Congruence and Green’s equivalence relation on ternary semigroup

Authors : Vn DIXIT

Pages : 0-0

Doi:10.1501/Commua1_0000000429

View : 16 | Download : 9

Publication Date : 1997-01-01

Article Type : Research Paper

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 insert ignore into journalissuearticles values(M-equivalent, L-equivalent); elements in a ternary semigroup T with an idempotent einsert ignore into journalissuearticles values(T‘); and y^, y^ are in T” such that lax^x^] = b and [by^yj = ainsert ignore into journalissuearticles values([XjaxJ = b and [y^by^] = a, [XjX^a] = b and [by^yj = a);, then the maps p |L and p |L insert ignore into journalissuearticles values(p |M and p |M , p |R and p |R ); are mutually inverse R-class insert ignore into journalissuearticles values(M-class, L-class); preserving bijections Ifom L to and ırom to insert ignore into journalissuearticles values(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.
Keywords : Congruence, Greens equivalence, ternary semigroup