order type

English

Alternative forms

• order-type, ordertype

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 $\omega$ denote the order type of $\mathbb {N}$ under the usual ordering and $\mathbb {Z}$ denote the order type of $\mathbb {Z}$ 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.

Translations

characteristic of being a member of an equivalence class of ordered sets

• Finnish: järjestystyyppi

See also

• order isomorphism
• ordinal number

Further reading

• Order isomorphism on Wikipedia.Wikipedia
• Ordinal number on Wikipedia.Wikipedia
• List of order structures in mathematics on Wikipedia.Wikipedia
• Order type on Encyclopedia of Mathematics
• Order Type on Wolfram MathWorld
• Order Isomorphic on Wolfram MathWorld