**Puzzle 75.**

>Suppose \\( (Y, \le_Y) \\) is a preorder, \\(X\\) is a set and \\(f : X \to Y\\) is any function. Define a relation \\(\le_X\\) on \\(X\\) by

>\[ x \le_X x' \textrm{ if and only if } f(x) \le_Y f(x') .\]

>Show that \\( (X, \le_X ) \\) is a preorder.

Our relation turns \\( f \\) into a *monotone map*. For all \\( x \in X \\) we have \\( f(x) \le_Y f(x)\\) and thus \\( x \le_X x\\) satisfying reflexivity. Similarly, for all \\( x,y,z \in X \\), \\( f(x) \le_Y f(y)\\) and \\( f(y) \le_Y f(z)\\) implies \\( f(x) \le_Y f(z)\\) and thus \\( x \le_X y\\) and \\( y \le_X z\\) implies \\( x \le_X z\\) satisfying transitivity. This gives us a preorder on X.

**Puzzle 76.**

>Now suppose \\( (Y, \le_Y) \\) is a poset. Under what conditions on \\(f\\) can we conclude that \\( (X, \le_X ) \\) defined as above is a poset?

Since we don't want to induce any equivalent elements in \\(X\\), \\( f \\) must be injective.

**Puzzle 77.**

>Now suppose that \\( (Y, \le_Y, \otimes_Y, 1_Y) \\) is a monoidal preorder, and \\( (X,\otimes_X,1_X ) \\) is a monoid. Define \\(\le_X\\) as above. Under what conditions on \\(f\\) can we conclude that \\( (X,\le_X\otimes_X,1_X) \\) is a monoidal preorder?

We need to assure that our induced preorder structure is compatible with our monoidal structure. To this end we require our *monotone map* \\( f \\) to be a *monoidal monotone* for which \\( 1_Y \le_Y f(1_X) \\) and \\( f(x) \otimes_Y f(y) \le_Y f(x \otimes_X y) \\)

Regarding Puzzle 71, does this mean we simply need to find injective *monoidal monotones* to other commutative monoidal posets (e.g \\( (\mathbb{R}, \le, +, 0 )\\)) or do we need stricter requirements to preserve the commutative sturcture (e.g \\( 1_Y = f(1_X) \\) and \\( f(x) \otimes_Y f(y) = f(x \otimes_X y) \\))?

I'm off to bed, so maybe someone else can continue my thought process...

>Suppose \\( (Y, \le_Y) \\) is a preorder, \\(X\\) is a set and \\(f : X \to Y\\) is any function. Define a relation \\(\le_X\\) on \\(X\\) by

>\[ x \le_X x' \textrm{ if and only if } f(x) \le_Y f(x') .\]

>Show that \\( (X, \le_X ) \\) is a preorder.

Our relation turns \\( f \\) into a *monotone map*. For all \\( x \in X \\) we have \\( f(x) \le_Y f(x)\\) and thus \\( x \le_X x\\) satisfying reflexivity. Similarly, for all \\( x,y,z \in X \\), \\( f(x) \le_Y f(y)\\) and \\( f(y) \le_Y f(z)\\) implies \\( f(x) \le_Y f(z)\\) and thus \\( x \le_X y\\) and \\( y \le_X z\\) implies \\( x \le_X z\\) satisfying transitivity. This gives us a preorder on X.

**Puzzle 76.**

>Now suppose \\( (Y, \le_Y) \\) is a poset. Under what conditions on \\(f\\) can we conclude that \\( (X, \le_X ) \\) defined as above is a poset?

Since we don't want to induce any equivalent elements in \\(X\\), \\( f \\) must be injective.

**Puzzle 77.**

>Now suppose that \\( (Y, \le_Y, \otimes_Y, 1_Y) \\) is a monoidal preorder, and \\( (X,\otimes_X,1_X ) \\) is a monoid. Define \\(\le_X\\) as above. Under what conditions on \\(f\\) can we conclude that \\( (X,\le_X\otimes_X,1_X) \\) is a monoidal preorder?

We need to assure that our induced preorder structure is compatible with our monoidal structure. To this end we require our *monotone map* \\( f \\) to be a *monoidal monotone* for which \\( 1_Y \le_Y f(1_X) \\) and \\( f(x) \otimes_Y f(y) \le_Y f(x \otimes_X y) \\)

Regarding Puzzle 71, does this mean we simply need to find injective *monoidal monotones* to other commutative monoidal posets (e.g \\( (\mathbb{R}, \le, +, 0 )\\)) or do we need stricter requirements to preserve the commutative sturcture (e.g \\( 1_Y = f(1_X) \\) and \\( f(x) \otimes_Y f(y) = f(x \otimes_X y) \\))?

I'm off to bed, so maybe someone else can continue my thought process...