Open Access Open Access  Restricted Access Subscription Access

U-Ternary Semigroups And V-Ternary Semigroup


 

In this paper the term U-ternary semigroup is introduced.  It is proved that a ternary semigroup T is a U-ternary semigroup if either T has a left (lateral, right) identity or T is generated by an idempotent.  It is proved that a ternary semigroup is U-ternary semigroup if and only if (1) every proper ideal of T is contained in a proper prime ideal of T, (2) every ideal A of T is semiprime ideal of T, (3) every ideal A of T is the intersection of all prime ideal of T contains A, (4) T\A is an n-system of T or empty where A is an ideal of T, (5) T\A is an m-system of T where A is an ideal of T.  Further it is proved that if T be a U-ternary semigroup.  Then T = T3 and hence every maximal ideal is prime.  Conversely if {P𝛂} is a collection of all prime ideals in T and if P is a maximal element in this collection, then P is a maximal ideal of T.  The term dimension n is introduced and it is proved that if A is a proper ideal of the finite dimensional U-ternary semigroup T.  Then A is contained in maximal ideal.

The term V-ternary semigroup is introduced and proved that a ternary semigroup T is a
V-ternary semigroup if and only if T has at least one proper prime ideal and if {P
𝛂} is the family of all proper prime ideals, then < x > = T for x T\P𝛂 or T is a simple ternary semigroup.

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


Keywords

U-ternary semigroup, ideal, prime ideal, semiprime ideal, n-system
User
Notifications
Font Size

Abstract Views: 158

PDF Views: 0




  • U-Ternary Semigroups And V-Ternary Semigroup

Abstract Views: 158  |  PDF Views: 0

Authors

Abstract


In this paper the term U-ternary semigroup is introduced.  It is proved that a ternary semigroup T is a U-ternary semigroup if either T has a left (lateral, right) identity or T is generated by an idempotent.  It is proved that a ternary semigroup is U-ternary semigroup if and only if (1) every proper ideal of T is contained in a proper prime ideal of T, (2) every ideal A of T is semiprime ideal of T, (3) every ideal A of T is the intersection of all prime ideal of T contains A, (4) T\A is an n-system of T or empty where A is an ideal of T, (5) T\A is an m-system of T where A is an ideal of T.  Further it is proved that if T be a U-ternary semigroup.  Then T = T3 and hence every maximal ideal is prime.  Conversely if {P𝛂} is a collection of all prime ideals in T and if P is a maximal element in this collection, then P is a maximal ideal of T.  The term dimension n is introduced and it is proved that if A is a proper ideal of the finite dimensional U-ternary semigroup T.  Then A is contained in maximal ideal.

The term V-ternary semigroup is introduced and proved that a ternary semigroup T is a
V-ternary semigroup if and only if T has at least one proper prime ideal and if {P
𝛂} is the family of all proper prime ideals, then < x > = T for x T\P𝛂 or T is a simple ternary semigroup.

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


Keywords


U-ternary semigroup, ideal, prime ideal, semiprime ideal, n-system