It is well-known that the set of noncrossing partitions of a finite set is a lattice; see e.g. Wikipedia or the 1991 article by Simion and Ullman.
What about the infinite case? Is the set of noncrossing partitions of an infinite set also a lattice?
Literature I have seen on the lattice structure of sets of noncrossing partitions seems to focus only on the finite case.
Any references to literature on this topic would be appreciated.
Yes, there is no difficulty adapting the argument to infinite sets. Specifically, let $S$ be a totally ordered set. Then define a noncrossing partition of $S$ to be a partition $P$ of $S$ such that there do not exist distinct $A,B\in P$ and $a,b\in A,x,y\in B$ such that $a<x<b<y$. Suppose $(P_i)$ is a family of partitions of $S$ and $P$ is their meet (in the complete lattice of partitions of $S$). Then I claim that if each $P_i$ is noncrossing, so is $P$. Indeed, suppose $A,B\in P$ are distinct with $a,b\in A,x,y\in B$ such that $a<x<b<y$. Then $A$ is an intersection of sets $A_i\in P_i$ for each $i$, and similarly $B$ is an intersection of sets $B_i\in P_i$. Since $A\neq B$, there is some $i$ such that $A_i\neq B_i$. But then we have $a,b\in A_i$ and $x,y\in B_i$, so $P_i$ is not noncrossing.
So, the set of noncrossing partitions is closed under meets in the complete lattice of all partitions. It follows that the set of noncrossing partitions is itself a complete lattice.