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} \$$.

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**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\$$.