The semiring is said to be an idempotent semiring if the two reducts are bands, that is, semirings where every element is an idempotent.
Zhao, Green's D-relation for the multiplicative reduct of an idempotent semiring, Arch.
the variety of all idempotent semirings S in which the multiplicative reduct (S; *) of S belongs to V.
Zhao, Varieties of idempotent semirings with a commutative addition, Algebra Universal, to appear.
Shum, D-subvariety of idempotent semirings, Algebra Colloquium, 9 (2002), 15-28.
Zhao, Idempotent semirings with a commutative additive reduct, Semigroup Forum, 64(2002), 289-296.
Axioms (1)-(11) say that the structure is an idempotent semiring
under +, [center dot], 0, and 1, and the remaining axioms (12)-(17) say essentially that * behaves like the Kleene star operator of formal language theory or the reflexive transitive closure operator of relational algebra.