On Boolean algebras in pointfree topology
The context:
What are Boolean algebras?
Well, you know this. A Boolean algebra is a bounded distributive lattice in which every element has a complement, the power set of a set being a particularly straightforward example.
What are frames?
A frame is a complete lattice (that is, a partially ordered set in which all subsets have joins/suprema and meets/infima) in which finite meet distributes over arbitrary join. In particular, since we regard the empty set as finite, a frame has a bottom element and a top element. Our primary example consists of the open sets of a topological space: these form a frame, with arbitrary join being union and finite meet being intersection. The bottom element is the empty set and the top element is the entire underlying set of the space. Apologies: I mention this because the word “frame” is one of those enormously useful English words that gets used to mean many different things, even in mathematics, and I wanted to avoid confusion with other uses. The term “pointfree topology” describes our area of interest pretty well. The idea is that, when you look at a topological space, what you can see are the open sets, these are the fundamental objects of interest, not the points in the underlying set.
What are partial frames?
Our specific context will be that part of pointfree topology dealing with partial frames. Partial frames are meet-semilattices where, in contrast with frames, not all subsets need have joins. A selection function, \(\mathcal{S}\), specifies, for all meet-semilattices, certain subsets under consideration, which we call the “designated” ones; an \(\mathcal{S}\)-frame then must have joins of (at least) all such subsets and binary meet must distribute over these. A small collection of axioms suffices to specify our selection functions; these axioms are sufficiently general to include as examples of partial frames, bounded distributive lattices, σ-frames, κ-frames and frames.
The specifics:
In the theory of (full) frames, one can factor out by congruences, to form onto morphisms, or one can use certain closure operators called nuclei. We explain why, for partial frames, the former are an appropriate tool, the latter not.
Given a full frame \(L\), one can form a least dense quotient of \(L\). Here a link with Booleanness arises: the codomain is Boolean, and this is, in fact, the unique dense Boolean quotient of \(L\). We discuss what does and does not work for partial frames, and what this tells us about Booleanness.
Given a partial frame, we will consider two associated full frames:
- its free frame
- its congruence frame We relate these three objects to each other and, by looking at appropriate universal maps, see where, surprisingly, Booleanness pops up again.