Congruence and green's equivalence relation on ternary semigroup

Abstract

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 $e(T^e)$ and $x_1$ , $x_2$, $y_1$ , $y_2$ are in $T^e$ such that $[ax_1x_2]$ = b and $[by_1y_2 ]$ = a,($[x_1ax_2 ]$ = b and $[y_1by_2]$ = a, $[x_1x_2a]$ = b and $[by_1y_2]$ = a), then the maps $rho_{x_1x_2}|L_a$ and $rho_{y_1y_2}|L_b (rho_{x_1x_2}|M_a$ and $rho_{y_1y_2}|M_b$ , $rho_{x_1x_2}|R_a$ and $rho_{y_1y_2}|R_b$ ) are mutually inverse R-class (M-class, L-class) preserving bijections from $L_a$ to $L_b$ and from $L_b$ to $L_a$ ($M_a$ to $M_b$ and $M_b$ to $M_a$ , $R_a$ to $R_b$ and $R_b$ to $R_a$ ). Further we prove Green's theorem -If H is a H-class in a ternary semigroup T, then either [HHH] $cap$ H = $Phi$ or [HHH] = H and H is a ternary subgroup of T.