Question

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.

Answer 1

A standard extention is to the lattice of subspaces of an n dimensional space over a field with q elements. Many nice properties of the Boolean lattice extend while others do not. Many extremal combinatorics result extend and some are even simpler.

Answer 2

One can consider the set of functions $S \to P$ where $P$ is any poset with $k$ elements. The reason the Boolean lattice is special is that the "truth poset" happens to lie at the intersection of a lot of important ideas (logic, set theory, universal algebra, etc.) and I don't think the same can be said about just any old poset.

If you want to generalize most of the important properties of the Boolean lattices I'm not convinced cardinality is the right think to be looking at.

Answer 3

The generalization you seek exists when k itself is a power of 2 (but gives no additional examples). This is because, as Q Yuan points out, the important properties of 2 that you seem to require are that is it a "truth" poset 2={false,true}, and when k=2^m, then there is a Boolean algebra of size k that can serve a similar purpose.

That is, the suggestion is that you should replace 2 with an arbitrary Boolean algebra B. For example, if you look at functions f:S to B, you can still perform lattice operations and complements pointwise. Perhaps this is the generalization you seek.

But you won't get any new examples this way, since they will still just be (larger) Boolean algebras.

Answer 4

Alfred Foster worked on "n-ality" in some structures. While not quite a generalization of Boolean algebras, he did work on structures with a notion that generalized duality. In one of his papers, he found (something like, I am operating with faulty memory here) a substructure that resembled a ring of idempotents that helped carry the notion. He was also interested in generalizations of the Chinese Remainder Theorem for structures.

That and n-valued logics are the closest I can think of toward the "n" part of your question.

Gerhard "Ask Me About System Design" Paseman, 2010.03.01

Answer 5

Just to answer the question with a few more questions:

• What is the definition of generalization that you have in mind here?

• Can you think of other definitions of generalization that are commonly used?

• For example, is working over $\mathbb{R}$ more general or less general than working over $\mathbb{B}$?

Addendum

There are many ways of defining an order of generalization, one of them being to align it with an order of abstraction. There are in fact several different kinds of abstraction, but the simplest among them is probably the idea that concrete things have many properties and abstraction is a process of "abstracting away" from the many toward the few.

In that light, the real domain $\mathbb{R}$ appears more concrete and more special while the boolean domain $\mathbb{B} = \lbrace 0, 1 \rbrace$ appears more abstract and more general.