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Primary Ideals in Ternary Semigroups

1G. Hanumanta Rao, 2A. Anjaneyulu and 3D. Madhusudhana Rao

1Department of Mathematics, S.V.R.M. College, Nagaram, Guntur (dt) A.P. India. Email : ghr@svrmc.edu.in

2, 3Department of Mathematics, V.S.R & N.V.R.College, Tenali, A.P. India. Email: anjaneyulu.addala@gmail.com, raoag1967@gmail.com

ABSTRACT

In this paper, the terms left primary ideal, lateral primary ideal, right primary ideal, primary ideal, left primary ternary semigroup, lateral primary ternary semigroup , right primary ternary semigroup, primary ternary semigroup are introduced. It is proved that A be an ideal in a ternary semigroup T and if X, Y, Z are three ideals of T such that 1) XYZA and Y ⊈ A, Z ⊈ A, implies X  A A iff x, y, z T, < x > < y > < z > A and y A, z A implies x  A . 2) XYZ A and X ⊈A, Z ⊈A, implies Y A if and only if x, y, z T, < x >< y >< z > A and xA, zA implies y  A . 3) XYZ A and X ⊈ A, Y ⊈ A, implies Z  A iff x, y, z T, < x >< y >< z > A and xA, yA implies z A . Further it is proved that if T be a commutative ternary semigroup and A be an ideal of T, then the conditions, 1) A is left primary ideal, 2) X, Y, Z are three ideals of T such that XYZA and Y ⊈ A, Z ⊈ A, implies X  A , 3) x, y, z T, < x >< y >< z > A and y A, z A implies x  A are equivalent. It is also proved that if T be a commutative ternary semigroup and A be an ideal of T, then the conditions, 1) A is lateral primary ideal, 2) X, Y, Z are three ideals of T such that XYZ  A and X ⊈ A, Z ⊈A, implies Y A , 3) x, y, z T, < x >< y >< z > A and x A, z A implies y  A . Further the conditions for an ideal in a commutative ternary semigroup T, 1) A is right primary ideal, 2) X, Y, Z are three ideals of T such that XYZ A and X ⊈A, Y ⊈A, implies Z  A , 3) x, y, z T, x y z A and x A, y A implies z  A are equivalent. It is proved that every ideal A in a ternary semigroup T, 1) T is a left primary if and only if every ideal A satisfies X, Y, Z are three ideals of T such that XYZ  A and Y ⊈A, Z ⊈ A, implies X  A , 2) T is a lateral primary if and only if every ideal A satisfies X, Y, Z are three ideals of T such that XYZ  A and X ⊈A, Z ⊈A, implies Y A , 3) T is a right primary if and only if every ideal A satisfies X, Y, Z are three ideals of T such that XYZ  A and X ⊈A, Y ⊈A, implies Z  A . It is proved that T be a ternary semigroup with identity and M be the unique maximal ideal in T. If A = M for some ideal A in T, then A is a primary ideal. Further it proved that if T is a ternary semigroup with identity and M is the unique maximal ideal of T , then for any odd natural number n, M n is a primary ideal of T. It is proved that if A is an ideal of quasi commutative ternary semigroup T, then 1) A is primary, 2) A is left primary, 3) A is lateral primary and 4) A is right primary are equivalent.

Subject Classification(2010): 20M07, 20M11, 20M12.

Key Words: Left primary ideal, Lateral primary ideal, right primary ideal, primary ideal, left primary ternary semigroup, lateral primary ternary semigroup , right primary ternary semigroup, primary ternary semigroup.

International eJournal of Mathematics and Engineering

Volume 4, Issue 3, Pages:  2145 - 2159