Let $X$ be the set $\{1,\ldots,n\}$, where $n$ is a positive integer, and let $P(X)$ be the set of partitions of $X$. A partial order $\leq$ on $P(X)$ is given by reversed refinement: $\pi\leq\sigma$ for any $\pi$ and $\sigma$ in $P(X)$ if and only if for each $R$ in $\pi$ there is a $S$ in $\sigma$ such that $R\subseteq S$.
The pair $(P(X),\leq)$ is a lattice, and explicit constructions of the join and meet in this lattice have been discussed here.
A partition $\pi$ in $P(X)$ for which there are integers $a$, $b$, $c$ and $d$ such that $a < b < c < d$ and $a,c\in R$ and $b,d\in S$ for distinct blocks $R,S\in\pi$ is said to be crossing. A noncrossing partition is a partition that is not crossing.
The meet in the lattice of noncrossing partitions in general is the same as the meet in the lattice of partitions, but the same does not hold for the join. An example from Nica and Speicher's book (page 146) is the following: for $n=4$, $\{\{1, 3\}, \{2\}, \{4\}\}$ and $\{\{1\}, \{2, 4\}, \{3\}\}$ are noncrossing partitions, and in the lattice of noncrossing partitions their join is $\{1, 2, 3, 4\}$, but in the lattice of partitions their join is the crossing partition $\{\{1, 3\}, \{2, 4\}\}$.
What is an explicit construction of the join in the lattice of noncrossing partitions?
I had the following idea for a proof, but I do not know if it's correct: in case of any mistakes I'd much appreciate to hear about it!
My claim is that the join in the lattice of noncrossing partitions can be constructed in the same way as in the lattice of general partitions, but with "nondisjoint" replaced by "nondisjoint or crossing".
Let's make this more concrete.
In the case of general partitions, if $\pi$ and $\sigma$ are partitions of $X$, then a block in the join $\pi\vee\sigma$ of $\pi$ and $\sigma$ is constructed by starting with a block $P_0$ in $\pi$, enlarging it to a set $P_1$ by taking its union with all blocks in $\sigma$ that are nondisjoint with $P_0$, repeating this starting instead with $P_1$ and enlarging it to a set $P_2$ by taking its union with all blocks in $\pi$ that are nondisjoint with $P_1$, and continue in this way adding blocks from $\sigma$ and $\pi$ until nothing further is added. The resulting set is one block in the join of $\pi$ and $\sigma$, by doing the same for all other blocks in $\pi$ all blocks in $\pi\vee\sigma$ are obtained (possibly with some duplicates). This is the approach described here.
In other words, let $\approx$ be the relation on the union $\pi\cup\sigma$ of $\pi$ and $\sigma$ defined for any $P$ and $Q$ in $\pi\cup\sigma$ by $P\approx Q$ if and only if $P$ and $Q$ are nondisjoint, and let $\sim$ be the transitive closure of $\approx$. Then $\sim$ is an equivalence relation, and if $[P]$ is an equivalence class of $\sim$ then $\bigcup[P]$ is a block in $\pi\vee\sigma$.
In the case of noncrossing partitions $\pi$ and $\sigma$, their join is obtained by letting $\approx$ be the relation on $\pi\cup\sigma$ defined for any $P$ and $Q$ in $\pi\cup\sigma$ by $P\approx Q$ if and only if $P$ and $Q$ are nondisjoint or crossing: if $[P]$ is an equivalence class of the transitive closure of $\approx$ then $\bigcup[P]$ is a block in $\pi\vee\sigma$.
Let's see if this construction indeed yields the join in the lattice of noncrossing partitions.
Let $\pi$ and $\sigma$ be noncrossing partitions of $X$ and let $\approx$ be defined as above.
Each block in $\pi\vee\sigma$ is a union of elements in $\pi\cup\sigma$ and therefore nonempty, and the union of all blocks in $\pi\vee\sigma$ equals the union of all elements in $\pi\cup\sigma$, which equals $X$. Let $U$ and $V$ be distinct blocks in $\pi\vee\sigma$. Then there are $P$ and $Q$ in $\pi\cup\sigma$ such that $U=\bigcup[P]$ and $V=\bigcup[Q]$. Suppose that $U$ and $V$ are nondisjoint, then some $R$ in $[P]$ must be nondisjoint with some $S$ in $[Q]$, hence $R\approx S$ and therefore $[P] = [Q]$ and $U = V$, which is a contradiction. So $U$ and $V$ must be disjoint. Suppose that $U$ and $V$ are crossing, i.e., there are $a$, $b$, $c$ and $d$ in $X$ such that $a < b < c < d$ and $a,c\in U$ and $b,d\in V$. Then there must be $a'$ and $c'$ in $X$, $a'< b < c' < d$, and an element $R$ in $[P]$ such that $a',c'\in R$, and there must be $b'$ and $d'$ in $X$, $a < b' < c < d'$, and an element $S$ in $[Q]$ such that $b',d'\in S$. If $b' \leq c'$ then $R$ and $S$ are nondisjoint or crossing, implying that $[P] = [Q]$ and $U = V$, which is a contradiction, and if $c' < b'$ then we can repeat the above for $a' < b < c' < b'$ instead of $a < b < c < d$. Since $X$ is finite, eventually we will run into a contradiction, hence $U$ and $V$ cannot cross. So $\pi\vee\sigma$ is a noncrossing partition of $X$.
Since each block $P$ in $\pi$ is a subset of a block $\bigcup[P]$ in $\pi\vee\sigma$, and each block $Q$ in $\sigma$ is a subset of a block $\bigcup[Q]$ in $\pi\vee\sigma$, we see that $\pi\vee\sigma$ is an upper bound of both $\pi$ and $\sigma$.
Suppose that $\upsilon$ is a noncrossing partition of $X$ and an upper bound of both $\pi$ and $\sigma$, and let $P$ be any element in $\pi\cup\sigma$. Then there is a $U$ in $\upsilon$ such that $P\subseteq U$. If $Q$ is an element in $[P]$ that crosses or is nondisjoint with $P$ then there is a $V$ in $\upsilon$ such that $Q\subseteq V$, but since $P$ and $Q$ are crossing or nondisjoint the same must hold for $U$ and $V$, implying that $V = U$ and consequently $Q\subseteq U$. If $Q$ is any element in $[P]$, then there is a sequence $(Q_i)^n_{i=1}$ in $[P]$ such that $Q_1 = P$, $Q_n = Q$, and $Q_i$ and $Q_{i+1}$ are crossing or nondisjoint for $i=1,\ldots,n-1$, but then $Q_i\subseteq U$ for $i=1,\ldots,n$, and in particular $Q\subseteq U$. So $\bigcup[P]\subseteq U$. As this holds for any block $P$ in $\pi$ or $\sigma$ we conclude that $\pi\vee\sigma$ is the least upper bound of $\pi$ and $\sigma$ in the lattice of noncrossing partitions of $X$.