Characterization of various k-regular in b-semirings

The notions of weak1 right k1 -ideals, weak1 right k2 -ideals, weak2 right k1 -ideals, weak2 right k2-ideals, weak1 left k1 -ideals, weak1 left k2 -ideals, weak2 left k1 -ideals and weak2 left k2 -ideals are introduced. We introduce the notions of k1 -regular and k2 -regular b -semirings. The necessary and sufficient condition for the b - semirings to be k1 -regular is established. We characterize k2 -regular b -semirings by weak2 right k2 -ideals and weak2 left k2 -ideals. We characterize k -regular b -semirings by right k -ideals and left k -ideals. Examples are provided to illustrate our results.