ordered

Ordered is an adjective meaning arranged according to a sequence or a defined rule. It describes items placed in a specific order, or the result of arranging them. It is also the past participle of the verb order, used in phrases such as "the items were ordered." In mathematics and related disciplines, "ordered" signals the existence of an order relation or a prescribed sequence.

In mathematics, an order relation on a set allows comparison of elements. A relation ≤ is a partial order if it is reflexive, antisymmetric, and transitive; it is a total (linear) order if any two elements are comparable. A well-order is a total order in which every non-empty subset has a least element. Examples include the natural numbers with ≤ and the real numbers with ≤; the lexicographic order on tuples is another common instance. An ordered pair (a,b) distinguishes the first component from the second.

Structures that incorporate order include ordered fields, where the order is compatible with addition and multiplication, and order topology, which is derived from the order rather than a metric.

In computing and data handling, order matters. Lists and arrays are ordered collections; an insertion-order dictionary preserves the sequence of insertion. Sorting arranges elements according to a comparator, and stable sorts retain the relative order of equal elements.

In everyday speech, to order can mean to request goods or services; to be ordered can mean arranged according to plan or rule.