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.

> 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.