> 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:

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

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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?