I have been wondering how this might generalize to Lawvere metric spaces.

A \$$[0-\infty]\$$-feasibility relation \$$\Lambda : X \nrightarrow Y\$$ would still be between two posets. Remembering that order is reversed for Lawvere spaces, we would have the following two rules:

\begin{align} x' \leq x & \implies \Lambda(x',y) \leq \Lambda(x,y)\\\\ y \leq y' & \implies \Lambda(x,y') \leq \Lambda(x,y) \end{align}

My intuition here is that boolean feasibility relationships are a special case. Specifically \$$\text{true} = 0\$$ and \$$\text{false} = \infty\$$ because of the flipped order.

Now composition would be defined as

$(\Lambda\Omega)(x,z) = \inf_{y \in Y} \Lambda(x,y) + \Omega(y,z)$

We could alternatively write \$$(\Lambda\Omega)(x,z) = \bigwedge_{y \in Y} \Lambda(x,y) + \Omega(y,z)\$$. Again \$$\wedge\$$ and \$$\vee\$$ are swapped because order is reversed.

The identity I wrote for boolean feasibility should generalize as an identity for \$$[0-\infty]\$$-feasibility relations:

$\mathbf{1}_X(x,y) = \begin{cases} 0 & x\leq y \\\\ \infty & \text{otherwise} \end{cases}$