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.

ie \\(\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:

ie \\(\text{false}\wedge x = \text{false}\\) for all \\(x\in\textbf{Bool}\\) and \\(\infty + x = \infty\\) for all \\(x\in\textbf{Cost}\\)

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

ie \\(\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:

ie \\(\text{false}\wedge x = \text{false}\\) for all \\(x\in\textbf{Bool}\\) and \\(\infty + x = \infty\\) for all \\(x\in\textbf{Cost}\\)