# How to write the set of intersections of sets in a collection of partitions

Let $$A$$ be a non empty set and $$X$$ be the set of all partitions of $$A$$. If $$T \subseteq X$$, I want to write the set of intersections of sets in the partitions of $$T$$. Let me explain myself:

1. If $$T = \{ S_1, S_2 \}$$, I want the set $$S = \{C \cap D: C \in S_1, D \in S_2\}$$
2. If $$T = \{ S_1, S_2, S_3 \}$$, I want the set $$S = \{C \cap D \cap E: C \in S_1, D \in S_2, E \in S_3\}$$

So my question is: if $$T \subseteq X$$, how can I write $$S$$?.

My take is this: we can write $$T = \{S_i: i \in I\}$$ for fome index $$I$$. Now, every $$T_i$$ is a colection of sets itself, so we can write, for every $$i \in I$$, $$T_i = \{C_{i,j}: j \in J_i\}$$, for some index $$J_i$$. So in my head $$S = \{N: N = \bigcap_{i \in I} (\bigcup_{j \in J_i} C_{i,j})\}$$. Is this correct, or there is something to change? Thank you.

• You probably also want to require that the intersections are non-empty. – Asaf Karagila Jun 22 '20 at 18:28
• @AsafKaragila Yes, of course. I didn't add that because I was writing that in a tablet, so I wanted put the minimum necessary. I just want to know if it's written correctly. – Iovita Kemény Jun 22 '20 at 23:02

This is slightly easier to write if you think about the actual cells in the partitions. Ultimately, I suppose, the goal is to find the coarsest partition that refines all the partitions in your collection.

So given $$a\in A$$, what will be the cell containing $$a$$? Well, exactly $$\{b\in A\mid\forall i\in I\,\forall S\in S_i:a\in S\to b\in S\}$$. So call this $$T(a)$$, and now the refinement is simply $$\{T(a)\mid a\in A\}$$. This also ensures that the result is a partition (i.e., it will not have any empty sets).

This answer assumes that $$T$$ is already defined as $$T = \left\{T_{i} \mid i \in I\right\}$$ for some index set $$I$$.

Take some $$B \in S$$. By definition of $$S$$, we have

$$B = \bigcap_{i \in I} B_{i}$$

for some collection of sets $$\left\{B_{i} \mid i \in I\right\}$$ such that for each $$i \in I$$, we have $$B_{i} \in T_{i}$$. Hence, we can define $$S$$ simply as

$$S = \left\{\bigcap_{i \in I} B_{i} \mid B_{i} \in T_{i}\right\}$$

Update

Assume that we must continue from the definitions already provided by the OP.

Take some $$N \in S$$. By definition of $$S$$, we have

$$N = \bigcap_{i \in I} C_{i, j}$$

for some collection of sets $$\left\{C_{i,j} \mid i \in I\right\}$$ such that for each $$i \in I$$, we have $$j \in J_{i}$$. Hence, $$S$$ is simply defined as

$$S = \left\{\bigcap_{\substack{i \in I \\ j \in J_{i}}} C_{i, j}\right\}$$

without any further characterization in the definition.