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July 5, 2013 / porton

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.

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One Comment

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  1. porton / Jul 5 2013 20:39

    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.

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