Mathematical symbol to show one mathematical structure represents another

Question

Is there a particular symbol that can be used to show that one mathematical structure is a representation of another?

For example, I want to state that the relation $R$ is represented by the connection matrix $M_{R}$. Is there a symbol I can use to communicate this in a concise manner? For instance, if the symbol were #, then I would say $R$ # $M_{R}$. Or does it simply suffice to say $R = M_{R}$ or $R \Leftrightarrow M_{R}$, although I don't think these statements are completely accurate.

Answer 1

The short answer is no: there is no standard symbol for this.

The long answer is that you could use a symbol (such as $M_R \sim R$ as suggested by Torsten in the comments), or introduce some other kind of notation such as $[R] = M_R$, but if you wanted to do this then it would be a good idea to specify explicitly what the notation means. For example:

In what follows, we will write $M \sim R$ to denote the assertion that the relation $R$ is represented by the connection matrix $M$.

Answer 2

There is no universal such symbol as far as I know - and in my opinion, there shouldn't be: there are many different ways one object can "represent" another, and we should distinguish between them. I think being overly concise can be a negative here.

Remember also that you can always introduce new terminology/notation in a paper, e.g. "When we have $aRb$ iff the $\langle a,b\rangle$-entry of $M$ is $1$, we write "$R\sim M$.""

Answer 3

I too know of no convention for this. I'd suggest that depending on the nature of the relationship between the object represented and the object doing the representing, another good symbol might be—or be based on—the isomorphism symbol $\cong$.