Let $I$ be a monoid and $G$ be an $I$-graded monoid, with multiplication
$$ ( - \cdot - ) : G_i \times G_j \to G_{i+j}. $$
I'm interested in the following property of $G$:
P: for any two indices $i, j \in I$, and any $g \in G_{i+j}$, there are unique elements $g_i \in G_i$, $g_j \in G_j$ such that $g = g_i \cdot g_j$.
In other words, multiplication is bijective when restricted to specific degrees.
Examples include:
- Any monoid indexed by itself.
- The free monoid on an alphabet $A$, indexed by degree in $\mathbb{N}$.
Has this property been studied anywhere? It's a strong condition; can it be reformulated in more familiar terms, perhaps as a certain freeness property?
Note: I am also interested in answers to this question in other monoidal categories.