Definition. Partial order [002d]
A partial order is just a preorder with an additional property called antisymmetry. A partial order on a set \(P\) is a binary relation \(\leq \) that satisfies the following three properties for all elements \(a, b, c \in P\):
- \(a \leq a\) (Reflexivity): Every element is related to itself.
- \(a \leq b \land b \leq c \to a \leq c\) (Transitivity): If an element \(a\) is related to \(b\), and \(b\) is related to \(c\), then \(a\) is also related to \(c\).
- \(a \leq b \land b \leq a \to a = b\) (Antisymmetry): If an element \(a\) is related to \(b\), and \(b\) is related to \(a\), then \(a\) and \(b\) must be the same element.
More accurately a partial order is a reflexive, weak, non-strict partial order. Also known as skeletal preorder. [0]