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Semiprimary Ideals in Ternary Semigroups

1G. Hanumanta Rao, 2A. Anjaneyulu and 3A. Gangadhara 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 semiprimary ideal, semiprimary ternary semigroup and natural ordering in ternary semigroups are introduced. It is proved that in a ternary semigroup T, 1) every left primary ideal is a semiprimary ideal, 2) every lateral primary ideal is a semiprimary ideal, 3) every right primary ideal is a semiprimary ideal. It is proved that if A is a semiprime ideal of a ternary semigroup T, then the following are equivalent 1) A is a prime ideal, 2) A is a primary ideal, 3) A is a left primary ideal, 4) A is a lateral primary ideal, 5) A is a right primary ideal and 6) A is a semiprimary ideal. Finally it is proved that a ternary semi group T is semiprimary if and only if prime ideals of T form a chain under set inclusion. If T is a duo semiprimary ternary semigroup, then globally idempotent principal ideals form a chain under set inclusion. It is proved that, if E is the set of all idempotent elements of ternary semigroup T. Then the natural ordering is a partial ordered relation, In a duo semiprimary ternary semigroupt, the idempotents in T forms a chain under natural ordering. Further it is proved that, the principal ideals of ternary semigroup T form a chian iff ideals in T form a chain . It is proved that, If T is a commutative regular ternary semigroup with identity , then i) every principal ideal of T generated by an idempotent, ii) principal ideals of T form a chain if and only if idempotents of T form a chain under natural ordering. Furter it is proved that, in a semIsimple ternary semigroup The following are equivalent.1. Every ideal in T is a prime ideal. 2. T is a primary ternary semigroup. 3. T is a left primary ternary semigroup. 4. T is a lateral primary ternary semigroup. 5. T is a right primary ternary semigroup. 6. T is a semiprimary ternary semigroup. 7. Prime ideals of T form a chain. Further it is proved that, In a commutative semisimple ternary semigroup T with identity, then the following are equivalent. 1. Every ideal in T is a prime ideal. 2. T is a primary ternary semigroup. 3. T is a left primary ternary semigroup. 4. T is a lateral primary ternary semigroup. 5. T is a right primary ternary semigroup. 6. T is a semiprimary ternary semigroup. 7. Prime ideals of T form a chain 8. Ideals of T form a chain. 9. Principle ideals of T form a chain. 10. Idempotents of T forms a chain under natural ordering. Further it is proved that, Every ideal of a ternary ternary semigroupT is prime if and only if T is semisimple primary ternary semigroup. It is also proved that, in a ternary semigroup T, the following are equivalent. 1. Every ideal in T is a prime ideal.

2. T is a semisimple and ideals of T form a chain. 3. T is a semisimple and prime ideals of T form a chain. Finally it is proved that, in a commutative semisimple ternary semigroup T, the following are equivalent. 1. Every ideal in T is a prime ideal. 2. T is a regular primary ternary semigroup. 3. T is a regular semiprimary ternary semigroup. 4. T is a regular ternary ternary semigroup and idempotents of T form a chain under natural ordering.

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

Key Words: Semiprimary ideal, semiprimary ternary semigroup, natural ordering in ternary semigroups.

International eJournal of Mathematical Sciences, Technology and Humanities

Volume 3, Issue 2, Pages:  1010 - 1025