Is there an accepted notation for the monoid of linear polynomials (with addition as the operation) with coefficients from some ring R?
Like $2p+3$, where $p$ and the identity generate the monoid over the integers?
It's not $Z[p]$, since that would imply a ring where higher order monomials can be present.
And what about multiple indeterminates? Like $2a+3b$, but I don't allow $ab$ in the monoid, as multiplication is not defined.
The examples I've given would be groups if I allowed any integer in the exponent, but the case I'm interested in would only allow positive integers (and zero) in the indeterminate exponents, while any integer would be allowed in the non-indeterminate. So $p-50$ would be in the monoid, but $-p$ would not.
The application I have in mind is a monoid ring that allows linear polynomials in exponents, so I'm working with things like $x^{p-50}$, but I don't do $x^{p^2}$.