As the title implies, I am looking for the right name for an algebraic structure, which is exactly as an idempotent semiring, apart from the fact that multiplication does not right-distribute over addition.


The name I would imagine is something like "idempotent left semiring", since multiplication still left-distributes over addition, but I'm unable to find a proper reference in the literature.


To make things clear, by semiring I mean an algebraic structure in which there is an associative and commutative additive operator ("$+$") as well as an associative multiplicative operator ("$\*$"), which is both left- and right-distributive over addition. An idempotent semiring is a semiring in which both operators are idempotent.


(I also posted this question on "Mathematics", but, on second thoughts, I feel mathoverflow is the right forum for it. Apologies if my assumption is wrong.)



Name of an algebraic structure that is an idempotent semiring but does not have right distributivity
I'm pretty sure you are looking for a [near-semiring](https://en.wikipedia.org/wiki/Near-semiring). You could call it an "idempotent near-semiring" using left-right if necessary.


There might be other/more terms suggested in Gondran and Minoux's book:



> 
> Gondran, Michel, and Michel Minoux. Graphs, dioids and semirings: new models and algorithms. Vol. 41. Springer Science & Business Media, 2008.
> 
> 
> 


I think they call a commutative semigroup equipped with another semigroup operation that is left distributive a *left pre-semiring*. They do not even require an absorbing element or identity for the second operation. One could tack on "idempotent" to this as well.


I can't really vouch for the impact of this book on terminology. But, terms for ring-like structures this general are a bit of a mess, and I can't complain much about an attempt to outline some sort of consistent system in a single book.


