Discrete Preorder

definition example

Every Set $X$ can be considered as a discrete preorder $(X,=)$: the only order relations are $x\le x$; if $x\ne y$ neither $x\le y$ nor $y\le x$ holds. Its Hasse Diagram is a collection of points. It is already a Partial Order.

Sources: 7 Sketches Example 1.32, Exercises 1.41, 1.44, 1.55, 1.67, 1.73; Kittenlab Lecture 5; CTfS Example 3.4.3.5, Exercises 3.4.3.6–3.4.3.7, Example 5.1.1.5

• Two elements are comparable iff they are equal (7S Exercise 1.44).
• Every function out of a discrete preorder is monotone (7S Exercise 1.67); thus $\mathrm{Disc}:\mathbf{Set}\to \mathbf{Preord}$ is a Functor, left adjoint to the underlying-set functor (Kittenlab Lecture 5 lists it alongside the Codiscrete Preorder functor).
• Dually (CTfS Exercise 3.4.3.6), a monotone map into a discrete preorder is a function that is constant on each connected component of the domain; monotone maps into the indiscrete preorder are arbitrary functions, and monotone maps out of it land in a clique. “The smallest preorder structure that can be put on $S$” is the discrete one, “the largest” the indiscrete one, and they are the left and right adjoints of the underlying-set functor (CTfS Example 5.1.1.5).
• The upper sets of a discrete preorder form the whole Power Set (7S Exercise 1.55).
• A skeletal Dagger Preorder is discrete, hence “can be identified with” a set (7S Exercise 1.73, Remark 1.74).
• As a category, a discrete preorder is a Discrete Category.