Summary

In case you couldn’t be bothered to read the whole thing, here’s a concise summary.

Definitions

The set of prods \(\mathbb{\Pi}\) is defined as:

1. \(0 \in \mathbb{\Pi}\)

2. \(x_1, ..., x_n \in \mathbb{\Pi} \implies [x_1, ..., x_n] \in \mathbb{\Pi}\)

3. \([x_1, ..., x_n, 0] = [x_1, ..., x_n]\)

By convention \([]:= [0] = [0, 0, ...]\) and prods are automatically padded to the same length for comparisons.

You can interpret any prod as a natural number by:

\[\begin{align*} I(0) &= 0 \\ I([x_1, ..., x_n]) &= \prod_{i=1}^n p_i^{I(x_i)} \text{ , where } p_i \text{ is the ith prime} \end{align*}\]

Trees are useful for visualizing prods.

The operation \(\sqcup: \mathbb{\Pi} \times \mathbb{\Pi} \to \mathbb{\Pi}\), called graft, is defined as:

\[\begin{align*} 0 \sqcup x &= x = x \sqcup 0 \\ [x_1, ..., x_n] \sqcup [y_1, ..., y_n] &= [x_1 \sqcup y_1, ..., x_n \sqcup y_n] \end{align*}\]

The dual operation \(\sqcap: \mathbb{\Pi} \times \mathbb{\Pi} \to \mathbb{\Pi}\), called prune, is defined similarly as:

\[\begin{align*} 0 \sqcap x &= 0 = x \sqcap 0 \\ [x_1, ..., x_n] \sqcap [y_1, ..., y_n] &= [x_1 \sqcap y_1, ..., x_n \sqcap y_n] \end{align*}\]

Results

1. \(I: \mathbb{\Pi} \to \mathbb{N}\) is a bijection (Theorem 5).

2. \(\sqcap\) induces a partial order equivalent to \(x \sqsubseteq y \iff x = 0 \lor (x_1 \sqsubseteq y_1 \land ... \land x_n \sqsubseteq y_n)\) (this section)

3. \(x \sqsubseteq y \implies I(x) | I(y)\), but not vice-versa (Theorem 6)

4. \(\mathbb{\Pi}, \sqcup, \sqcap\) form a distributive lattice (this section).

5. For any \(x \in \mathbb{\Pi}\), the downset \(\downarrow x := \{y | y \sqsubseteq x\}\) is a Heyting algebra (Theorem 9).

6. \(\downarrow x - \{0\}\) is a Boolean algebra iff \(x\)’s only components are \(0, []\) (Theorem 10). The condition is analogous to square-free integers.