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Exercise 60 - Chapter 2

edited June 2018

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

1. 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.

2. Make up a interpretation, like that in Exercise 2.59, for how to imagine enrichment in $$W$$.

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.
Comment Source: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.