What is a semiring without commutativity requirement for addition?

A semiring is defined by two operations $+$ and $\times$ on a set $R$, such that:

• $(R, +)$ is a commutative monoid
• $(R, \times)$ is a monoid
• $\times$ left-distributes and right-distributes over $+$
• The identity element for $+$ is an absorbing element for $\times$

What is the name of a structure where $(R, +)$ is only required to be a monoid? I'm interested in this because it may make sense in the study of provenance semirings in relational databases. However, beyond this specific context, I think this is a natural subset of the semiring axioms.

The same question has been asked here for rings (where $(R, +)$ is required to be a commutative group), here and here. In this context, the commutativity requirement can be explained (as in this answer) because commutativity of $+$ is implied by distributivity, the existence of a neutral element for $\times$, associativity of $+$, and cancellativity of $+$. However, in a semiring, $(R, +)$ is only a monoid, so in general it is not cancellative: hence, the same explanation does not work.

A generalization of semirings that removes the commutativity requirement are near-semirings, but they additionally remove the requirement of a neutral element for $\times$ and one of the distributivity requirements (e.g., right-distributivity). Hence, my question is equivalent to asking whether there is a name for near-semirings with multiplicative identity and left- and right-distributivity of $\times$ over $+$.

Another related question here asks about just removing the identity requirement for $\times$. I checked quickly the references from this answer to see whether my concept had a name, but apparently Golan's Semirings and Their Applications and Gondran and Minoux's Graphs, Dioids and Semirings: New Models and Algorithms both assume commutativity of $+$ straight away.

• Just to give a simple example of such a structure with noncommutative addition, you can take $\{-\infty, 0, \infty\}$ with $-\infty + \infty = -\infty$ and $\infty + -\infty = \infty$. This can probably be applied to the extended reals as well (I haven't checked the details).
– S.C.
Dec 26, 2023 at 21:33