Exercise 2-45

  1. Is \((\mathbb{N},\leq,1,*)\) a symmetric monoidal preorder?

  2. If so, does there exist a monoidal monotone \((\mathbb{N},\leq,0,+) \rightarrow (\mathbb{N},\leq,1,*)\)

  3. Is \((\mathbb{Z},\leq,*,1)\) a symmetric monoidal preorder?

Solution(1)

  1. Yes. Monotonicity holds, and multiplication by 1 is unital. The operator is symmetric and associative.

  2. Exponentiation (say, by \(2\)) is a strict monoidal monotone.

    • \(1 = 2^0\) and \(2^x * 2^y = 2^{x+y}\)

  3. No: monotonicity does not hold (multiplying negative numbers gives a larger number).