Consider $A \subset \{0,1\}^n$
I want $A$ to have $2$ properties.
$1.$ $A$ is increasing, i.e., If $x \in A$ and $x \subset y$ then $y \in A$ too.
$2.$ $A^c$ is equal to set $B=\{x | x^c \in A\}$
Is there any characterization for such a set? I have to example for it. But I want to find an IFF condition for such sets...
$e1)$ $A=\{x|$ first coordinate of $x$ is $1\}$
$e2)$ Fix an odd number of coordinates. $A= \{x| x$ contains at least half of coordinates equal to $1\}$ [For even number there is a similar example]