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.