A quasiorder on a set is a (binary) relation on that is both irreflexive and transitive. That is:
A quasiordered set, or quoset, is a set equipped with a quasiorder.
Unlike with other notions of order, a set equipped with a quasiorder cannot be constructively understood as a kind of enriched category (at least, not as far as I know …). Using excluded middle, however, a quasiorder 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 quasiorder satisfies comparison (if , then or ), then it is a strict order, and additionally, if it is a linear relation, it is a linear order.
There are also certainly examples of quasiordered 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 quasiordered with meaning that always. Except in degenerate cases, it's quite possible to have , , and simultaneously.
Last revised on December 7, 2022 at 16:10:09. See the history of this page for a list of all contributions to it.