I'm going to suggest an approach to transitive closures which will yield the usual definition, in the special case of a 'simple' digraph having an adjacency matrix with entries in {0,1}. Namely: the transitive closure can be regarded as a maximum value over weights of walks, where we regard multi-arcs as arcs with greater weight. We can then formulate two different definitions according to how you would like to define the weight of a walk.

Throughout, *A(D)* denotes the adjacency matrix of a (multi-)digraph *D*.

**A. 'Max-min' approach**

If you would like the weight of a walk to be the minimum weight of any edge (according to the principle of a chain being only as strong as the weakest link), you would then define

$\begin{align}
A(T)\_{a,b} \;\;=\quad \max_{\ell \in \mathbb N} \; \max_{\substack{v \in V(D)^{\ell+1} \\\\ (v_0, v_{\ell+1}) = (a,b)}} \min_{0 \le j \le \ell}\; \;\Bigl[ A(D)\_{v_j,v_{j+1}} \Bigr]\;,
\end{align}$

fairly straightforwardly.

**B. 'Combinatoric' approach**

If you would rather concieve of walk-weifghts as a product of the constituent arc-weights, as happens in sum-over-paths descriptions of probabilistic processes, you should rather define
$\begin{align}
A(T)\_{a,b} \;\;=\quad \sup_{\ell \in \mathbb N} \; \max_{\substack{v \in V(D)^{\ell+1} \\\\ (v_0, v_{\ell+1}) = (a,b)}} \;\; \prod_{j=0}^\ell \Bigl[ A(D)\_{v_j,v_{j+1}} \Bigr] \;,
\end{align}$

where the supremum may be replaced by a maximum in the case where (as with probabilistic mixing) all arc-weights are between −1 and 1, if the network does not contain any directed cycles, or similar conditions.

(If your digraph is not acyclic, and some vertex-pairs in some strongly-connected component contains multiple arcs between them, and you don't like the idea of digraphs with countably infinitely many arcs between vertices, you may wish to replace the maximum over tuples with a maximum over non-repeating sequences, in which case the supremum also becomes a maximum. This corresponds to taking maximum wieghts of paths, rather than walks.)