Order type

In mathematics, especially in set theory, two ordered sets X,Y are said to have the same order type just when they are order isomorphic, that is, when there exists a bijection (each element matches exactly one in the other set) f: X → Y such that both f and its inverse are strictly increasing (order preserving i.e. the matching elements are also in the correct order). In the special case when X is totally ordered, monotonicity of f implies monotonicity of its inverse.

For example, the set of integers and the set of even integers have the same order type, because the mapping ${\displaystyle n\mapsto 2n}$ preserves the order and so does the inverse map, ${\displaystyle 2n\mapsto n}$ . But the set of integers and the set of rational numbers (with the standard ordering) are not order isomorphic, because, even though the sets are of the same size (they are both countably infinite), there is no order-preserving bijective mapping between them. To these two order types we may add two more: the set of positive integers (which has a least element), and that of negative integers (which has a greatest element). The open interval (0,1) of rationals is order isomorphic to the rationals (since

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