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IDEALS IN TERNARY SEMIGROUPS

Y. Sarala1, A.Anjaneyulu2, D.Madhusudhana Rao3

Dept. of Mathematics, Nagarjuna University, Guntur, A.P. India. Email ID: Saralayella1970@gmail.com

Dept. of Mathematics, V S R & N V R College, Tenali, A.P. India. Email ID: Anjaneyulu.addala@gmail.com

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

ABSTRACT

In this paper the terms ideal, trivial ideal, proper ideal, maximal ideal are introduced.  It is proved that the union and intersection of any family of ideals of ternary semigroup T is an ideal of T. It is also proved that union of all proper ideals of ternary semigroup T is the unique maximal ideal of T.  The terms ideal of ternary semigroup T generated by A, principal ideal generated by an element are introduced.  It is proved that the ideal of a ternary semigroup T generated by a non-empty subset A is the intersection of all ideals of T containing A.  It is also proved that T is a ternary semigroup and a T then
J(a) = a aTT TTa TaT TTaTT . The terms, simple ternary semigroup, globally idempotent ideal are introduced. In any ternary semigroup T, principal ideals of T form a chain and ideals of T form a chain are equivalent.  It is proved that a ternary semigroup T is simple ternary semigroup if and only if TTaTT = T for all a T.   It is also proved that if T is a globally idempotent ternary semigroup having maximal ideals then T contains semisimple elements.

Mathematics Subject Classification: 20M12, 20N10, 60F03, 20N99

Key words: Ideal, trivial ideal, proper ideal, chain, ideal of ternary semigroup T generated by A, principal ideal generated by a, simple ternary semigroup T.

 

 

 


International eJournal of Mathematics and Engineering

Volume 4, Issue 1, Pages:  1950 - 1968