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Quoting the Proofwiki definition of mononomials:

A mononomial in the indexed set $\displaystyle {\{X_j:j\,{\in}\,J}\}$ is a possibly infinite product

$$\displaystyle \prod_{j \mathop \in J} X_j^{k_j}$$

with integer exponents $k_jā‰„0$ such that $k_j=0$ for all but finitely many $j$.

How to understand this definition? It talks about a product of elements $X_j$ - how is this operation defined?

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    Such an opeartion is simply a map assigning a power $k_j$ to all indeterminates $X_j$. In other words a monomial is simply a map $J \to \Bbb{N}$ with finite support (only finitely many $j$ have $k_j \neq 0$). ā€“ 2017-01-12
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    So the question is rather about what are the $X_j$'s and how is the multiplication is defined? **Polynomial functions** are defined over (commutative) rings, whereas **polynomials** are defined abstractly and are usually defined as sequences with finitely many nonzero elements. ā€“ 2017-01-12
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    I'd never seen it written as "mononomial" before (despite the linguistic explanation); in mathematics, at least, I've always seen it as "monomial". ā€“ 2017-01-15

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