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N(A)-TERNARY SEMIGROUPS

Ch. Manikya Rao1, P. Koteswara Rao2 and D. Madhusudhana Rao3

1 Department of Mathematics, Bapatla Arts & Sciences College, Bapatla, Guntur (Dt), A. P. India. Email : cmrao1998@gmail.com

2Acharya Nagarjuna University, Nagarjuna Nagar, Guntur

1Department of Mathematics, V. S. R & N. V. R. College, Tenali, A. P. India. Email : dmrmaths@gmail.com

ABSTRACT

In this paper, the terms, ‘A-potent’, ‘left A-divisor’, ‘right A-divisor’, ‘A-divisor’ elements, ‘N(A)-ternary semigroup’ for an ideal A of a ternary semigroup are introduced. If A is an ideal of a ternary semigroup T then it is proved that
(1)
(2) N0(A) = A2, N1(A) is a semiprime ideal of T containing A, N2(A) = A4 are equivalent, where No(A) = The set of all A-potent elements in T, N1(A) = The largest ideal contained in No(A), N2(A) = The union of all A-potent ideals. If A is a semipseudo symmetric ideal of a ternary semigroup then it is proved that N0(A) = N1(A) = N2(A).  It is also proved that if A is an ideal of a ternary semigroup such that N0(A) = A then A is a completely semiprime ideal.  Further it is proved that if A is an ideal of ternary semigroup T then R(A), the divisor radical of A, is the union of all A-divisor ideals in T.  In a N(A)- ternary semigroup it is proved that R(A) = N1(A). If A is a semipseudo symmetric ideal of a semigroup T then it is proved that S is an N(A)- ternary semigroup iff
R(A) = N0(A).  It is also proved that if M is a maximal ideal of a ternary semigroup T containing a pseudo symmetric ideal A then M contains all A-potent elements in T or T\M is singleton which is A-potent.

Mathematical  subject classification (2010) : 20M07; 20M11; 20M12.

KEY WORDS:  Pseudo symmetric ideal, semipseudo symmetric ideal, prime ideal, semiprime ideal, completely prime ideal, completely semiprime ideal, semisimple element, A-potent element, A-potent ideal, A-divisor, N(A)- ternary semigroup.

International eJournal of Mathematical Sciences, Technology and Humanities

Volume 3, Issue 3, Pages:  1080 - 1090