Schröder–Bernstein property
class of mathematical properties
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
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 subset,
"Similar" is interpreted as equinumerous.
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.
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