**InternationaleJournals**

**Ternary
Semigroups**

^{1}Y. Sarala, ^{2}A.Anjaneyulu and ^{3}D.Madhusudhana Rao

** ^{1}**Dept. of Mathematics, Nagarjuna University, Guntur,
A.P. India. Email: Saralayella1970@gmail.com

** ^{2}**Dept. of Mathematics, V S R & N V R College,
Tenali, A.P. India. Email: anjaneyulu.addala@gmail.com

** ^{3}**Dept. of Mathematics, V S R & N V R College,
Tenali, A.P. India. Email: dmrmaths@gmail.com

**ABSTRACT**

In
this paper the terms; quasi commutative ternary semigroups, normal ternary
semigroups, pseudo commutative ternary semigroups are introduced. Unital
element and zero element in a ternary semigroups are introduced. It is proved
that every commutative ternary semigroup is a normal ternary semigroup and a
Pseudo commutative ternary semigroup. It is proved that a ternary semigroup T
has atmost one identity element and atmost one zero element. If A be a
non-empty subset of a ternary semigroup
T then it is proved that <A> = the intersection of all ternary
subsemigroup of T containing A. The terms idempotent element , regular element,
completely regular element, intra regular element, semisimple elemenet are
introduced. It is proved that *a* be a left regular (or) lateral regular
(or) right regular element of a ternary
semigroup T then *a* is semisimple lelment of T. The terms arthimedean,
strongly arthemedean ternary semigroups are introduced. Finally it is proved
that every strongly a ternary semigroup
T is an arthemedean ternary semigroup.

**Mathematics Subject Classification** :
20M12 , 20N10, 60F05,20N99

**Key words**
: Quasi commutative, normal, pseudo commutative, ternary subsemigroup of T
generated by A, semisimple elements, regular ternary semigroup, completely
regular ternary semigroup, ideal, principal ideal, idempotent, archimedean
semigroup.

**International
eJournal of Mathematical Sciences, Technology and Humanities**

**Volume** **2**,
**Issue** **5**, **Pages**: ** **848 - 859