First we have obtained equivalent conditions for a regular semigroup and is equivalent to N = N1 It is observed that every regular semigroup is weakly separative and C ? S and on a completely regular semigroup S ? N and S is partial order . It is also obtained that a band (S, .) is normal iff C = N . It is also observed that on a completely regular semigroup (S, .), C = S = N iff (S, .) is locally inverse semigroup and the restriction of C to E(S) is the usual partial order on E(S). Finally it is obtained that, if (S, .) is a normal band of groups then C = S = N .