In my research I have come across the following condition on a monoid.
Every element $x$ satisfies the following property: there exists a natural number $n$ such that for any $m \geq n$ and any decomposition $x = x_1 \cdot x_2 \cdot \cdots \cdot x_m$ of $x$ it must be that at least one of $x_i$ is an idempotent.
Free monoids obviously satisfy this property, and every monoid which has only idempotent elements. And of course natural numbers under multiplication due to prime factorization.
A non-example is any monoid with an elementa unit which has an inverse, but is not its own inverse,the identity and things like positive rationals or reals under multiplicationaddition.
Decompositions of monoid elements seem to be studied in certain areas such as language theory, however I have not found a reference for this exact notion of "finitely decomposable", so I would like to know whether such a condition (or a condition close to it) has been encountered before, and if so, when.
If it helps in answering the question, all the monoids I am considering are commutative.