nLab
quasiorder

Definitions

A quasiorder on a set $S$ is a (binary) relation $<$ on $S$ that is both irreflexive and transitive. That is:

• $x\nless x$ always;
• If $x<y<z$, then $x<z$.

A quasiordered set, or quoset, is a set equipped with a quasiorder.

Properties

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 $x\le y$ literally to mean that $x<y$ or $x=y$, while $x<y$ conversely means that $x\le y$ but $x\ne y$.

Accordingly, quasiorders in general should usually be replaced by partial orders when generalising mathematics to other categories. However, if a quasiorder satisfies comparison (if $x<z$, then $x<y$ or $y<z$), 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 $\le$ (while related and so denoted with similar symbols) don't correspond as above. For example, if $A$ is any inhabited set and $B$ is any linearly ordered set, then the function set $B^A$ is partially ordered with $f\le g$ meaning that $f(x)\le g(x)$ always and quasiordered with $f<g$ meaning that $f(x)<g(x)$ always. Except in degenerate cases, it's quite possible to have $f\ne g$, $f\nless g$, and $f\le g$ simultaneously.