# Partially ordered set

## Mathematical set with an ordering / From Wikipedia, the free encyclopedia

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In mathematics, especially order theory, a **partial order** on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word *partial* is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable.

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Y indicates that the column's property is always true the row's term (at the very left), while ✗ indicates that the property is not guaranteed in general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation $R$ be transitive: for all $a,b,c,$ if $aRb$ and $bRc$ then $aRc.$ |

Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A **partially ordered set** (**poset** for short) is an ordered pair $P=(X,\leq )$ of a set $X$ (called the *ground set* of $P$) and a partial order $\leq$ on $X$. When the meaning is clear from context and there is no ambiguity about the partial order, the set $X$ itself is sometimes called a poset.