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Consider the following definition from section 2.2 of this research paper

We write $\mathcal{P}(𝑃)$ for the power set of the set $𝑃$. An $π‘Ÿ-$partition $Ξ©=\{𝑃_1,\cdots,𝑃_π‘Ÿ\}$ of a set $𝑃$ is a disjoint family of $π‘Ÿ$ nonempty subsets of $𝑃$ with $𝑃=𝑃_1βˆͺ \cdots βˆͺ 𝑃_π‘Ÿ$. Let $Ξ›βŠ†\mathcal{P}(𝑃)$ be a family of subsets of $𝑃$. For a permutation $𝜎$ on $𝑃$, we define $𝜎(Ξ›)=\{𝜎(𝐴):π΄βˆˆΞ›\}βŠ†\mathcal{P}(𝑃)$. A family of subsets $Ξ›βŠ†\mathcal{P}(𝑃)$ is said to be $Ξ©-$partite if $𝜎(Ξ›)=Ξ›$ for every permutation $𝜎$ such that $𝜎(𝑃_𝑖)=𝑃_𝑖$ for every $𝑃_π‘–βˆˆΞ©$. We say that $Ξ›$ is $π‘Ÿ-$partite if it is $Ξ©-$partite for some $π‘Ÿ-$partition $Ξ©$. These concepts can be applied to access structures, which are actually families of subsets.

My doubt lies in $𝜎(Ξ›)=\{𝜎(𝐴):π΄βˆˆΞ›\}βŠ†\mathcal{P}(𝑃)$ because the paper does not define the permutation function on a set, but used it. I didn't came through any standard defifnition of permutation function on set which gives another set which is different from the input set because the order of elements in a set is immaterial.

What I am clear from the context is

$$\sigma: P \rightarrow P$$ $$ \sigma(A \in \mathcal{P}(P)) = B \in \mathcal{P}(P)$$

but I don't know what exactly $B$ is.

What is the definition of a permutation function over a set?

In addition, can you give some example of the above $Ξ©-$partite ?

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A permutation on a set is a bijection from the set to itself; thus $\sigma$ is a (bijective) map from $P$ to itself, not from $p(P)$ to itself (though it does induce a map from $p(P)$ to itself in a natural manner).

Also, if $f$ is a map from $A$ to $B$ and $X$ is a subset of $A$, then $f(X) = \{f(x), x \in X\}$. This is completely standard notation.

For a collection $\Lambda$ of subsets of $P$, $\sigma(\Lambda) = \{\sigma(A), A \in \Lambda\}$ is the collection of subsets of $P$ which is obtained by applying $\sigma$ to all the elements of $\Lambda$.

Example: $P = \{1,2,\dots,8\}$, $\Omega = \{\{1,2\},\dots,\{7,8\}\}$. Then $\Lambda = \{\{1,2,3,4\},\{5,6,7,8\}\}$ is $\Omega$-partite since clearly every permutation of $P$ that fixes each $P_i$ also fixes each element of $\Lambda$, so in fact we even have $\sigma(A) = A$ for all $A$. On the other hand, $\Lambda' = \{\{1,2,3\},\{4,5,6\},\{7,8\}\}$ is not; for instance $\sigma = (34)$ yields $\sigma(\Lambda') = \{\{1,2,4\},\{3,5,6\},\{7,8\}\} \ne \Lambda'$.

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  • $\begingroup$ My observation is $\mathcal{P}(P)\longrightarrow \mathcal{P}(P)$ . Domain is $\mathcal{P}(P)$ and codomain is $\mathcal{P}(P)$. but you are telling $P\longrightarrow P$, How it is? Can you give an explanation $\endgroup$
    – Natwar
    Dec 21, 2019 at 7:46

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