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 ) 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 on a set is a partial order iff all of the following:

- ;
- ;
- .

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 such that iff for every there exists such that and rotating the set radians counter-clockwise produces a set which intersects with .

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