Anindya wrote:

> I can see two ways in which they "play the same role":

>• \$$\text{false}\in\textbf{Bool}\$$ and \$$\infty\in\textbf{Cost}\$$ are both bottom elements with respect to the partial order.

> i.e. \$$\text{false}\leq x\$$ for all \$$x\in\textbf{Bool}\$$ and \$$\infty\leq x\$$ for all \$$x\in\textbf{Cost}\$$

>• they are both zero elements with respect to the monoidal product:

> i.e. \$$\text{false}\wedge x = \text{false}\$$ for all \$$x\in\textbf{Bool}\$$ and \$$\infty + x = \infty\$$ for all \$$x\in\textbf{Cost}\$$

Great! I think the first one is more fundamental in what we're doing, but they're somehow connected.

By the way, instead of 'zero element' I prefer to say 'absorbing element': an [**absorbing element**](https://en.wikipedia.org/wiki/Absorbing_element) \$$x\$$ for a binary operation \$$\cdot\$$ is one with \$$x\cdot a = x = a \cdot x\$$ for all \$$a\$$.

For example, zero is not an absorbing element for addition in \$$\mathbb{R}\$$ or \$$[0,\infty] \$$, but it makes me uncomfortable to say zero is not a zero element. 