Varieties of idempotent distributive semirings with regular multiplicative reduct

Abstract

The multiplicative reduct of an idempotent distributive semiring $$S$$ is a regular band if and only if $$S$$ is a spined product of a semiring satisfying $$xy+xyx \approx xyx+xy$$ and a semiring satisfying $$yx+xyx \approx xyx+yx$$ with respect to a semiring satisfying $$xy+yx \approx yx+xy$$ . In a similar way, we characterize idempotent distributive semirings whose multiplicative reduct is a normal band.