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Original Article
Int. J. Chem. Sci, Volume: 15( 4)

Structure of M (I): Ternary Gamma-Semigroups

*Correspondence:
Madhusudhana Rao D, Associate Professor, Department of Mathematics, VSR & NVR college, Tenali, Guntut (Dt), Andhra Pradesh, India, Tel: 9440358718; E-mail: [email protected]

Received: October 25, 2017; Accepted: November 20, 2017; Published: November 23, 2017

Citation: Vasantha M, Madhusudhana Rao D. Structure of M (I): Ternary Gamma-Semigroups. Int J Chem Sci. 2017;15(4):224

Abstract

The terms, ‘I-dominant’, ‘left I-divisor’, ‘right I-divisor’, ‘I-divisor’ elements, ‘M (I)-ternary Γ-semigroup’ for a ternary Γ-ideal I of a ternary Γ-semigroup are introduced and we characterized M (I)-ternary gamma semigroups.

Keywords

Completely prime ternary Γ -ideal; I-dominant element; I-dominant ternary Γ-ideal; I-divisor; M (I)-ternary Γ -semigroup

Introduction

In [1] introduced the concepts of A-potent elements, A-divisor elements and N (A)-semigroups for a given ideal A in a semigroup and characterized N (A)-semigroups for a pseudo symmetric ideal A. He proved that if M is a maximal ideal containing a pseudo symmetric ideal A, then either M contains all A-dominant elements or M is trivial. In this paper we extent these notions and results to M (I)-ternary Γ-semigroups.

Experimental

Preliminaries

Definition 2.1: Let T and Γ be two non-empty set. Then T is said to be a Ternary Γ-semigroup if there exist a mapping from T × Γ × T × Γ × T to T which maps (x1,α,x2,β,x3) → [x1αx2βx3] satisfying the condition:Equation ∀ xi ∈ T 1≤ i ≤ 5 and α,β,γ,δ∈Γ. A nonempty subset A of a ternary Γ-semigroup T is said to be ternary Γ-ideal of T if b,c∈T, α,β∈Γ, a∈A implies bαcβa∈ A,bαaβc∈ A,aαbβc∈ A. A is said to be a completely prime Γ-ideal of T provided x, y, z ∈ T and xΓyΓz ⊆A implies either x∈ A or y∈A or z∈ A. and A is said to be a prime Γ-ideal of T provided X,Y,Z are Ternary Γ-ideal of T and XΓYΓZ⊆A⇒X⊆A or Z⊆A. A ternary Γ-ideal A of a ternary Γ-semigroup T is said to be a completely semiprime Γ-ideal provided Equation for some odd natural number n>1 implies Equation Similarly, A ternary Γ-ideal A of a ternary Γ-semigroup T is said to be semiprime ternary Γ-ideal provided X is a ternary Γ- ideal of T and Equation for some odd natural number n implies X ⊆ A [2-6].

Definition 2.2: A ternary Γ-ideal I of a ternary Γ-semigroup T is said to be pseudo symmetric provided x, y, z ∈T, Equation I impliesEquation for all s, t ∈T and I is said to be semi pseudo symmetric provided for any odd natural number n,x∈T,Equation

Theorem 2.3: Let I be a semi-pseudo symmetric ternary Γ-ideal of a ternary Γ-semigroup T. Then the following are equivalent.

1) I1=The intersection of all completely prime ternary Γ-ideals of T containing I.

2) I11 =The intersection of all minimal completely prime ternary Γ-ideals of T containing I.

3) 111I =The minimal completely semiprime ternary Γ-ideal of T relative to containing I.

4) I2={x ∈ T: (xΓ)n-1 x ⊆I for some odd natural number n}

5) I3=The intersection of all prime ternary Γ-ideals of T containing I.

6) I31 =The intersection of all minimal prime ternary Γ-ideals of T containing I.

7) I113 =The minimal semiprime ternary Γ-ideal of T relative to containing I.

8) I4={x ∈ T: (<x>Γ)n-1< x > ⊆ I for some odd natural number n}.

Theorem 2.4: If I is a ternary Γ-ideal of a semi simple ternary Γ-semigroup T, then the following are equivalent. 1) I is completely semiprime.

2) I is pseudo symmetric.

3) I is semi-pseudo symmetric.

Results and Discussion

M (i)-ternary gamma-semigroup

We now introduce the terms I-dominant element and I-dominant ternary Γ-ideal for a ternary Γ-ideal of a ternary Γ- semigroup [7].

Definition 3.1: Let I be a ternary Γ-ideal in a Ternary Γ-semigroup T. An element x∈T is said to be I-dominant provided there exists an odd natural number n such that Equation A ternary Γ-ideal J of T is said to be I-dominant ternary Γ- ideal provided there exists an odd natural number n such that Equation

Note 3.2: If I is a ternary Γ-ideal of a ternary Γ-semigroup T, then every element of I is a I-dominant element of T and I itself an I-dominant ternary Γ-ideal of T.

Definition 3.3: Let I be a ternary Γ-ideal of a ternary Γ-semigroup T. An I-dominant element x is said to be a nontrivial Idominant element of T if x ∉ I.

Notation 3.4: Mo (I)=The set of all I-dominant elements in T.

M1 (I)=The largest ternary Γ-ideal contained in Mo (I).

M2 (I)=The union of all I-dominant ternary Γ-ideals.

Theorem 3.5: If I is a ternary Γ-ideal of a ternary Γ-semigroup T, the Equation

Proof: Since I is itself an I-dominant ternary Γ-ideal, and M2 (I) is the union of all I-dominant ternary Γ-ideals. Therefore, I⊆M2 (I). Let Equationbelongs to at least one I-dominant ternary Γ-ideals Equation is an I-dominant element. Hence, x∈M0 (I). Therefore,Equation Clearly M2 (I) is a ternary Γ-ideal of T. Since M1 (I) is the largest ternary Γ-ideal contained in Mo (I), we have Equation Hence,Equation

Theorem 3.6: If I is a ternary Γ-ideal in a ternary Γ-semigroup T, then the following are true.

1. M0 (I)=I2.

2. M1 (I) is a semiprime ternary Γ-ideal of T containing I.

3. M2 (I)=I4.

Proof: (1) Mo (I)=The set of all I-dominant elementsEquation

(2) Suppose that Equation for some odd natural number n. Suppose, if possible Equation M1 (I), < x > are the ternary Γ-ideals implies Equation is a ternary Γ-ideal. Since M1 (I) is the largest ternary Γ-ideal in M0 (I), We haveEquation Hence, there exists an element y such thatEquation NowEquation for some odd natural number Equation It is a contradiction. Therefore, x ∈ M1 (I). Hence, M1 (I) is a semiprime ternary Γ-ideal of T containing I.

(3) Let x∈M2 (I). Then there exists an I-dominant ternary Γ-ideal J such that x∈J.

J is I-dominant ternary Γ-ideal implies there exists an odd natural number n such that EquationEquation for some oddEquation Therefore,EquationEquation for some odd n∈ N. So < x > is an I-dominant ternary Γ-ideal in T and hence,EquationEquation Therefore,Equation Hence,Equation It is natural to ask whether M1 (I)=I3. This is not true.

Example 3.7: In the free ternary Γ-semigroup T over the alphabet x, y, z. For the ternary Γ-ideal EquationEquation andEquation ButEquation is a prime ternary Γ-ideal, let I, J, K are three ternary Γ-ideals of T such that Equation implies all words containingEquationor all words containing Equation or all words containingEquation orEquation Therefore,Equation is a prime ternary Γ-ideal. We have Equation so Equation Therefore, we can remark that the inclusions in Equation may be proper in an arbitrary ternary Γ-semigroup [8-11].

Theorem 3.8: If I is a semi pseudo symmetric ternary Γ-ideal in a ternary Γ-semigroup T, then M0 (I)=M1 (I)=M2 (I).

Proof: Suppose I is a semi pseudo symmetric ternary Γ-ideal in a ternary Γ-semigroup T. By theorem 3.7, M0 (I)=I2 and M2 (I)=I4. Also by theorem 2.10, we have I2=I4. Hence, M0 (I)=M2 (I). By the theorem 3.5, Equation We haveEquation Now let Equation Therefore,Equation Hence,Equation Therefore,Equation

Theorem 3.9: For any semi pseudo symmetric ternary Γ-ideal I in a ternary Γ-semigroup T, a nontrivial I-dominant element Equation cannot be semi simple [12,13].

Proof: Since x is a nontrivial I-dominant element, there exists an odd natural number n such that Equation Since I is semi pseudo symmetric ternary Γ-ideal, we have Equation If x is semi simple, thenEquationEquation and hence,Equation this is a contradiction. Thus, x is not semi simple.

Theorem 3.10: If I is a ternary Γ-ideal in a ternary Γ-semigroup T, such that M0 (I)=I, then I is a completely semiprime ternary Γ-ideal and I is a pseudo symmetric ternary Γ-ideal.

Proof: Let Equation andEquation SinceEquation Thus, there exists an odd natural number n such thatEquation Therefore, I is a completely semiprime ternary Γ-ideal. By corollary 2.11, A is pseudo symmetric ternary Γ-ideal. Hence, I is completely semiprime and pseudo symmetric ternary Γ-ideal.

Theorem 3.11: If I is a semi pseudo symmetric ternary Γ-ideal of a ternary semi simple Γ-semigroup then I=M0 (I).

Proof: Clearly, Equation LetEquation If Equation then x is a nontrivial I-dominant element. By theorem 3.9, x cannot be semi simple. It is a contradiction. Therefore, Equation and hence,Equation ThusEquation

We now introduce a left I-divisor element, lateral I-divisor element, right, I-divisor element and I-divisor element corresponding to a ternary Γ-ideal A in a ternary Γ-semigroup.

Definition 3.12: Let I be a ternary Γ-ideal in a ternary Γ-semigroup T. An element Equation is said to be a left I-divisor (a lateral I-divisor, right I-divisor) provided there exist two elements Equation such thatEquation

Definition 3.13: Let I be a ternary Γ-ideal in a ternary Γ-semigroup T. An element x∈T is said to be two-sided A-divisor if x is both a left I-divisor and a right, I-divisor element.

Definition 3.14: Let I be a ternary Γ-ideal in a ternary Γ-semigroup T. An element x∈T is said to be I-divisor if a is a left Idivisor, a lateral I-divisor and a right, I-divisor element.

We now introduce a left I-divisor ternary Γ-ideal, lateral I-divisor ternary Γ-ideal, right I-divisor ternary Γ-ideal and I-divisor ternary Γ-ideal corresponding to a ternary Γ-ideal I in a ternary Γ-semigroup.

Definition 3.15: Let I be a ternary Γ-ideal in a ternary Γ-semigroup T. A ternary Γ-ideal J in T is said to be a left I-divisor ternary Γ-ideal (lateral I-divisor ternary Γ-ideal, right I-divisor ternary Γ-ideal, two sided I-divisor ternary Γ-ideal) provided every element of J is a left I-divisor element (a lateral I-divisor element, a right I-divisor element, it is both a left I-divisor ternary Γ-ideal and a right I-divisor ternary Γ-ideal).

Definition 3.16: Let I be a ternary Γ-ideal in a ternary Γ-semigroup T. A ternary Γ-ideal J in T is said to be I-divisor ternary Γ-ideal provided if it is a left I-divisor ternary Γ-ideal, a lateral I-divisor ternary Γ-ideal and a right I-divisor ternary Γ-ideal of a ternary Γ-semigroup T.

Notation 3.17: Rl (I)=The union of all left I-divisor ternary Γ-ideals in T.

Rr (I)=The union of all right I-divisor ternary Γ-ideals in T.

Rm (I)=The union of all lateral I-divisor ternary Γ-ideals in T.

Equation We call R (I), the divisor radical of T.

Theorem 3.18: If I is any ternary Γ-ideal of a ternary Γ-semigroup T, then Equation

Proof: Let Equation SinceEquation we haveEquation for some odd natural number n. Let n be the least odd natural number such that Equation If n=1 then x∈I and hence, Equation

If n >1, then Equation whereEquation

Hence, x is an I-divisor element. Thus, Equation Therefore,Equation

Theorem 3.19: If I is a ternary Γ-ideal in a ternary Γ-semigroup T, then R (I) is the union of all I-divisor ternary Γ-ideals in T.

Proof: Suppose I is a ternary Γ-ideal in a ternary Γ-semigroup T.

Let J be I-divisor ternary Γ-ideal in T. Then J is a left I-divisor, a lateral I-divisor and a right I-divisor ternary Γ-ideal in T. Thus Equation andEquation

Equation

Therefore, R (I) contains the union of all I-divisor ternary Γ-ideals in T. Let Equation ThenEquation SoEquation

Hence, Equation is I-divisor ternary Γ-ideal. So, R (I) is contained in the union of all divisor ternary Γ-ideals in T. Thus R (I) is the union of all divisor ternary Γ-ideals of T.

Corollary 3.20: If I is a pseudo symmetric ternary Γ-ideal in a ternary Γ-semigroup T, then R (I) is the set of all I-divisor elements in T.

Proof: Suppose I is a pseudo symmetric ternary Γ-ideal in T. Let x be I-divisor element in T. ThenEquation where y, zEquation I is pseudo symmetric

Equation is I-divisor ternary Γ-ideal Equation

Equation Hence, R (I) is the set of all I-divisor elements in T. We now introduce the notion of M (I)-ternary Γ- semigroup.

Definition 3.21: Let I be a ternary Γ-ideal in a ternary Γ-semigroup T. T is said to be a M (I)-ternary Γ-semigroup provided every I-divisor is I-dominant.

Notation 3.22: Let T be a ternary Γ-semigroup with zero. If I={0}, then we write R for R (I) and M for M0 (I) and M-ternary Γ-semigroup for M (I)-ternary Γ-semigroup.

Theorem 3.23: If T is an M (I)-ternary Γ-semigroup, then R (I)=M1 (I).

Proof: Suppose T is an M (I)-ternary Γ-semigroup. By theorem 3.18, Equation

Let Equation is an I-divisorEquation is an I-dominantEquation

Hence, Equation

Theorem 3.24: Let I be a semipseudo symmetric ternary Γ-ideal in a ternary Γ-semigroup T. Then T is an M (I)-ternary Γ- semigroup iff R (I)=M0 (I).

Proof: Since I is a semi-pseudo symmetric ternary Γ-ideal, by theorem 3.8, M0 (I)=M1 (I)=M2 (I). If Tan M (I)-ternary Γ- semigroup, then by theorem 3.23, R (I)=M1 (I). Hence, R (I)=N0 (I). Conversely suppose that R (I)=M0 (I). Then clearly every I-divisor element is an I-dominant element. Hence, T is an M (I)-ternary Γ-semigroup.

Corollary 3.25: Let I be a pseudo symmetric ternary Γ-ideal in a ternary Γ-semigroup T. Then T is an M (I)-ternary Γ- semigroup if and only if R (I)=M0 (I).

Proof: Since every pseudo symmetric ternary Γ-ideal is a semi-pseudo symmetric ternary Γ-ideal, by theorem 3.24, R (I)=M0 (I).

Corollary 3.26: Let T be a ternary Γ-semigroup with 0 and < 0 > is a pseudo symmetric ternary Γ-ideal. Then R=M iff T is an M-ternary Γ-semigroup.

Proof: The proof follows from the theorem 3.24.

Theorem 3.27: If N is a maximal ternary Γ-ideal in a ternary Γ-semigroup T containing a pseudo symmetric ternary Γ-ideal I, then N contains all I-dominant elements in T or T\N is singleton which is I-dominant.

Proof: Suppose N does not contain all I-dominant elements.

Let Equation be any I-dominant element and y be any element in T\N.

Since N is a maximal ternary Γ-ideal, Equation

Since Equation we have Equation Let n be the least positive odd integer such that Equation Since I is a pseudo symmetric ternary Γ-ideal then I is a semipseudo symmetric ternary Γ-ideal and hence, Equation

Therefore Equation and hence, y is I-dominant element. Thus, every element in T\N is I-dominant.

Similarly, we can show that if m is the least positive odd integer such that Equation Therefore, there exists an odd natural number p such thatEquation for allEquation

Let x, y, z ∈ T\N. Since N is maximal ternary Γ-ideal, we have Equation

So Equation So Equation and hence, x ∈ sΓyΓt for some s, t ∈ T1. Now since I is a pseudo symmetric ternary Γ-ideal,

we have, Equation If y ≠ x then s, t ∈ T. If s, t ∈ N then Equation

Which is not true. In both the cases we have a contradiction. Hence, x=y.

Similarly, we show that z=x.

Corollary 3.28: If N is a nontrivial maximal ternary Γ-ideal in a ternary Γ-semigroup T containing a pseudo symmetric ternary Γ-ideal I. Then M0 (I) ⊆N.

Proof: Suppose in Equation Then by above theorem 3.27, N is trivial ternary Γ-ideal. It is a contradiction. Therefore, M0 (I) ⊆N.

Corollary 3.29: If N is a maximal ternary Γ-ideal in a semi simple ternary Γ-semigroup T containing a semipseudo symmetric ternary Γ-ideal I. Then M0 (I) ⊆N.

Proof: By theorem 3.11, I is pseudo symmetric ternary Γ-ideal. If Equation is I-dominant, then x cannot be semi simple. It is a contradiction. Therefore, M0 (I) ⊆ N.

Conclusion

According to theorem 3.11, I is pseudo symmetric ternary Γ-ideal. If x ∈ T\N is I-dominant, then x cannot be semi simple. Hence, is a contradiction. Therefore, M0 (I) ⊆ N.

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