orderordinal

Orderordinal is a term encountered mainly in informal discussions of ordered sets. It is not a standard, universally defined object in mainstream mathematics, where the focus is on ordinal numbers and the concept of an order type. When used, orderordinal usually refers to the ordinal index that identifies the position of an element within a well-ordered sequence.

Formally, in a well-ordered set (A, <), the orderordinal of an element x ∈ A is the unique ordinal α such that x is the α-th element of A under <. For finite sets this reduces to natural numbers, and in common practice a 0-based indexing is often assumed (the first element has orderordinal 0). For infinite well-orders, the orderordinal can be a transfinite ordinal, reflecting the element's place in an infinitely long sequence. The notion is conceptually equivalent to the idea of ranking an element by its position, but with the indexing carried by ordinals rather than plain integers.

Because the orderordinal depends on the chosen order, the same element can have different orderordinal values under different orderings of the underlying set. In introductory contexts it is typically introduced only after the formal notion of ordinals and well-orders has been established, and it is more of a descriptive label than a separate mathematical construct.

Examples: In the finite list [apple, banana, cherry] with the order defined by the list, the orderordinal of apple is 0, banana is 1, and cherry is 2. In the natural numbers with the usual order, the orderordinal of n is n.

See also: ordinal number, well-order, order type, indexing, ordinal arithmetic.