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.)