Our short paper “Domain Semirings United” coauthored with Uli Fahrenberg, Georg Struth, and Krzysztof Ziemiánski, was accepted by the journal Acta Cybernetica, but will appear only next year (probably in Vol. 25 Nr. 3). In the meantime you can read the paper on the arXiv.
Domain operations on semirings have been axiomatised in two different ways: by a map from an additively idempotent semiring into a boolean subalgebra of the semiring bounded by the additive and multiplicative unit of the semiring, or by an endofunction on a semiring that induces a distributive lattice bounded by the two units as its image. This note presents classes of semirings where these approaches coincide.