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# Exercise 4 - Chapter 4

edited June 2018

Suppose we have the preorders $$\mathcal{X} := \left[ \text{monoid} \rightarrow \text{category} \leftarrow \text{preorder} \right]$$ $$\mathcal{Y} := \left[ \text{nothing} \rightarrow \text{this book} \right]$$

1. Draw the Hasse diagram for the preorder $$\mathcal{X}^{op} \times \mathcal{Y}$$.
2. Write down a profunctor $$\Lambda : \mathcal{X} \nrightarrow \mathcal{Y}$$ and, reading $$\Lambda(x, y) = \text{true}$$ as “my uncle can explain $$x$$ given $$y$$”, give an interpretation of the fact that the preimage of $$\text{true}$$ forms an upper set in $$\mathcal{X}^{op} \times \mathcal{Y}$$.

Definition 4.2

Let $$\mathcal{X} = (X, \le_X ) \text{ and } \mathcal{Y} = (Y, \le_Y )$$ be preorders. A feasibility relation for $$\mathcal{X}$$ given $$\mathcal{Y}$$ is a monotone map $$\tag{4.3} \Phi : \mathcal{X}^{op} \times \mathcal{Y} \rightarrow \textbf{Bool}$$ We denote this by $$\Phi : \mathcal{X} \nrightarrow \mathcal{Y}$$. Given $$x \in X \text{ and } y \in Y$$, if $$\Phi(x, y) = \text{true}$$ we say $$x$$ can be obtained given $$y$$.

• Options
1.

1) Draw the Hasse diagram for the preorder $$\mathcal{X}^{op} \times \mathcal{Y}$$.

Comment Source:> 1) Draw the Hasse diagram for the preorder \$$\mathcal{X}^{op} \times \mathcal{Y} \$$. ![drawing](https://docs.google.com/drawings/d/e/2PACX-1vRRo3ojmQS2vWow_yZiqlqYlLGFYJpgvHtLSy57lCsr7V2WwPFR-dzhP63T88i7TzqR03iuIT42RZFo/pub?w=365&h=251) 
• Options
2.
edited June 2018

2) Write down a profunctor $$\Lambda : \mathcal{X} \nrightarrow \mathcal{Y}$$ and, reading $$\Lambda(x, y) = \text{true}$$ as “my uncle can explain $$x$$ given $$y$$”, give an interpretation of the fact that the preimage of $$\text{true}$$ forms an upper set in $$\mathcal{X}^{op} \times \mathcal{Y}$$.

I am not entirely sure what it means to "write down a profunctor" but I will make a guess.

$$\begin{array}{c c | c} x & y & p \\ \text{monoid} & \text{this book} & \text{true} \\ \text{monoid} & \text{nothing} & \text{false} \\ \text{preorder} & \text{this book} & \text{true} \\ \text{preorder} & \text{nothing} & \text{false} \\ \text{category} & \text{this book} & \text{true} \\ \text{category} & \text{nothing} & \text{false} \end{array}$$ My uncle can explain categories, monoids and preorders if he has studied the book.

As will be seen later this can also be expressed as...

$$\begin{array}{c c | c} \Lambda & \text{this book} & \text{nothing} \\ \text{monoid} & \text{true} & \text{false} \\ \text{preorder} & \text{true} & \text{false} \\ \text{category} & \text{true} & \text{false} \end{array}$$ Another way to think about this is to convert the $$\le$$ into "better". On the consumer side, consuming less it "better", while on the producing side more is "better". Thus consuming less and/or producing more are both "better". Converting the consuming side means $$X_{op}$$.

Comment Source:> 2) Write down a profunctor \$$\Lambda : \mathcal{X} \nrightarrow \mathcal{Y} \$$ and, reading \$$\Lambda(x, y) = \text{true} \$$ as “my uncle can explain \$$x\$$ given \$$y\$$”, give an interpretation of the fact that the preimage of \$$\text{true} \$$ forms an upper set in \$$\mathcal{X}^{op} \times \mathcal{Y} \$$. I am not entirely sure what it means to "write down a profunctor" but I will make a guess. $\begin{array}{c c | c} x & y & p \\\\ \text{monoid} & \text{this book} & \text{true} \\\\ \text{monoid} & \text{nothing} & \text{false} \\\\ \text{preorder} & \text{this book} & \text{true} \\\\ \text{preorder} & \text{nothing} & \text{false} \\\\ \text{category} & \text{this book} & \text{true} \\\\ \text{category} & \text{nothing} & \text{false} \end{array}$ My uncle can explain categories, monoids and preorders if he has studied the book. As will be seen later this can also be expressed as... ![fig1](https://docs.google.com/drawings/d/e/2PACX-1vTx8gGPnMistT2dtZpDf5LzQKCARUbEWNSqzrRb0OcAT3syCH5pjoGUnzVMnEPFdc81JGqPRmuWU4O6/pub?w=483&h=191) $\begin{array}{c c | c} \Lambda & \text{this book} & \text{nothing} \\\\ \text{monoid} & \text{true} & \text{false} \\\\ \text{preorder} & \text{true} & \text{false} \\\\ \text{category} & \text{true} & \text{false} \end{array}$ Another way to think about this is to convert the \$$\le \$$ into "better". On the consumer side, consuming less it "better", while on the producing side more is "better". Thus consuming less and/or producing more are both "better". Converting the consuming side means \$$X_{op} \$$.