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# Exercise 28 - Chapter 2

edited June 2018

Show there is a monoidal structure on $$( \mathbb{N} , \le )$$ where the monoidal product is $$*$$, i.e. $$6 * 4 = 24$$. What should the monoidal unit be?

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1.
edited June 2018

Monoidal unit is 1.

The $$*$$ operation is associative and commutative, so it satisfies properties (c) and (d).

Multiplication of any $$x \in \mathbb{N}$$ by 1 yields back $$x$$, so (b) holds as well.

Multiplication by any $$z \in \mathbb{N}$$ in $$(\mathbb{N}, \leq)$$ is monotone, i.e. it holds that for any $$x, y \in \mathbb{N}$$, if $$x \leq y$$ then $$x * z \leq y * z$$, so (a) is satisfied as well and $$(\mathbb{N}, \leq, 1, *)$$ is a symmetric monoidal preorder.

Comment Source:Monoidal unit is 1. The \$$* \$$ operation is associative and commutative, so it satisfies properties (c) and (d). Multiplication of any \$$x \in \mathbb{N} \$$ by 1 yields back \$$x\$$, so (b) holds as well. Multiplication by any \$$z \in \mathbb{N}\$$ in \$$(\mathbb{N}, \leq) \$$ is monotone, i.e. it holds that for any \$$x, y \in \mathbb{N} \$$, if \$$x \leq y \$$ then \$$x * z \leq y * z \$$, so (a) is satisfied as well and \$$(\mathbb{N}, \leq, 1, *) \$$ is a symmetric monoidal preorder.