It looks like you're new here. If you want to get involved, click one of these buttons!

- All Categories 2.2K
- Applied Category Theory Course 338
- Applied Category Theory Exercises 149
- Applied Category Theory Discussion Groups 48
- Applied Category Theory Formula Examples 15
- Chat 470
- Azimuth Code Project 107
- News and Information 145
- Azimuth Blog 148
- Azimuth Forum 29
- Azimuth Project 190
- - Strategy 109
- - Conventions and Policies 21
- - Questions 43
- Azimuth Wiki 707
- - Latest Changes 699
- - - Action 14
- - - Biodiversity 8
- - - Books 2
- - - Carbon 9
- - - Computational methods 38
- - - Climate 53
- - - Earth science 23
- - - Ecology 43
- - - Energy 29
- - - Experiments 30
- - - Geoengineering 0
- - - Mathematical methods 69
- - - Meta 9
- - - Methodology 16
- - - Natural resources 7
- - - Oceans 4
- - - Organizations 34
- - - People 6
- - - Publishing 4
- - - Reports 3
- - - Software 20
- - - Statistical methods 2
- - - Sustainability 4
- - - Things to do 2
- - - Visualisation 1
- General 38

Options

## 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.`