How do you name these semirings?

Question

I came across a family of semirings with no invertible elements. Do these semirings have a special name?

Thanks!

Answer 1

I am not aware of a specific terminology for this family of semirings. However, idempotent semirings (semirings in which every element satisfies $x^2 = x$) form an important subfamily.

Indeed, suppose that an idempotent element $x$ has an inverse $y$. Then $1 = xy = x^2y = x(xy) = x$. Thus $1$ is the unique invertible element.