Does meet of two partitions of a set always exist? Let $\Omega$ be any set. Let $\mathcal{P_1}$ and $\mathcal{P}_2$ be partitions of $\Omega$. By $P_i(\omega)$ we denote cell of partition $i$ containing $\omega$.
Meet of partitions $\mathcal{P}_1$ and $\mathcal{P}_2$ is defined in the following way:
$\mathcal{P}_1\wedge \mathcal{P}_2$ is the finest partition of $\Omega$ such that $P_i(\omega)\subset (\mathcal{P}_1\wedge \mathcal{P}_2)(\omega)$ for $i=1,2$ and every $\omega\in\Omega$. 
By $(\mathcal{P}_1\wedge \mathcal{P}_2)(\omega)$ we denote a cell of $\mathcal{P}_1\wedge \mathcal{P}_2$ containing $\omega$.
My question is: 
Is existence of $\mathcal{P}_1\wedge \mathcal{P}_2$ always guaranteed? Is it unique? ($\Omega$ can be finite, countable or uncountable). 
In the worst case (when there are no finer partitions), $\{\Omega\}$ is always meet. Is that enough to prove existence? How can i prove uniqness?
 A: For $a,b\in\Omega$ say $a\sim b$ if there exist finitely many elements $x_0=a,x_1, \ldots , x_n=b$ such that $x_{i+1}\in \mathcal P_1(x_i)\cup \mathcal P_2(x_i)$ for $0\le i<n$. Then $\sim$ is an equivalence relation on $\Omega$ and the corresponding partition has the desired property. Also, any other partition with the desired property must be coarser because for any sequence $x_0,\ldots, x_n$ as above we need that all $x_i$ are in the same cell.
A: If you think about a partition as being induced from an equivalence relation, then we know that if $E_1,E_2$ are two equivalence relations then there is no reason for $E_1\cup E_2$ to be an equivalence relation.
However, $E_1\cup E_2$ is a reflexive and symmetric relation. So if we take $E$ to be its transitive closure, then $E$ is the smallest equivalence relation which includes both $E_1$ and $E_2$.
And it's not hard to see why $E$ induces the wanted partition. Of course that for $i=1,2$, if $a\mathrel{E_i}b$ then $a\mathrel{E}b$; and if $E'$ is an equivalence relation which has the above property then it must contain $E$, so $E$ is indeed the smallest.
This is effectively what Hagen von Eitzen suggested, although I agree it's not as clear when thinking about it in terms of partitions.
