# order type

## English

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### Noun

order type (plural order types)

1. (set theory) In the context of sets equipped with an order (especially, the context of totally ordered sets), the characteristic of being a member of some equivalence class of such sets under the equivalence relation "existence of an order-preserving bijection".
• 1965 [John Wiley], Raymond L. Wilder, Introduction to the Foundations of Mathematics: 2nd Edition, 2012, Dover, page 116,
Another way of putting this is to state that the order type is that aspect of the arrangement of the elements of a simply ordered set, which remains unchanged when any two elements are exchanged. [] As in the case of cardinal numbers, order types may be denoted by suitable symbols called ordinal numerals.
• 2005, Egbert Harzheim, Ordered Sets, Springer, page 332,
In [13] Chajoth studied how the order type of a chain can alter if we change the position of elements in a linearly ordered set, resp. if we introduce a new element in a linearly ordered set.
• 2011, Douglas Cenzer, Valentina Harizanov, Jeffrey B. Remmel, Effective Categoricity of Injection Structures, Benedikt Löwe, Dag Normann, Ivan Soskov, Alexandra Soskova (editors, Models of Computation in Context: 7th Conference on Computability in Europe, CiE 2011, Proceedings, Springer, LNCS 6735, page 51,
We let ${\displaystyle \omega }$  denote the order type of ${\displaystyle \mathbb {N} }$  under the usual ordering and ${\displaystyle \mathbb {Z} }$  denote the order type of ${\displaystyle \mathbb {Z} }$  under the usual ordering.

#### Usage notes

One says that two sets have the same order type if they are members of the same equivalence class, as described in the definition.

In the case of well-ordered sets, the order types are identified as the ordinal numbers. Strictly speaking, an ordinal number (as constructed by John von Neumann) is a representative member of some equivalence class of well-ordered sets. In particular, each ordinal number has a characteristic cardinality (size) that also characterises every set in the equivalence class it represents.