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}

\]

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}

\]