Number of elements in a set that forms a semiring

Question

I wonder wether there is any possible rule to determine the number of elements in a set $ S$ under certain binary operations if it forms a semiring or, can we find the number of elements in a set using the rules of semiring? I think it is not possible, because the set of real numbers $\Bbb R$ is semiring under usual addition and multiplication, but its number of elements cannot be determined.
Motivation: We can see that if $G$ is a finite graph with $ m$ vertices and $n$ edges then, it is very difficult to find all the possible sub graphs of $G$, and let's denote the set of all possible sub graphs by $ P(G)$. But we see that $(P(G), \cup, \cap)$ is a semiring, where $G'\cup G''=(V'\cup V'', E'\cup E'')$. So, if the number elements in a set can be found by rules of semiring, then it will be a good idea to find the number of all possible sub graphs of $G$.