Your building blocks are known as the atoms in the Boolean algebra or field of sets generated by the $A_i$. Each building block will consist of points that have all the same pattern of answers for membership in the $A_i$. To see this, observe first that by maximality any building block will respect this equivalence and therefore be a union of such atoms. Conversely, if a set has points with two different patterns of answers, then it will contain points from two $A_i$ without being contained in their intersection. (Specifically, if $x,y\in B$ and $x\in A_i$ but $y\notin A_i$, then pick $j$ such that $y\in A_j$, and observe that $B$ meets both $A_i$ and $A_j$, buit is not contained in $A_i\cap A_j$.) So the building blocks are the atoms.

If your family is finite as you indicated, then the Boolean algebra generated by the $A_i$ consists precisely of the unions of blocks. This is the representation theorem showing that every finite Boolean algebra is isomorphic to a finite power set---the power set of the atoms.

In the infinite case, should you entertain infinitely many $A_i$, then unfortunately, you don't get a partition, for there are atomless Boolean algebras. Nevertheless, these Boolean algebras are fascinating.

Your building blocks are known as the atoms in the Boolean algebra or field of sets generated by the $A_i$. Each building block will consist of points that have the same pattern of answers for membership in the various $A_i$. To see this, observe first that by maximality any building block will respect this equivalence and therefore be a union of such atoms. Conversely, if a set has points with two different patterns of answers, then it will contain points from two $A_i$ without being contained in their intersection. (Specifically, if $x,y\in B$ and $x\in A_i$ but $y\notin A_i$, then pick $j$ such that $y\in A_j$, and observe that $B$ meets both $A_i$ and $A_j$, buit is not contained in $A_i\cap A_j$.) So the building blocks are the atoms.

If your family is finite as you indicated, then the Boolean algebra generated by the $A_i$ consists precisely of the unions of blocks. This is the representation theorem showing that every finite Boolean algebra is isomorphic to a finite power set---the power set of the atoms.

In the infinite case, however, the Boolean algebra generated by the $A_i$ may be atomless---it may have no atoms at all, and this is a fascinating case. Nevertheless, your blocks still form a partition, and are precisely the atoms in the infinitary-generated field of sets, still determined by my argument above by the patttern-of-answers to membership in the $A_i$.

Your building blocks are known as the atoms in the Boolean algebra generated by the $A_i$. Each building block will consist of the points that have all the same pattern of answers for membership in the $A_i$. It is clear by maximality that any building block will respect this equivalence and therefore be a union of such atoms. Conversely, if a set has points with two different patterns of answers, then it will contain points from two $A_i$ without being contained in their intersection.

If your family is finite as you indicated, then the Boolean algebra generated by the $A_i$ consists simply of the unions of blocks. If your original family is infinite, however, then you don't get a partition, for there are atomless Boolean algebras, and things become very interesting.

Your building blocks are known as the atoms in the Boolean algebra or field of sets generated by the $A_i$. Each building block will consist of points that have all the same pattern of answers for membership in the $A_i$. To see this, observe first that by maximality any building block will respect this equivalence and therefore be a union of such atoms. Conversely, if a set has points with two different patterns of answers, then it will contain points from two $A_i$ without being contained in their intersection. (Specifically, if $x,y\in B$ and $x\in A_i$ but $y\notin A_i$, then pick $j$ such that $y\in A_j$, and observe that $B$ meets both $A_i$ and $A_j$, buit is not contained in $A_i\cap A_j$.) So the building blocks are the atoms.

If your family is finite as you indicated, then the Boolean algebra generated by the $A_i$ consists precisely of the unions of blocks. This is the representation theorem showing that every finite Boolean algebra is isomorphic to a finite power set---the power set of the atoms.

In the infinite case, should you entertain infinitely many $A_i$, then unfortunately, you don't get a partition, for there are atomless Boolean algebras. Nevertheless, these Boolean algebras are fascinating.