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Exercise 28 - Chapter 2

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?

The symmetric monoidal preorder definition.

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Comments

  • 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.
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