Is there a name for a set equipped with an infinite sequence of Boolean lattices?
What kinds of structure can you give to this class?
For example, is there a natural notion of a product?
Let's talk about the notion of an interpretation without parameters as an example. Somewhat surprisingly (to me), we can talk about whether a structure is interpretable in another structure without paying that much attention to its signature. We can "reduce" a first-order structure to its carrier and a sequence of Boolean lattices corresponding to its lattices of definable sets associated with formulas with varying numbers of free variables.
This notion of a set-equipped-with-a-sequence-of-lattices appears to have a natural notion of exponentiation by a positive integer.
However, this notion of exponentiation does not line up with a naive idea of a binary product.
So, that makes me wonder:
- Do these things have a name?
- Is there a natural structure we can equip their class with?
Suppose $M$ is a first-order structure. $M$ is naturally associated with a Boolean lattice $\Lambda(M) \subset 2^M$, its lattice of parameter-free definable sets (of $1$-tuples).
$M$ is also naturally associated with $\Xi(M)$, a sequence of Boolean algebras. Let $\Xi(M, i) = \Xi(M)_i$ be the lattice of parameter-free definable sets associated with well-formed formulas with exactly $i$ free variables where $i \ge 0$.
A $k$-interpretation $\mathfrak{I}$ from $M$ to $N$ can be phrased as follows.
- $\mathfrak{I}$ is a partial function from $M^k$ to $N$.
- $\mathfrak{I}$ is a surjection onto $N$.
- For all $X$ in $\Xi(N, 1)$, $\mathfrak{I}^{\leftarrow}(X)$ is in $\Xi(M, k)$.
This definition is in some sense nice because it lets us restrict attention, for a structure $M$, to $(M, \Xi(M))$, the pair of its carrier and its lattices of definable sets.
This definition also has the following property.
Let $\mathcal{M}$ be the pair $(M, \Xi(M))$. We can define $\mathcal{M}^k$ in the following way.
$\mathcal{M}^k = (M, \Xi(M))^k = (M^k, \{ \Xi(M, a) : a \in \mathbb{N} \land k | a \})$
With this definition, a $k$-interpretation from $\mathcal{M}$ to $\mathcal{N}$ is just a $1$-interpretation from $\mathcal{M}^k$ to $\mathcal{N}$.
Defining a binary product $\mathcal{A} \times \mathcal{B}$, though, componentwise in the most naive possible way give us:
$$ \mathcal{A} \times \mathcal{B} = (A, \Xi(A)) \times (B, \Xi(B)) = (A \times B, \Xi(A) \times \Xi(B)) $$
In the above, $\Xi(A) \times \Xi(B)$ is defined componentwise.
This produces too few definable sets in the case of $\mathcal{M} \times \mathcal{M}$ though. For example, it can't represent $ \{(x, y) : x = y\}$ because the product doesn't "know" that the two $\mathcal{M}$'s are the same.