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Exercise 91 - Chapter 2

Let \(S\) be a set and recall the powerset monoidal preorder \( ( \mathbb{P}(S), \subseteq, S, \cap ) \) from Exercise 2.32 . Is it a quantale?

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  • 1.

    As exercise 2.32 shows that the powerset preorder is a symmetric monoidal preorder, it suffices to check whether it has all joins.

    Let \(A\subseteq\mathbb{P}(S)\). \(\vee A=\cup A\). The existence of \(\cup A\) is guaranteed by the Axiom of Union. So, the only thing to check is that \(\cup A\subseteq S\). Let \(x\in\cup A\). Then (by the Axiom of Union) there is a set \(D\in A\) such that \(x\in D\). For every \(D'\in A\), \(D'\subseteq S\), so in particular, \(D\subseteq S\). Thus, \(x\in D\Rightarrow x\in S\) (since \(x\) was arbitrary, this holds for every \(x\in\cup A\)). So, \(\cup A\subseteq S\), as was needed to prove that the powerset symmetric monoidal preorder is a quantale.

    Comment Source:As exercise 2.32 shows that the powerset preorder is a symmetric monoidal preorder, it suffices to check whether it has all joins. Let \\(A\subseteq\mathbb{P}(S)\\). \\(\vee A=\cup A\\). The existence of \\(\cup A\\) is guaranteed by the [Axiom of Union](https://en.wikipedia.org/wiki/Axiom_of_union). So, the only thing to check is that \\(\cup A\subseteq S\\). Let \\(x\in\cup A\\). Then (by the Axiom of Union) there is a set \\(D\in A\\) such that \\(x\in D\\). For every \\(D'\in A\\), \\(D'\subseteq S\\), so in particular, \\(D\subseteq S\\). Thus, \\(x\in D\Rightarrow x\in S\\) (since \\(x\\) was arbitrary, this holds for every \\(x\in\cup A\\)). So, \\(\cup A\subseteq S\\), as was needed to prove that the powerset symmetric monoidal preorder is a quantale.
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