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 .
Accordingly, quasiorders in general should usually be replaced by partial orders when generalising mathematics to other categories. However, if a quasiorder satisfies comparison (if , then or ), then it is a linear order (at least on some quotient set), which is an important concept.
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.