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DUO Chained r-Semigroup

A. Gangadhara Rao1, A. Anjaneyulu2, D. Madhusudhana Rao3

Dept. of Mathematics, V S R & N V R College, Tenali, A.P. India.

1raoag1967@gmail.com, 2 anjaneyulu.addala@gmail.com, 3 dmrmaths@gmail.com

 


ABSTRACT

In this paper we introduce the terms left α-cancelltive, right α-cancellative, α-cancellative, left Γ-cancellative, right Γ-cancellative, Γ-cancellative, strongly left cancellative, strongly right cancellative, strongly cancellative elements, α-inverse,
Γ-inverse, complete inverse of an element, unit in a Γ-semigroup and a duo chained
Γ-semigroup.  It is proved that if P is a prime -ideal of a duo chained Γ-semigroup S and  then .  It is also proved that every duo chained  
Γ-semigroup is a semiprimary -semigroup. It is proved that (1) if  is a semisimple elements of a duo chained Γ-semigroup S , then .  (2) if a duo chained Γ-semigroup S has no 횪;-idempotent elements, then for any a S, < a > w = or < a > w is a prime 횪;-ideal. In a duo chained Γ-semigroup S if SSS then S\SS = { x } for some x  S.  Further it is proved that  in a duo chained -semigroup S,  if SSS such that S\SS = { x } for some x  S, then  (1) S = x S1 = S1 x and S S = x S = S x is the unique maximal  -ideal of S.  (2)  If aS and a  < x >w then  for some natural number n > 1. (3) If S contains strongly cancelable elements then x is a strongly cancelable element and < x >w is either empty or a prime -ideal of S.  It is proved that, if S is a duo chained Γ-semigroup, then S is an archemedian Γ-semigroup without Γ-idempotents if and only if  < a >w  = for every
a S.  It is proved that if S be a strongly cancellative archemedian duo chained
Γ-semigroup with   for some a S, then S is a Γ-group.  Also it is proved that, let S be a duo chained Γ-semigroup containing strongly cancellative elements and  < a >w  =   for every a S, then S is a strongly cancellative Γ-semigroup.

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

KEY WORDS : chained Γ-semigroup, duo chained Γ-semigroup, α-cancellative element, Γ-cancellative element, strongly cancellative Γ-semigroup, α-inverse,
Γ-inverse, complete inverse of an element, unit in a Γ-semigroup,  

 

 

 


International eJournal of Mathematical Sciences, Technology and Humanities

Volume 2, Issue 3, Pages:  520 - 533