A SchröderâBernstein property is any mathematical property that matches the following pattern:
The name SchröderâBernstein (or CantorâSchröderâBernstein, or CantorâBernstein) property is in analogy to the theorem of the same name (from set theory).
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 non- continuous bijection (right). | ||
In order to define a specific SchröderâBernstein property one should decide:
In the classical (Cantorâ)SchröderâBernstein 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 SchröderâBernstein 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 SchröderâBernstein 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 SchröderâBernstein 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 SchröderâBernstein property (for a given class and two relations) holds or not, is called a SchröderâBernstein problem. A theorem that states a SchröderâBernstein property (for a given class and two relations), thus solving the SchröderâBernstein problem in the affirmative, is called a SchröderâBernstein theorem (for the given class and two relations), not to be confused with the classical (Cantorâ) SchröderâBernstein theorem mentioned above.
The SchröderâBernstein theorem for measurable spaces [1] states the SchröderâBernstein property for the following case:
In the SchröderâBernstein 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 SchröderâBernstein theorem for operator algebras is in some sense a noncommutative counterpart of the SchröderâBernstein theorem for measurable spaces.
The Myhill isomorphism theorem can be viewed as a SchröderâBernstein theorem in computability theory. There is also a SchröderâBernstein theorem for Borel sets. [4]
Banach spaces violate the SchröderâBernstein property; [5] [6] here:
Many other SchröderâBernstein problems related to various spaces and algebraic structures (groups, rings, fields etc.) are discussed by informal groups of mathematicians (see External Links below).
A SchröderâBernstein property is any mathematical property that matches the following pattern:
The name SchröderâBernstein (or CantorâSchröderâBernstein, or CantorâBernstein) property is in analogy to the theorem of the same name (from set theory).
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 non- continuous bijection (right). | ||
In order to define a specific SchröderâBernstein property one should decide:
In the classical (Cantorâ)SchröderâBernstein 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 SchröderâBernstein 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 SchröderâBernstein 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 SchröderâBernstein 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 SchröderâBernstein property (for a given class and two relations) holds or not, is called a SchröderâBernstein problem. A theorem that states a SchröderâBernstein property (for a given class and two relations), thus solving the SchröderâBernstein problem in the affirmative, is called a SchröderâBernstein theorem (for the given class and two relations), not to be confused with the classical (Cantorâ) SchröderâBernstein theorem mentioned above.
The SchröderâBernstein theorem for measurable spaces [1] states the SchröderâBernstein property for the following case:
In the SchröderâBernstein 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 SchröderâBernstein theorem for operator algebras is in some sense a noncommutative counterpart of the SchröderâBernstein theorem for measurable spaces.
The Myhill isomorphism theorem can be viewed as a SchröderâBernstein theorem in computability theory. There is also a SchröderâBernstein theorem for Borel sets. [4]
Banach spaces violate the SchröderâBernstein property; [5] [6] here:
Many other SchröderâBernstein problems related to various spaces and algebraic structures (groups, rings, fields etc.) are discussed by informal groups of mathematicians (see External Links below).