> I'm really just getting ready to explain how they show up in logic: first in good old classical "subset logic", and then in the weird new "partition logic".

I am pretty excited to read what's next!

I wanted to a few puzzles I ran into a while ago related to these topics.

First, a some definitions...

**Definition.** A poset \\((A,\leq,\wedge,\vee)\\) with a join \\(\vee\\) and a meet \\(\wedge\\) is called a **lattice**. (Note: Lattices ***must*** obey the anti-symmetry law!)

**Definiton.** The **product poset** of two posets \\((A,\leq_A)\\) and \\((B,\leq_B)\\) is \\((A \times B, \leq_{A\times B})\\) where

$$
(a_1,b_1) \leq_{A\times B} (a_2,b_2) \Longleftrightarrow a_1 \leq_A a_2 \text{ and } b_1 \leq_B b_2
$$

**Definition.** Let \\((A,\leq)\\) be a poset. The **diagonal function** \\(\Delta : A \to A\times A\\) is defined:

$$
\Delta(a) := (a,a)
$$

---------------------------

Let \\(A\\) be a lattice.

**MD Puzzle 1**: Show that \\(\Delta\\) is monotonically increasing on \\(\leq_{A\times A}\\)

**MD Puzzle 2**: Find the *right adjoint* \\(r : A\times A \to A\\) to \\(\Delta\\) such that:

$$
\Delta(x) \leq_{A\times A} (y,z) \Longleftrightarrow x \leq_{A} r(y,z)
$$

**MD Puzzle 3**: Find the *left adjoint* \\(l : A\times A \to A\\) to \\(\Delta\\) such that:

$$
l(x,y) \leq_{A} z \Longleftrightarrow (x,y) \leq_{A\times A} \Delta(z)
$$

**MD Puzzle 4**: Consider \\(\mathbb{N}\\) under the partial ordering \\(\cdot\ |\ \cdot\\), where

$$
a\ |\ b \Longleftrightarrow a \text{ divides } b
$$

What are the adjoints \\(l\\) and \\(r\\) in this case?