On band orthodox semirings

A semiring S is called a band orthodox semiring if $$E^+(S)$$ of all additive idempotents of S is a band semiring. Here we show that every band orthodox semiring is a subdirect product of a left band orthodox semiring and a right band orthodox semiring. Also we introduce the concept of quasi-spined product of a left normal band semiring and a d-inversive semiring with respect to a distributive lattice to study the structure of left normal band orthodox semirings.