Let $X$ be a countably infinite set. Let $\prod$ denote the free abelian semigroup generated by $X.$ Apparently $\prod$ consists of all the 'formal products' of the form $\prod_{P \in X} P^{a_P}$ there $a_P$ are non-negative integers, and 0 for all but finitely many $P.$ Why can't we call these elements as simply products. What distinguishes a product from a formal one?
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What would it mean for them to be products? When we talk about e.g. products of real numbers we're referring to an already existing operation. But here we haven't defined the operation yet; to define it requires a domain for it to be defined on, which is what's being defined here. – Qiaochu Yuan Nov 28 '18 at 08:03
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So would this be analogous to a 'formal derivative' of a polynomial where we might not have defined what a derivative is but we sort of mimic the process? – green frog Nov 28 '18 at 08:07
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Yes, it's a similar use of "formal." – Qiaochu Yuan Nov 28 '18 at 08:42