I'm looking for the names of algebraic structures that are more general than a near-semiring. My preferred structure requires both + and · only be magmas; two semigroups is still over-constraining (associativity is not necessary, but distributivity is required).
I'm trying to pin down the semantic requirements for the C++ algorithm inner_product. In short, inner_product computes a result not unlike a dot product, but where the binary operations can be very general. Based on the algorithm's specification, associativity is not necessary, and I'm particularly interested in the case where both operations are magmas.
Does such a structure have a name? If not, I'm inclined to call this a magmaring, since both operations are magmas.
There is an algebraic object known as a double magma that is a nonempty set $S$ with two binary operations. However, there is an additional requirement that the two binary operations satisfy an interchange law $$(w \cdot x) \star (y \cdot z) = (w \cdot y) \star (x \cdot z)$$
It can be found in the following:
Edmunds, C. (2015). Constructing Double Magma on Groups Using Commutation Operations. Canadian Mathematical Bulletin, 58(3), 497-506.
In ten minutes of browsing, this seems to be the most general structure of interest and that has its own name.