# Partial order funcoids and reloids

Partial order funcoids and reloids formalize such things as “infinitely small” step rotating a circle counter-clockwise.

This is “locally” a partial order as every two nearby “small” sets (where we can define “small” for example as having the diameter (measuring along the circle) less than $\pi$) are ordered: which is before in the order of rotating the circle counter-clockwise and which is after.

The definition for partial order funcoid (and similarly partial order reloid) is a trivial generalization of the classical definition of partial order.

The endo-funcoid $f$ on a set $A$ is a partial order iff all of the following:

1. $f\sqsupseteq\mathrm{id}^{\mathsf{FCD}}_A$;
2. $f\sqcap f^{-1}\sqsubseteq\mathrm{id}^{\mathsf{FCD}}_A$;
3. $f\circ f\sqsubseteq f$.

This can also be defined for reloids entirely analogous to funcoids.

What are possible applications of partial order funcoids and partial order reloids? I yet don’t know.

## One comment

1. This “infinitely small counter-clockwise step” can be defined as the funcoid $f$ such that $X[f]Y$ iff for every $\epsilon>0$ there exists $\epsilon'\ge 0$ such that $\epsilon'<\epsilon$ and rotating the set $X$ $\epsilon'$ radians counter-clockwise produces a set which intersects with $Y$.

Exercise: Prove that the funcoid $f$ exists and that is a partial order funcoid.