### African Journal ofMathematics and Computer Science Research

• Abbreviation: Afr. J. Math. Comput. Sci. Res.
• Language: English
• ISSN: 2006-9731
• DOI: 10.5897/AJMCSR
• Start Year: 2008
• Published Articles: 254

## N(A)-ternary semigroups

• Department of Mathematics, V. S. R and N. V. R. College, Tenali, A. P. India

•  Accepted: 08 July 2013
•  Published: 31 July 2013

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 ternary 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.

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 Γ-ideal, A-divisor, N(A)- ternary semigroup.