# Example of the $Ξ©$βpartite

Consider the following definition from section 2.2 of this research paper

We write $$\mathcal{P}(π)$$ for the power set of the set $$π$$. An $$π-$$partition $$Ξ©=\{π_1,\cdots,π_π\}$$ of a set $$π$$ is a disjoint family of $$π$$ nonempty subsets of $$π$$ with $$π=π_1βͺ \cdots βͺ π_π$$. Let $$Ξβ\mathcal{P}(π)$$ be a family of subsets of $$π$$. For a permutation $$π$$ on $$π$$, we define $$π(Ξ)=\{π(π΄):π΄βΞ\}β\mathcal{P}(π)$$. A family of subsets $$Ξβ\mathcal{P}(π)$$ is said to be $$Ξ©-$$partite if $$π(Ξ)=Ξ$$ for every permutation $$π$$ such that $$π(π_π)=π_π$$ for every $$π_πβΞ©$$. We say that $$Ξ$$ is $$π-$$partite if it is $$Ξ©-$$partite for some $$π-$$partition $$Ξ©$$. These concepts can be applied to access structures, which are actually families of subsets.

My doubt lies in $$π(Ξ)=\{π(π΄):π΄βΞ\}β\mathcal{P}(π)$$ 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 ?

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 $$\sigma(\Lambda') = \{\{1,2,4\},\{3,5,6\},\{7,8\}\} \ne \Lambda'$$.
• 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 Dec 21, 2019 at 7:46