Or to be more precise,
Is there anything behind the “$m + N$” notation for elements in the factor structure $M/N$ other than a hint at the set-theoretical realization?,
for example, if $M$ and $N$ are abelian groups.
Why I’m asking: I try not to think of composed structures in terms of their set-theoretic realization, but maybe there are other, more structural reasons to maintain this notation instead of any other (like $[m]_N$ for instance).
Also, do you know any other alternative notations for elements in the factor structure?