.
It has a noncommutative counterpart, the Schröder–Bernstein theorem for operator algebras;<ref>{{harvnb|Kadison|Ringrose|1986}}, see Proposition 6.2.4 (on page 406).</ref> here
 
In the Schröder–Bernstein theorem for operator algebras,<ref>{{harvnb|Kadison|Ringrose|1986}}, see Proposition 6.2.4 (on page 406).</ref>
* objects are projections in a given von Neumann algebra;
* objects are projections in a given von Neumann algebra;
* "a part" is interpreted as a subprojection (that is, ''E'' is a part of ''F'' if ''F''–''E'' is a projection);
* "a part" is interpreted as a subprojection (that is, ''E'' is a part of ''F'' if ''F'' – ''E'' is a projection);
* "''E'' is similar to ''F''" means that ''E'' and ''F'' are the initial and final projections of some partial isometry in the algebra (that is, ''E'' = ''V*V'' and ''F'' = ''VV*'' for some ''V'' in the algebra).
* "''E'' is similar to ''F''" means that ''E'' and ''F'' are the initial and final projections of some partial isometry in the algebra (that is, ''E'' = ''V*V'' and ''F'' = ''VV*'' for some ''V'' in the algebra).
Taking into account that commutative von Neumann algebras are closely related to measurable spaces,<ref>{{harvnb|Kadison|Ringrose|1986}}, see Theorem 9.4.1 (on page 666).</ref> 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.


[[Banach space]]s violate the Schröder–Bernstein property;<ref name=Ca>{{harvnb|Casazza|1989}}</ref><ref name=Go>{{harvnb|Gowers|1996}}</ref> here
[[Banach space]]s violate the Schröder–Bernstein property;<ref name=Ca>{{harvnb|Casazza|1989}}</ref><ref name=Go>{{harvnb|Gowers|1996}}</ref> here
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==References==
==References==
 
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A 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).

is often called a Schröder–Bernstein (or Cantor–Schröder–Bernstein, or Cantor–Bernstein) property 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,

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.

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:

  • objects are measurable spaces,
  • "a part" is interpreted as a measurable subset treated as a measurable space,
  • "similar" is interpreted as isomorphic.

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

  • objects are projections in a given von Neumann algebra;
  • "a part" is interpreted as a subprojection (that is, E is a part of F if F – E is a projection);
  • "E is similar to F" means that E and F are the initial and final projections of some partial isometry in the algebra (that is, E = V*V and F = VV* for some V in the algebra).

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.

Banach spaces violate the Schröder–Bernstein property;[4][5] here

  • objects are Banach spaces,
  • "a part" is interpreted as a subspace[4] or a complemented subspace[5],
  • "similar" is interpreted as linearly homeomorphic.

Many other Schröder–Bernstein problems related to various spaces and algebraic structures (groups, rings, fields etc) are discussed by informal groups of mathematicians (see the external links page).

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. ↑ 4.0 4.1 Casazza 1989
  5. ↑ 5.0 5.1 Gowers 1996

References

  • Srivastava, S.M. (1998), A Course on Borel Sets, Springer, ISBN 0387984127.
  • Kadison, Richard V. & John R. Ringrose (1986), Fundamentals of the theory of operator algebras, vol. II, Academic Press, ISBN 0-12-393302-1.
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