Suppose certain input data (or signals) have to flow from the source (or sender) $A$ to the target (or receiver) $B$ through a node (or station) $S$. Suppose that at an instant $t$, $S$ being a set of inputs from $A$ forms a semiring under two binary operations $(+)$ and $(\cdot)$ that is $(S, +, \cdot, 0, 1)$ is a semiring. Now, my question is what might be a possible advantage of such data (signals) forming an algebraic structure (particularly, a semiring) to get a desired output in $B$? Or, what are the rules of semirings that can be used to process the data in $S$? Any good references in the related works if possible would be highly helpful.
Edit: Suppose, an organization $P$ (source) wishes to pass a resolution based on $n$ number of criterion, which is to be implemented at the target $Q$. Each participant of the organization gives his/her opinions based on the given $n$ criterion. Consider that the decision made by any individual of the organization forms a graph (for simplicity, undirected simple graphs) on the set of vertex of $n$ criterion (nodes). Each edge being a certain relationship/dependecies between two criterion (defined/ associated by the individual). In this scenario, the decision of each participant will be a simple undirected graph namely, $g_1, g_2,..., g_n$ on the same set of vertices (i.e., $n $) vertices. The problem starts as on how to accept the decisions made by each individual for further consideration giving equal importance to each decision. Assuming that the head of the organization is obliged to give a uniform justification to each participant by accepting their decision fairly. The onus is on the head to make the final resolution anonymously and make it conceded by all. One way of doing it to take the union of all the graphs in which identical vertices or edges from any two graphs are merged by the union operator. The resulting graphs union will be a simple undirected graph $G$ with $n$ vertices (it may be a complete graph at the most). Further, the target $Q$ will randomly choose a decision for implementation at a time from number of decisions. The target $Q$ is designed in such a way that it may take a decision with a single criteria at a time or a decision with all criterion at a time, and so forth, then the number of possibilities of accepting a decision at $Q$ is equal to the number of sub graphs of $G$. Henec, decompose the graph $G$ into all possible sub graphs. Let $S$ be the collection of all such sub graphs. Then we notice that $S$ equipped with the graph operations viz. union $\cup$ and intersection $\cap$ forms a commutative and idempptent semiring with additive identity being an empty graph and multiplicative identity is the whole graph $G$.
How should the output at $Q$ be defined so that some rule of semiring can be applied to get a desired result?