Edited for clarity
(The original statement of the problem I was explaining the problem as a redefinition of multiplication, as can see from the original comments; but I think the algorithmic explanation below might have more insight without big claims of redefinitions)
In the problem I am working on, it is algorithmically identical to expanding a product of similar structured terms and then replacing any exponent greater than one with a one.
Example: $(1+ax)(1+bx)$ after expanding would be $1+ax+bx+abx^2$, and replacing all higher exponents with ones would yield $1+ax+bx+abx$. Additionally, this can also be factored as $1+(a+b+ab)x$. Given $\{a,b,x\}$ a numerical answer can be given.
In this particular problem all products have the same structure. There is a known indicator matrix which gives all the information needed in the problem. Each column is a new variable, and each row is a new product. All products are in the form of $(1+r_i\prod c_j )$ where $r_i$ is the coefficient for that row and $c_j$ is whenever the indicator is 1 in the $(i,j)$ spot.
Example: \begin{bmatrix} 1&0&0\\ 1&1&1 \\ 0&0&1 \\ 1&0&1 \\ \end{bmatrix}
Would represent
$(1+r_1c_1)(1+r_2c_1c_2c_3)(1+r_3c_3)(1+r_4c_1c_3)$
If this was regular multiplication it would be trivial to program vectorized multiplication across the indicator matrix. (Currently using Matlab). If you assume $r=[.1, .2, .3, .4]$ and $c=[.5, .3, .2]$ the answer looking for is 1.1822; whereas normal multiplication would give 1.1645.
Currently I am checking every term in the expansion and simplifying the exponents and then summing the terms. However, even up to 20 rows it is now over one million terms have to check.
For small cases there exists a closed-form expression. If only one column of all ones, then
$\prod(1+r_ic_1) \rightarrow 1+c_1 (\prod(1+r_i)-1)$
This is equivalent to the first example as $1+((1+a)(1+b)-1)x$
The question is: Given an indicator matrix $M$, vectors $r$ and $c$ (all known numerically). Is there a scalable way to evaluate the final answer without having to evaluate all terms?
I don't think there's a general algebraic way to express the formula for arbitrary indicator matrix, but is there an efficient algorithmic way other than checking all $2^{n_r}$ terms? ($n_r$=number of rows)