New Green’s*relations L*G and R*G were introduced on a semigroup S.Though they are different from the old Green’s*relations L* and R*,yet are related to each other by Green’s relation on some over semigroups and it holds that L*L*G and R*R*G.Another class of semigroups,the Gabundant (or adequate) semigroups and their Gadequate transversals were defined and the properties and structures of Gabundant semigroups with some Gadequate transversals were discussed.