What would be a proper term to use to call elements of a monoid $M$ in a corresponding monoid ring $RM$ (where $R$ is a ring) or in a monoid algebra $kM$ (where $k$ is a field)? Calling them monomials wouldn't be appropriate, or would it?
I am considering an ordered monoid $M$ and its monoid algebra $kM$, an I would like to talk about "monomial decomposition" of an element of $kM$, "the leading monomial" of an element of $kM$, etc.
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I would reserve the use of monomial to the case of a graded monoid, that is, a monoid equipped with a monoid morphism $d$ from $M$ into the additive monoid $\mathbb{N}$. This would allow one to define the degree of a monomial.
For instance if $M$ is the free monoid $A^*$, you obtain the ring of polynomials over the set of non-commutative variables $A$ and you can take as gradation $d$ the length of a word $u$.
I would just call it the “leading order term”, taking care to define it using the monoid order.
When you write $\sum r_m m$ to denote an element of a semigroup ring, nobody refers to it as a “basic element decomposition.” Everyone just takes it for granted that elements have this form. There doesn’t need to be a special word for the elements of $M$ in $kM$. Because we just refer to them as “elements of $M$” by identification.
I think it’s natural to call the term with the largest generator (with respect to the monoid order) the leading term. This fits completely with what we already do with the most famous monoid ring $\mathbb R[x]$.
The only reason we use “monomial” in polynomial rings is because we don’t introduce what a monoid is until much later (or never at all) so most people don’t have that framework to o think of them as elements of a monoid.