I am trying to figure out what the correct notation for a multiplicative algebra should be. I've seen a bunch of weird and conflicting ways for writing these things in the literature, but they are all either too specialized or bring in extra baggage.
To be more specific, given a ring R and an R-module, M, with a finite basis B, we can identify the elements of M with functions of the form $f, g : B \to R$. Then we construct the multiplicative algebra on M according to the following rule:
$$ (f \, g)(x) = f(x) g(x) $$
My question is simple: What is the standard notation for this construction? I've seen a bunch of weird forms. For example, in the case where R is the field of complex numbers, we can write something like [; C^k(B) ;] for this algebra. Or if we are dealing with matrices, we can talk about Hadamard products.
But can we write something simpler? I would really just like to do something along the lines of say; define R(B) to be this multiplicative algebra and be done with it. The reason for this is that if we are working with something like Pontryagin duality or representation theory, the set B may be a group/monoid/whatever, and there could be multiple algebras we defined over it.
To illustrate why this is an issue, let $B = \mathbb R$ be the group of reals under addition, then we can define both a convolution algebra, $R[ \mathbb R ]$ and a multiplicative algebra $ R( \mathbb R )$ which are identified with the same R-module, but have totally different structures. (Though they are related by Pontryagin duality in the case where $R = \mathbb C$.)