[To Do] Basics of Iterating Relations
I want to outline some elementary construction that can start a possible systematic theory of iterated (non-functional) relations. Some basic examples of monoids of relations come naturally from the theory of metric spaces and the "apartness up to some \(\epsilon\in \mathbb R\)" relation or by physical measurements
[*]. Other more complex examples may be, I hope, relevant in the iteration of multivalued functions and of iterated (non-integer) solutions of functional equations, i.e. fractional calculus and non-integer ranks.

Definition (relational action) Let \(R:{\bf B} M\to {\rm Rel} \) a functor. It defines a monoid of relations under relations composition.
$$xR_{s+t}y \iff xR_sR_ty\iff \exists w.\,xR_sw\land wR_ty$$

Even if not well behaved as the iteration of functions we can still study and make explicit the network of "wires" that glues points of \(X\) and that is induced simultaneously by all the relations \(R_{r\in M}\).

Definition (Associated pullbacks bundles) Let \(R_s\times_XR_t\) the pulback of \(\pi_1:R_s\to X\) and \(\pi_0:R_t\to X\) then

$$(x,w,y):R_s\times_XR_t\implies xR_sR_ty \iff xR_{s+t}y$$

in particular we have a surjection \(\rho_{s,t}:R_s\times_X R_t\to R_sR_t=R_{s+t}\) defined by central projection and that gives a bundle s.t. its fibers are of the form $${\boldsymbol \rho}_{s,t}(x,y)\cong\{w:X.\,xR_sw\land wR_ty\}$$

Definition (associated graph) We can form a graph over \( X\) it by considering the left and right projections over the pullback. They are induced by the projections \(R_{s+t}\to X\) by precomposing with \(\rho_{s,t}\).

Definition (\(r\)-intermediate points) We define all the possible \(r\in M\)-intermediates of two points as $${\boldsymbol \rho}_r(x,y)=\coprod_{s+t=r} {\boldsymbol \rho}_{s,t}(x,y)\cong\{w:X.\,\exists s,t\in M.\,s+t=r\land xR_sw\land wR_ty\}$$

these sets are the fibers of the following bundle

Definition (Associated total pulback bundle over the time \(r\in M\)) Define it as

$${\boldsymbol \rho}_r:\coprod_{s+t=r}R_s\times_X R_t\to R_r$$

Many graph structures arise over \(X\) one for every \(r\in M\). We can define the total graph of the functor \(R\) at the time \(r\in M\) by precomposing the two left and right projections \(R_r\to X\) by \({\boldsymbol \rho}_r\)
$$\coprod_{s+t=r}R_s\times_X R_t\to  X$$

More basic examples and proof of basic properties are needed. I just leave this as a a todo-thread for me. Any idea/comment is welcome.

[*]Consider \(x\leq_t y :\iff |y-x|\leq t \) then clearly \(x\leq_t w\), i.e. \(|w-x|\leq t\), and \(w\leq_s y\), i.e. \(|y-w|\leq s\), then \(x\leq_{s+t}y\) because by triangle inequality \(|y-x|\leq|w-x|+|y-w|\leq s+t\). Vice versa if \(|y-x|\leq s+t\) then we take a point \(w\) in the intersections of two balls \(B_s(x)\cap B_t(y)\). It must be non-empty, we thus have opposite implication.

MSE MphLee
Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)
S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)

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