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partial order
A partial order (often simply referred to as an order or ordering) is a relation $\leq\>\subset A\times A$ that satisfies the following three properties:
1. Reflexivity: $a\leq a$ for all $a\in A$
2. Antisymmetry: If $a\leq b$ and $b\leq a$ for any $a,b\in A$, then $a=b$
3. Transitivity: If $a\leq b$ and $b\leq c$ for any $a,b,c\in A$, then $a\leq c$
A total order is a partial order that satisfies a fourth property known as comparability:

Comparability: For any $a,b\in A$, either $a\leq b$ or $b\leq a$.
Remark. In some literature, especially those dealing with the foundations of mathematics, a partial order $\leq$ is defined as a transitive irreflexive binary relation (on a set). As a result, if $a\leq b$, then $b\nleq a$, and therefore $\leq$ is antisymmetric.
Mathematics Subject Classification
06A06 no label found35C10 no label found35C15 no label found5501 no label found5500 no label found Forums
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A partial order is a relation that satisfies the following 3 conditions



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