nLab strict preorder

Definitions

A strict preorder or strict quasiorder on a set $S$ is a (binary) relation $\lt$ on $S$ that is both irreflexive and transitive. That is:

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

A strictly preordered set, or strict proset, is a set equipped with a strict preorder.

Properties

Theorem

Strict preorders orders are asymmetric.

Proof

Transitivity of $\lt$ says that for all $x \in S$ and $y \in S$, $x \lt y$ and $y \lt x$ implies $x \lt x$. However, irreflexivity says that for all $x$, $x \leq x$ is always false. This implies that for all $x$ and $y$, $x \lt y$ and $y \lt x$ 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.

In constructive mathematics

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

Instead, the relation $\lt$ should be defined as an irreflexive comparison when generalising mathematics to other categories and to constructive mathematics.

If a strict preorder satisfies comparison (if $x \lt z$, then $x \lt y$ or $y \lt z$), 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 $\lt$ and $\leq$ (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 $f \leq g$ meaning that $f(x) \leq g(x)$ always and strictly preordered with $f \lt g$ meaning that $f(x) \lt g(x)$ always. Except in degenerate cases, it's quite possible to have $f \ne g$, $f \nless g$, and $f \leq g$ simultaneously.

 Related concepts

• preorder
• strict total order

References

• Wikipedia, Strict preorder

• Wikipedia, Strict partial order