I'm basically just going to copy Matthew Doty's puzzles but with lexicographical order:

**Definiton.** The **lexicographical order** of two posets \\((A,\leq_A)\\) and \\((B,\leq_B)\\) is \\((A \times B, \leq^{lex})\\) where

$$
(a_1,b_1) \leq^{lex} (a_2,b_2) \Longleftrightarrow a_1 \lt_A a_2 \text{ or } (a_1 =_A a_2 \text{ and } b_1 \leq_B b_2)
$$

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

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

**AV Puzzle 1**: Show that \\(\Delta\\) is monotonically increasing on \\(\leq^{lex}\\)

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

$$
\Delta(x) \leq^{lex} (y,z) \Longleftrightarrow x \leq_{A} r(y,z)
$$

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

$$
l(x,y) \leq_{A} z \Longleftrightarrow (x,y) \leq^{lex} \Delta(z)
$$

I think there are solutions but I could be wrong.