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Consider the preorder \( W := ( \mathbb{N} \cup {\infty}, \le, \infty, min) \).

Draw a small graph labeled by elements of \(W\) and compute the corresponding distance table. This will give you a feel for how \(W\) works.

Make up a interpretation, like that in Exercise 2.59, for how to imagine enrichment in \(W\).

## Comments

2 - Enrichment in \( W \) could capture some sort of notion of throughput.. If we imagine it as plumbing, given a water pipe \( \mathcal{X}(a, b) \) and a water pipe \( \mathcal{X}(b, c) \) the water pipe \( \mathcal{X}(a, c) \)'s throughput is given by \( min \), i.e. it's bottlenecked by the amount of water that can pass through the smallest pipe. I do feel that \( W^{op} \) would capture that notion a bit better; throughput should intuitively be

at mostthe minimum of those two pipes.`2 - Enrichment in \\( W \\) could capture some sort of notion of throughput.. If we imagine it as plumbing, given a water pipe \\( \mathcal{X}(a, b) \\) and a water pipe \\( \mathcal{X}(b, c) \\) the water pipe \\( \mathcal{X}(a, c) \\)'s throughput is given by \\( min \\), i.e. it's bottlenecked by the amount of water that can pass through the smallest pipe. I do feel that \\( W^{op} \\) would capture that notion a bit better; throughput should intuitively be _at most_ the minimum of those two pipes.`