In this paper we have defined the left, lateral and right congruence on a ternary semigroup. We discuss Green’s Equivalence relations L, M, R, H, D, J, on T. We give one new relation called M-equivalence relation. We also prove that under certain conditions a ternary semigroup reduces to an ordinary semigroup or even to a band. We prove the Green’s Lemma - Let a and b be R-equivalent (M-equivalent, L-equivalent) elements in a ternary semigroup T with an idempotent e(T‘) and y^, y^ are in T” such that lax^x^] = b and [by^yj = a([XjaxJ = b and [y^by^] = a, [XjX^a] = b and [by^yj = a), then the maps p |L and p |L (p |M and p |M , p |R and p |R ) are mutually inverse R-class (M-class, L-class) preserving bijections Ifom L to and ırom to (M^ to and M to M , R to R and R to R). Further we prove Green’s theorem -If H is a b aa aa b b a' H-class in a ternary semigroup T, then either [HHH] n H = 0 or [HHH] = H and H is a ternary subgroup of T.