What is the name of an algebraic structure that is an idempotent semiring but does not have right distributivity? As in the title, I am looking for the right name for this 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 locate 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.
 A: I've transferred my answer on matheoverflow over here: I did not realize it was crossposted, and it is a better question for this forum.

I'm pretty sure you are looking for a 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.
A: They are called idempotent near-semirings, but I disagree with your definition of an idempotent semiring. An idempotent semiring is a semiring in which the addition is idempotent. See also Semiring on wikipedia.
