Preorder
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special cases of a preorder: an antisymmetric (or skeletal) preorder is a partial order, and a symmetric preorder is an equivalence relation.
The name preorder comes from the idea that preorders (that are not partial orders) are 'almost' (partial) orders, but not quite; they are neither necessarily antisymmetric nor asymmetric. Because a preorder is a binary relation, the symbol
≤
{\displaystyle \,\leq \,}
can be used as the notational device for the relation. However, because they are not necessarily antisymmetric, some of the ordinary intuition associated to the symbol
≤
{\displaystyle \,\leq \,}
may not apply. On the other hand, a preorder can be used, in a straightforward fashion, to define a partial order and an equivalence relation.
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