# Monotonic Access Structure

I got confused with the monotonic access structure definition.

Following definition is one of the answers to the following question:

What are monotonic and non monotonic access structures in ABE ?

Let $$\{1,2,...,n\}$$ be a set of indices. An access structure is a collection $$\mathbb{A}$$ of non-empty subsets of $$\{1,2,3,...,n\}$$. We say a collection (or an access structure) $$\mathbb{A} \subseteq 2^{\{1,2,...,n\}}$$ is monotonic if for any $$B,C \in 2^{\{1,2,...,n\}}$$, if $$B \in \mathbb{A}$$ and $$B \subseteq C$$ then $$C \in \mathbb{A}$$.

According to the above definition, every access structure $$\mathbb{A}$$ which not include $$\{1,2,3,...,n\}$$ is not monotonic!, because every arbitrary $$B \in 2^{\{1,2,...,n\}}$$ always is subset of $${\{1,2,...,n\} }$$

Could you please clarify what I am going wrong?

The access structure itself need not be non-empty, so the universe doesn't have to be in the structure. Of course, any non-empty access structure will have $$\{1,2,\dots,n\}$$ as an element.
In the context of ABE, it suffices to consider monotone access structures if the scheme supports an expressive policy set, because any subset $$\mathbb{B}$$ of $$2^{\{1,2,\dots,n\}}$$ (i.e., any not necessarily monotone access structure of universe $$\{1,2,\dots,n\}$$) can be encoded as a monotone access structure $$\mathbb{A}$$ of the universe $$\{\pm1,\pm2,\dots,\pm n\}$$ as follows: \begin{aligned}\mathbb{A}'&{}=\Bigl\{{\{i\,|\,i\in X\}\cup\{-j\,|\,j\notin X\}}\,\Big|\,X\in\mathbb{B}\Bigr\},\\\mathbb{A}&{}=\Bigl\{A\,\Big|\,\mathbb{A}'\ni A'\subseteq A\subseteq\{\pm1,\pm 2,\dots,\pm n\}\Bigr\}.\end{aligned} In English, this means you prepare positive/negative literals for every member of $$\{1,2,\dots,n\}$$, then encode any subset of it by explicitly putting negative literals for non-members into it, which gives you $$\mathbb{A}'$$. Then you take the smallest monotone access structure $$\mathbb{A}$$ containing $$\mathbb{A}'$$.
There isn’t anything wrong. In most access structure applications if a group of users (say subset $$B$$) are authorized for access this means we don’t mind giving larger sets $$C$$ containing $$B$$ access as well.