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## Comments

Definition 2.87A

unital commutative quantaleis a symmetric monoidal closed preorder \(\mathcal{V}=(V,\leq,I,\otimes)\) that has all joins: \(\vee A\) exists for every \(A\subseteq V\). In particular, we denote the empty join \(0:=\vee\emptyset\).In exercise 81 it was shown that

Boolis monoidal closed. Ex. 2.24 shows that it is symmetric monoidal (and thus symmetric). So, it remains to show thatBoolhas all joins. Let \(A\subseteq\mathbb{B}\), then \(A\in\{\emptyset,\{\mathrm{true}\},\{\mathrm{false}\},\{\mathrm{true},\mathrm{false}\}\}\). The joins of these sets are \(\mathrm{false},\mathrm{true},\mathrm{false}\), and \(\mathrm{true}\), respectively. Since these joins all exist and lie in \(\mathbb{B}\),Boolis a quantale.`**Definition 2.87** A _unital commutative quantale_ is a symmetric monoidal closed preorder \\(\mathcal{V}=(V,\leq,I,\otimes)\\) that has all joins: \\(\vee A\\) exists for every \\(A\subseteq V\\). In particular, we denote the empty join \\(0:=\vee\emptyset\\). In exercise 81 it was shown that **Bool** is monoidal closed. Ex. 2.24 shows that it is symmetric monoidal (and thus symmetric). So, it remains to show that **Bool** has all joins. Let \\(A\subseteq\mathbb{B}\\), then \\(A\in\\{\emptyset,\\{\mathrm{true}\\},\\{\mathrm{false}\\},\\{\mathrm{true},\mathrm{false}\\}\\}\\). The joins of these sets are \\(\mathrm{false},\mathrm{true},\mathrm{false}\\), and \\(\mathrm{true}\\), respectively. Since these joins all exist and lie in \\(\mathbb{B}\\), **Bool** is a quantale.`