| Green’s relations play important roles in the the study of semiring algebra theory.Based on the Green’s relation of multiplicative and additive semigroups over semirings,we study the semirings which satisfy the identities xn≈x,(2n-1)x≈x,(x+y)n-1≈xn-1+yn-1,(xy)n-1≈xn-1yn-1(n∈Z,n>1).Let us denote by COS_n~· the class of semirings defined by these additional identities.And the main results are as follows:1.Green’s L relation on the member of COS_n~· are studied.The characterizations of the binary relations (?) related to the Green’s relation of the multiplicative semigroups(additive semigroups)of the semirings are given.2.The sufficient and necessary conditions which make (?) be congruences are obtained.Moreover,the classes of semirings which are determined by these congruences are proved to be semiring varieties.3.Green’s D relation on the member of COSn are studied.The characterizations of the binary relations (?) related to the Green’s relation of the multiplicative semigroups(additive semigroups)of the semirings are given.4.The sufficient and necessary conditions which make (?) be congruences are obtained.Moreover,the classes of semirings which are determined by these congruences are proved to be semiring varieties. |
