## Quick Reference

A relation defined between elements of some set and satisfying certain properties, discussed below. It is basically a convenient generalization of the usual comparison operators, such as > or <, that are typically defined on the integers or the real numbers. The generalization also captures the essential properties of the set operations such as “is a subset of”, the alphabetic ordering of strings, and so on. In denotational semantics, partial orderings are used to express some **approximation relation** between partially defined computational objects.

Two different but equivalent definitions of a partial ordering are possible. The first is a generalization of the usual ≤ operation in which the relation must be a transitive, antisymmetric, and reflexive relation defined on the set *S*. The second definition is a generalization of the usual < operation in which the relation must be a transitive, asymmetric, and irreflexive relation defined on *S*. A set with a partial ordering defined on it is called a **partially ordered set** or sometimes a **poset**.

*Subjects:*
Computing.

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