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 f: X → Y such that both f and its inverse are monotone (order preserving). (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 preserves the order. 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
provides a monotone bijection from the former to the latter); the half-closed intervals, and the closed interval, are three additional order type examples.
Since order-equivalence is an equivalence relation, it partitions the class of all ordered sets into equivalence classes.
Read more about Order Type: Order Type of Well-orderings, Rational Numbers, Notation
Famous quotes containing the words order and/or type:
“Remember, a woman has to work harder than a man and have more patience in order to achieve success.”
—Margaret Mary Morgan, U.S. suffragist, print shop owner, and politician. As quoted in Dianne Feinstein, ch. 5, by Jerry Roberts (1994)
“But the mother’s yearning, that completest type of the life in another life which is the essence of real human love, feels the presence of the cherished child even in the debased, degraded man.”
—George Eliot [Mary Ann (or Marian)