On distributive lattices of simple ordered semirings

In this paper, we study principal ordered ideals of ordered semirings, and give their descriptions with some properties. The characterization of simple ordered semirings is studied, and a relation between a prime ordered ideal and filter has been developed. Finally, we characterize the ordered semirings which are distributive lattices of simple ordered semirings. Here, we find that the equivalence relations MATHEMATICAL SCRIPT CAPITAL N and J induced, respectively, from principal filters and principal ordered ideals coincide as distributive lattice congruence in an ordered semiring S whenever S is a distributive lattice of simple ordered semirings.