A matrix with entries in \(\mathcal{V}\), where \(\mathcal{V}=(V, \leq, \otimes, I)\) is a quantale. (A \(\mathcal{V}\) matrix). Matrix multiplication between \(\mathcal{V}\) matrices
Need two sets, and function \(X \times Y \xrightarrow{M} V\). Call \(M(x,y)\) the (x,y)th entry.
We can multiply \(X \times Y \xrightarrow{M} V\) and \(Y \times Z \xrightarrow{N} V\) to get a new \(\mathcal{V}\) matrix \(X \times Z \xrightarrow{M*N} V\).
\((M*N)(x,z)(x,z)\) defined as \(\bigvee_y\ M(x,y)\otimes N(y,z)\)
Note that this is similar to the standard matrix multiplication, with \(\bigvee \mapsto \Sigma, \otimes \mapsto *\)*