Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order."
|Transitive binary relations|
|indicates that the column's property is required by the definition of the row's term (at the very left). For example, the definition of an equivalence relation requires it to be symmetric. All definitions tacitly require the homogeneous relation be transitive: for all if and then and there are additional properties that a homogeneous relation may satisfy.|
The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.