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 $latex \pi$) 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 $latex f$ on a set $latex A$ is a partial order iff all of the following:
- $latex f\sqsupseteq\mathrm{id}^{\mathsf{FCD}}_A$;
- $latex f\sqcap f^{-1}\sqsubseteq\mathrm{id}^{\mathsf{FCD}}_A$;
- $latex f\circ f\sqsubseteq f$.
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.
This “infinitely small counter-clockwise step” can be defined as the funcoid $latex f$ such that $latex X[f]Y$ iff for every $latex \epsilon>0$ there exists $latex \epsilon’\ge 0$ such that $latex \epsilon'<\epsilon$ and rotating the set $latex X$ $latex \epsilon'$ radians counter-clockwise produces a set which intersects with $latex Y$.
Exercise: Prove that the funcoid $latex f$ exists and that is a partial order funcoid.