About Idempotent Element Semi-ring Cluster Of A Sub-cluster

In the study of semirings theory, idempotent semiring is a very important classof semiring. We study a subvariety RoD of idemportent semiring variety. It is thesubvariety of idempotent semiring variety which is studied by Pastij and Zhao in thereference[12], and corressponds with the subvariety RoD of idempotent semiring variety which is studied by Sen, Guo and Shum. We prove that semiring S belongs toRoD if and only if Green's D-relation D on the multiplicative reduct (S, . ) of S is lattice congruence on S. We obtain the identities that the variety satisfies and givesome description of the member of RoD . Based on the study of multiplicative reduct (S, .) of idempotent semiring S in RoD , we research the subdirect product decomposition of the idempotent semiring in RoD , and study characters and properites of some significant subvarieties of RoD.