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.

Thanks to all!

A newcomer