Formal sums

Generally, formal sums are purely formal, i.e. notational, invocations of infinite sums, without considering convergence &c. Specifically, a formal sum fcalc

with coëfficients in and exponents in may be identified with a function

The formal variable is then simply a bookkeeping device. In most applications, is a commutative monoid, in which case is a commutative monoid under pointwise addition, and in many cases the structure is richer.