[To Do] Basics of Iterating Relations MphLee Long Time Fellow    Posts: 367 Threads: 28 Joined: May 2013 12/27/2022, 07:57 PM (This post was last modified: 12/27/2022, 10:19 PM by MphLee.) 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. $$R_1=\Delta_X$$ $$xR_{s+t}y \iff xR_sR_ty\iff \exists w.\,xR_sw\land wR_ty$$ 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 $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ « Next Oldest | Next Newest »

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