Gathering rights

As we saw in section , we can write $R=R_C \cup R_F$ with $R_C=U \times C$ and $R_F=U \times A \times F$.

We saw in section that we can reduce the size of the starting set, and thus the size of the function, using a subset of $F$ and the file system structure. We can do the same here.

For each file $f \in F$, we will adapt the set of rights. let $H_f$ be a subset of $F$ so that "/"$\in H_f$. We have, as defined in section ,

$$\psi_{H_f}~:~F \rightarrow H_f$$

$$x \mapsto \psi^{\alpha_{H_f}(x)}(x)$$

The binary relation $\mathcal{R}_{H_f}$ defined by

$$x \mathcal{R}_{H_f} y \Leftrightarrow \psi_{H_f}(x)=\psi_{H_f}(y)$$

is an equivalence relation.

If $x \in F$, we write $\tilde{x}=\{y \in F ~\vert~ y \mathcal{R}_{H_f} x\} \in \frac {F}{\mathcal{R}_{H_f}}$ the equivalence class of $x$. $\frac {F}{\mathcal{R}_{H_f}}$ is the quotient set, i.e. the set of all the equivalence classes.

Let's define the function

$$\Psi_{H_f}~:~ U \times C \cup U \times A \times H_f \rightarrow \mathcal{P}(R)$$

$$(u,c) \in U \times C \mapsto \{(u,c)\}$$

$$(u,a,h) \in U \times A \times H_f \mapsto \{(u,a)\} \times \tilde{h}$$

With a given $f \in F$, if we correctly choose $H_f$, we can factorize $\sigma'$ and $\mu'$. Let's call $\beta(f)$ the best set $H_f$ that can be found for $f$. Let's call $\gamma:f \mapsto \Psi_{\beta(f)}$. And let's call $\delta(f):G \rightarrow \beta(f)$ so that $\forall f \in F \quad\sigma(f)=\left(\gamma(f)\circ \delta(f)\circ \psi_G\right)(f)$. Here is our factorization.

Thus we have replaced the data of the graph of $\sigma ~:~ F \rightarrow R$ by the data of the graphes of $\beta ~:~ F \rightarrow \mathcal{P}(F)$ and

$$\delta ~:~ F \rightarrow \fonc(G,\beta(f))$$

$$f \mapsto \delta(f) ~:~ G \rightarrow \beta(f)$$

which is smaller in practical cases.