I'm interested in the following question:
In a monoid $\mathbf{M}$, given sums $\sum_{i \in I} a_i = \sum_{j \in J} b_j$, is it possible to find elements $c_{i,j}$ such that
- for all $i$, $a_i = \sum_{j \in J} c_{i,j}$,
- and for all $j$, $b_i = \sum_{i \in I} c_{i,j}$ ?
This amounts to find an $I \times J$ matrix such that $a_i$ (resp. $b_j$) is the sum of the $i$th row (resp. $j$th column).
In addition:
- Working within the assumption that $I$ and $J$ are (at most) countable is enough for me.
- The fact that the $a_i$ and $b_j$ describe (possibly) infinite sums carries the implicit assumption that these sums are well-defined. In practice we can restrict to complete monoids (and our real-life example is $\overline{\mathbf{R}_+} = \mathbf{R}_+ \cup \{\infty\}$).
I think my colleagues and I found a proof that this is always possible provided some good assumptions are satisfied (completeness, total ordering), but our proof is quite convoluted and doesn't seem very general. So that's my question: Is this problem known? Does it have a name? Has it been addressed somewhere? Does anybody know anything about this?
This has been studied under the name refinement monoid.