I am interested in the congruence lattices of algebraic structures.
Is there a brute-force way to produce an infinite algebraic structure whose lattice of congruences is isomorphic to some given finite lattice?
This MO post by Noah Schweber describes a brute force algorithm for building a finite algebraic structure with a given congruence lattice that doesn't succeed.
Namely, starting with a finite lattice $\Lambda$, you represent it as a set of partitions of some finite set $X$ picked in some unspecified manner that is order-isomorphic to $\Lambda$; call it $\Lambda((X))$.
Next, you create a new algebraic structure with domain $X$ that is endowed with every function that is well-defined when modded out by every partition associated with $X$; call this structure $\Lambda(X)$.
$\Lambda(X)$ is a clone; it has projections and is closed under composition.
There are a few things about this construction that I don't understand, but I'm willing to accept that it breaks in an instructive way on faith:
- How did we choose $X$?
- If $\Lambda$ is the congruence lattice of some finite algebra $A$, it certainly seems like we could extend $A$ with all functions that respect its congruences and end up with a clone $\Lambda(X)$ for some finite $X$ equipped with a set of partitions.
The fact that this fails for finite sets $X$ makes me wonder if there's a construction using infinite sets that succeeds.