Suppose we had a finite group G with elements $e_0, e_1 ... e_k$
Then consider objects from the set
$$ M = { a_0 e_0 + a_1 e_1 + a_2 e_2 ... a_n e_k }, a_i \in \Bbb{R}$$
whereas
$$ m + n, (m,n \in M) $$
Is just summation component wise for each element in the group. And
$$ m*n = m_0 n_0 e_0^2 + m_0 n_1 e_0 e_1 + m_0 n_2 e_0 e_2 ... $$
And each expression $e_i e_j$ is simplified using the underlying group structure.
A concrete example being the famous Quaternions:
How does one simplify infinite sums over such structures.
An example is over $\Bbb{C}$ it is challenging to resolve:
$$ \sum_{k=0}^{\infty}{\frac{1}{k^2}} = \frac{\pi^2 }{6} $$
But what if we consider the following over $\Bbb{H}$ (the Quaternions)
$$ \sum_{d=0}^{\infty}{\frac{1}{(d-j)(d+k)}} $$
It's not clear how to do this, as most of the conventional tools such as the trigonometric functions etc... don't seem to be well defined. Yet I could algorithmically find sequences of elements in the Quaternions that converge to whatever that sum comes down to.