Schröder–Bernstein property
Encyclopedia
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).
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.
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.
Banach space
s 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).
- 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
Cantor–Bernstein–Schroeder theorem
In set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions and between the sets A and B, then there exists a bijective function...
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 subsetSubsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
, - "similar" is interpreted as equinumerous.
Not all statements of this form are true. For example, assume that
- objects are triangleTriangleA triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
s, - "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 relationRelation (mathematics)In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
"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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
:
- 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
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
relation), and "be isomorphic" is an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. 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
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
. 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:
- 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,
- 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, 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
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s violate the Schröder–Bernstein property; here
- objects are Banach spaces,
- "a part" is interpreted as a subspace or a complemented subspace,
- "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 External Links below).
See also
- Cantor–Bernstein–Schroeder theoremCantor–Bernstein–Schroeder theoremIn set theory, the Cantor–Bernstein–Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions and between the sets A and B, then there exists a bijective function...
- Schroeder–Bernstein theorem for measurable spaces
- Schröder–Bernstein theorems for operator algebrasSchröder–Bernstein theorems for operator algebrasThe Schröder–Bernstein theorem, from set theory, has analogs in the context operator algebras. This article discusses such operator-algebraic results.- For von Neumann algebras :...
- Commutative von Neumann algebras
External links
- Theme and variations: Schroeder-Bernstein - Various Schröder–Bernstein problems are discussed in a group blog by 8 recent Berkeley mathematics Ph.D.
- When does Cantor Bernstein hold? - "Mathoverflow" discusses the question in terms of category theory: "Can we characterize Cantor-Bernsteiness in terms of other categorical properties?"