In an applied research problem I am currently working on, I am using non-commutative semiring convolution to formulate some interesting types of calculations on images and solid objects. For discrete groups, this is pretty easy to set up. Given a semiring, $(S, \oplus, \odot)$, and a discrete group $G$, define the semialgebra $S[G]$ as the collection of maps $S^G$ equipped with the bilinear operator $\star : S[G] \times S[G] \to S[G]$ such that for all $f, h \in S[G]$:

$(f \star h)(x) = \bigoplus \limits_{y \in G} f(y) \odot h(y^{-1}x)$

This works well, and is consistent with the usual definition of a group algebra when $S$ is a ring. However, in almost all of the interesting situations I can think of, I would like to take $G$ to be a Lie group. To make this happen, it seems that the right thing to do should be to define some kind of *'semiring valued Haar measure'*, $\mu : 2^G \to S$ which would allow one to compute $S$-valued integrals over $G$. Assuming this works just like it ought to in fields of characteristic 0, then convolution becomes some $S$-valued integral over $G$ like the following thing:

$(f \star h)(x) = \bigoplus \limits_{y \in G} f(y) \odot g(y^{-1}x) \odot d \mu(y)$

Where the symbol $\bigoplus$ means something like a 'semiring' Lebesgue integral over $G$. Unfortunately this definition is not very robust. One can easily pick plausible values of $S, G, \mu$ which catastrophically fail. For instance, if $S = \mathbb{Z} / n \mathbb{Z}$, then it seems to me that the convolution integral is divergent for any function with measurable support!

However, all is not lost. I can think of a few easy cases where one can construct a measure which is convergent and gives 'reasonable' results (and by reasonable, I mean that they are intuitively similar to the results one would get in the case of a discrete group). Consider, for example, the case where $S$ is the idempotent boolean semifield, $B = ( \mathbf{2}, OR, AND )$. Then take:

$\mu(\emptyset) = 0$

$\mu(S \neq \emptyset) = 1$

Now the elements $f, g \in B^G$ can be identified with subsets $X_f, X_g \subseteq G$, and moreover the convolution on $G$ over $B$ gives rise to a so-called generalized Minkowski product:

$f \star g \cong X_f \oplus X_g = \bigcup \limits_{x \in X_f} x X_g$

Which is indeed a useful calculation! Similarly, one can define a measure like this over the $(max, +)$ semiring via another ad-hoc construction. This begs the question: when does this convolution actually work? More specifically, what are the conditions on $S, G, \mu$ such that we can guarantee that convolution is convergent?

(Note: I am probably not being as careful as I should be with definitions. It is probably a good idea to restrict the elements of $S[G]$ to 'Haar-like' measurable functions, but I really don't have a good grasp of what this should mean until I know what the proper conditions on $\mu$ should be...)