A Boolean lattice has a number of rather nice properties which give it a central role in many parts of combinatorics. For instance, it's a lattice, it can be augmented with a ring structure, it can also be augmented with an associative algebra structure, it has a complementation operation, it can be "identified" in different ways with things like the hypercube or a powerset, it satisfies the Stone representability theorem, etc.

It's kind of obvious that Boolean lattices are pretty closely related to the number 2. Like, they have a duality, finite ones all have order a power of 2, etc. One way to see this is to look at a (finite) Boolean lattice as the set of all functions from a set $S \rightarrow \{0, 1\}$, with the lattice and complementation structures acting pointwise.

To what extent can you get an analogue of Boolean lattices (or even posets) with some other natural number k taking the place of 2? You could consider the set of all functions from $S \rightarrow \{0, 1, ..., k-1\}$, which again gives you a poset with a lattice structure, but we don't get a complementation map (although I guess we do get some sort of "k-ality.")

There are a number of questions that this raises: Is this the proper generalization of Boolean lattices in this direction? Does some weird analogue of the Stone theorem hold? What's true for Boolean lattices but isn't true for these guys? What are the right notions to replace "hypercube" and "power set?" Etc.