chap-micro.tex

\chapter{Micronization}

I defined ``micronization'' wrongly in my book and did some erroneous
proofs about it. Here is an attempt to salvage it.

\url{https://en.wikipedia.org/wiki/Transitive_reduction} is a special
case of micronization. (Hm, maybe them coincide only for finite sets?)

\begin{defn}
\emph{Micronization} $\mu(E)$ of a binary relation~$E$ is defined by
the formula:
\[
  \mu(E) = \bigsqcap^{\mathsf{RLD}} \setcond{f\in\mathsf{RLD}}{
    S^{\ast}(f)\sqsupseteq E \land f\asymp f^2.
  }
\]
\end{defn}

It's wrong (consider micronization of $\leq$ on real numbers
(which should be addition of infinite small).

\begin{question}
Under which conditions $S^\ast(\mu(E)) = E$?
\end{question}