I derived this definition by searching for a representation of a family of sets. I am quite sure that someone should have thought to this before, because it seems to be quite straightforward given a family of sets. However, I did not find anything suitable on google or on wikipedia.


Let a family of sets, say $A\_1, \ldots, A\_n$, be given. To avoid misunderstanding I will call them *modules*. This family induces a unique partition on the union set $A = \bigcup\_{i=1}^{n} A\_i$ in the following way: I call *building block* a maximal subset $B$ of $A$ such that do not exist 2 different modules $A\_i$ and $A\_j$ with:


$B \cap A\_i \not= \emptyset$,


$B \cap A\_j \not= \emptyset$ and


$B\not\subseteq A\_i \cap A\_j$.


For example, if my family consists of 2 different overlapping modules $A\_1$ and $A\_2$, I can partition the set $A = A\_1 \cup A\_2$ as:


(elements in $A\_1 \cap A\_2$);


(elements in $A\_1$ but not in $A\_2$);


(elements in $A\_2$ but not in $A\_1$).


I know that in logic there is something similar, but I am searching for something in set theory. Moreover, I want to underline the dependency of this uniquely derived partition from the family of sets I am given.


See also <http://en.wikipedia.org/wiki/Wikipedia:Reference_desk/Mathematics#Given_a_family_of_sets.2C_I_can_generate_a_partition_of_their_union_set.2C_by_looking_at_their_overlapping>...


Thanks to all!


A newcomer



I am searching for the name of a partition (if it already exists)
Your building blocks are known as the *atoms* in the [Boolean algebra](http://en.wikipedia.org/wiki/Boolean_algebras) or [field of sets](http://en.wikipedia.org/wiki/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 $A\_i$.


