Is the set of noncrossing partitions of an infinite set a lattice?

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.

Solution 1:

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.