A strict preorder or strict quasiorder on a set is a (binary) relation on that is both irreflexive and transitive. That is:
A strictly preordered set, or strict proset, is a set equipped with a strict preorder.
Strict preorders orders are asymmetric.
Transitivity of says that for all and , and implies . However, irreflexivity says that for all , is always false. This implies that for all and , and is always false, which is precisely the condition of asymmetry.
As a result, sometimes the term strict partial order is used for strict preorders, since they are always asymmetric. However, the term “strict partial order” is also used for other order relations.
Unlike with other notions of order, a set equipped with a strict preorder cannot be constructively understood as a kind of enriched category (at least, not as far as I know …). Using excluded middle, however, a strict preorder is the same as a partial order; interpret literally to mean that or , while conversely means that but .
Instead, the relation should be defined as an irreflexive comparison when generalising mathematics to other categories and to constructive mathematics.
If a strict preorder satisfies comparison (if , then or ), then it is a strict weak order, and additionally, if it is a connected relation, it is a strict total order.
There are also certainly examples of strictly preordered sets that are also partially ordered, where and (while related and so denoted with similar symbols) don't correspond as above. For example, if is any inhabited set and is any linearly ordered set, then the function set is partially ordered with meaning that always and strictly preordered with meaning that always. Except in degenerate cases, it's quite possible to have , , and simultaneously.
Wikipedia, Strict preorder
Wikipedia, Strict partial order
Last revised on December 26, 2023 at 05:10:11. See the history of this page for a list of all contributions to it.