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# Exercise 91 - Chapter 2

edited June 2018

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?

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.