# Quasiorders

## 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, then $x.

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 or $x=y$, while $x 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, then $x or $y), 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 $f\left(x\right)\le g\left(x\right)$ always and quasiordered with $f meaning that $f\left(x\right) always. Except in degenerate cases, it's quite possible to have $f\ne g$, $f\nless g$, and $f\le g$ simultaneously.

Revised on September 2, 2012 10:05:30 by Toby Bartels (98.23.143.147)