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 343
- Applied Category Theory Exercises 149
- Applied Category Theory Discussion Groups 48
- Applied Category Theory Formula Examples 15
- Chat 475
- 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 708
- - Latest Changes 700
- - - 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

Consider the preorder \( (P, \le ) \) with Hasse diagram \( \Big[ \text{ no } \le \text{ maybe } \le \text{ yes } \Big] \) .

We propose a monoidal structure with "yes" as the monoidal unit and "min" as the monoidal product.

1) Make sense of "min" by filling in the multiplication table with elements of P.

2) Check the axioms of Definition 2.2 hold for \( \mathbf{NMY} := (P, \le, \text{yes}, \text{min} ) \), given your definition of "min".

$$ \begin{array}{c|ccc} \text{min} & no & maybe & yes \\ \hline no & ? & ? & ? \\ maybe & ? & ? & ? \\ yes & ? & ? & ? \end{array} $$ Previous Next

## Comments

The multiplication table is

$$ \begin{array}{c|ccc} \text{min} & no & maybe & yes \\ \hline no & no & no & no \\ maybe & no & maybe & maybe \\ yes & no & maybe & yes \end{array} $$ The "min" operation is associative and commutative, so properties (c) and (d) of Definition 2.1 hold.

For any element \(x \in P, x \otimes yes = yes \otimes x = \min{(x, yes)} = x\), so the unit laws (b) also hold.

Suppose \(x_1\leq y_1, x_2\leq y_2\). Since \(P\) is a chain, we must have \(x_1\leq x_2\) and/or \(x_2\leq x_1\); consider the former case (the other case is dealt with similarly).

We need to show that \(x_1\leq y_1 \otimes y_2 =\min{(y_1, y_2)}\). This is true by assumption if \(y_1 =\min{(y_1, y_2)}\), so consider the case \(y_2 =\min{(y_1, y_2)}\).

We then have either \(x_1 \leq y_2\) or \(x_2 \leq y_2 < x_1 \leq y_1\). If the latter were true, then transitivity would imply \(x_2 < x_1 \) contrary to the assumption that \(x_1\leq x_2\). So we must have \(x_1 \leq y_2\). This establishes property (a) of Definition 2.1 and thus concludes the proof.

`The multiplication table is $$ \begin{array}{c|ccc} \text{min} & no & maybe & yes \\\\ \hline no & no & no & no \\\\ maybe & no & maybe & maybe \\\\ yes & no & maybe & yes \end{array} $$ The "min" operation is associative and commutative, so properties (c) and (d) of Definition 2.1 hold. For any element \\(x \in P, x \otimes yes = yes \otimes x = \min{(x, yes)} = x\\), so the unit laws (b) also hold. Suppose \\(x_1\leq y_1, x_2\leq y_2\\). Since \\(P\\) is a chain, we must have \\(x_1\leq x_2\\) and/or \\(x_2\leq x_1\\); consider the former case (the other case is dealt with similarly). We need to show that \\(x_1\leq y_1 \otimes y_2 =\min{(y_1, y_2)}\\). This is true by assumption if \\(y_1 =\min{(y_1, y_2)}\\), so consider the case \\(y_2 =\min{(y_1, y_2)}\\). We then have either \\(x_1 \leq y_2\\) or \\(x_2 \leq y_2 < x_1 \leq y_1\\). If the latter were true, then transitivity would imply \\(x_2 < x_1 \\) contrary to the assumption that \\(x_1\leq x_2\\). So we must have \\(x_1 \leq y_2\\). This establishes property (a) of Definition 2.1 and thus concludes the proof.`

It seems to be suggested that there's some logical connective correspondence for this

min(as in Example 2.10 and Exercise 2.11). Is that what question 2 means by "definition of min"? :-?`It seems to be suggested that there's some logical connective correspondence for this _min_ (as in Example 2.10 and Exercise 2.11). Is that what question 2 means by "definition of min"? :-?`