Several Classes of Additively Non-Regular Semirings

In this paper,we introduce Green's *-relations on semirings and define [left,right] adequate semirings to explore additively non-regular semirings.We characterize the semirings which are strong b-lattices of [left,right] skew-halfrings.Also,as further generalization,the semirings are described which are subdirect products of an additively commutative idempotent semiring and a [left,right] skew-halfring.We extend results of constructions of generalized Clifford semirings(given by M.K.Sen,S.K.Maity,K.P.Shum,2005) and the semirings which are subdirect products of a distributive lattice and a ring(given by S.Ghosh,1999) to additively non-regular semirings.