Links on this site may earn a commission.
Our affiliate programs include, but are not limited to;
Walmart, Bestbuy, eBay Partner Network, Amazon, Target, Etsy, Gamestop, & more.
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. Doing so, however, is not always useful or worthwhile, depending on the problem domain being studied.
In words, when
a
≤
b
,
{\displaystyle a\leq b,}
one may say that b covers a or that a precedes b, or that b reduces to a. Occasionally, the notation ← or → or
≲
{\displaystyle \,\lesssim \,}
is used instead of
≤
.
{\displaystyle \,\leq .}
To every preorder, there corresponds a directed graph, with elements of the set corresponding to vertices, and the order relation between pairs of elements corresponding to the directed edges between vertices.
Some links on this site may earn a small commission when you make a purchase.
Our affiliate programs include, but are not limited to; eBay, Amazon, Target, BestBuy, Etsy & more.
We are independently owned and the opinions expressed here are our own, except where indicated.