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 (x_{1},α,x_{2},β,x_{3}) → [x_{1}αx_{2}βx_{3}] satisfying the condition: ∀ x_{i} ∈ 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 for some odd natural number n>1 implies 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 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, I implies for all s, t ∈T and I is said to be semi pseudo symmetric provided for any odd natural number n,x∈T,

**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) I_{1}=The intersection of all completely prime ternary **Γ**-ideals of T containing I.

2) I_{1}^{1} =The intersection of all minimal completely prime ternary **Γ**-ideals of T containing I.

3) 1_{1}^{1I} =The minimal completely semiprime ternary **Γ**-ideal of T relative to containing I.

4) I_{2}={x ∈ T: (x**Γ**)^{n-1} x ⊆I for some odd natural number n}

5) I_{3}=The intersection of all prime ternary **Γ**-ideals of T containing I.

6) I_{3}^{1} =The intersection of all minimal prime ternary **Γ**-ideals of T containing I.

7) I^{11}_{3} =The minimal semiprime ternary **Γ**-ideal of T relative to containing I.

8) I_{4}={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 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

**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: **M_{o} (I)=The set of all I-dominant elements in T.

M_{1} (I)=The largest ternary Γ-ideal contained in M_{o} (I).

M_{2} (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

Proof: Since I is itself an I-dominant ternary Γ-ideal, and M_{2} (I) is the union of all I-dominant ternary Γ-ideals. Therefore, I⊆M_{2} (I). Let belongs to at least one I-dominant ternary Γ-ideals is an I-dominant element. Hence, x∈M_{0} (I). Therefore, Clearly M_{2} (I) is a ternary Γ-ideal of T. Since M_{1} (I) is the largest ternary Γ-ideal contained in M_{o} (I), we have Hence,

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

1. M_{0} (I)=I_{2}.

2. M_{1} (I) is a semiprime ternary Γ-ideal of T containing I.

3. M_{2} (I)=I_{4}.

**Proof:** (1) M_{o} (I)=The set of all I-dominant elements

(2) Suppose that for some odd natural number n. Suppose, if possible M_{1} (I), < x > are the ternary Γ-ideals implies is a ternary Γ-ideal. Since M_{1} (I) is the largest ternary Γ-ideal in M_{0} (I), We have Hence, there exists an element y such that Now for some odd natural number It is a contradiction. Therefore, x ∈ M_{1} (I). Hence, M_{1} (I) is a semiprime ternary Γ-ideal of T containing I.

(3) Let x∈M_{2} (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 for some odd Therefore, for some odd n∈ N. So < x > is an I-dominant ternary Γ-ideal in T and hence, Therefore, Hence, It is natural to ask whether M_{1} (I)=I_{3}. 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 and But is a prime ternary Γ-ideal, let I, J, K are three ternary Γ-ideals of T such that implies all words containingor all words containing or all words containing or Therefore, is a prime ternary Γ-ideal. We have so Therefore, we can remark that the inclusions in 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 M_{0} (I)=M_{1} (I)=M_{2} (I).

**Proof:** Suppose I is a semi pseudo symmetric ternary Γ-ideal in a ternary Γ-semigroup T. By theorem 3.7, M_{0} (I)=I_{2} and M_{2} (I)=I_{4}. Also by theorem 2.10, we have I2=I_{4}. Hence, M_{0} (I)=M_{2} (I). By the theorem 3.5, We have Now let Therefore, Hence, Therefore,

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

**Proof: **Since x is a nontrivial I-dominant element, there exists an odd natural number n such that Since I is semi pseudo symmetric ternary Γ-ideal, we have If x is semi simple, then and hence, 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 M_{0} (I)=I, then I is a completely semiprime ternary Γ-ideal and I is a pseudo symmetric ternary Γ-ideal.

**Proof:** Let and Since Thus, there exists an odd natural number n such that 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=M_{0} (I).

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

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 is said to be a left I-divisor (a lateral I-divisor, right I-divisor) provided there exist two elements such that

**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: **R_{l} (I)=The union of all left I-divisor ternary Γ-ideals in T.

R_{r} (I)=The union of all right I-divisor ternary Γ-ideals in T.

R_{m} (I)=The union of all lateral I-divisor ternary Γ-ideals in T.

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

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

If n >1, then where

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

**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 and

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

Hence, 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. Then where y, z I is pseudo symmetric

is I-divisor ternary Γ-ideal

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 M_{0} (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)=M_{1} (I).

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

Let is an I-divisor is an I-dominant

Hence,

**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)=M_{0} (I).

**Proof: **Since I is a semi-pseudo symmetric ternary Γ-ideal, by theorem 3.8, M_{0} (I)=M_{1} (I)=M_{2} (I). If Tan M (I)-ternary Γ- semigroup, then by theorem 3.23, R (I)=M_{1} (I). Hence, R (I)=N_{0} (I). Conversely suppose that R (I)=M_{0} (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)=M_{0} (I).

**Proof: **Since every pseudo symmetric ternary Γ-ideal is a semi-pseudo symmetric ternary Γ-ideal, by theorem 3.24, R (I)=M_{0} (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 be any I-dominant element and y be any element in T\N.

Since N is a maximal ternary Γ-ideal,

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

Therefore 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 Therefore, there exists an odd natural number p such that for all

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

So So and hence, x ∈ sΓyΓt for some s, t ∈ T_{1}. Now since I is a pseudo symmetric ternary Γ-ideal,

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

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 M_{0} (I) ⊆N.

**Proof: **Suppose in Then by above theorem 3.27, N is trivial ternary Γ-ideal. It is a contradiction. Therefore, M_{0} (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 M_{0} (I) ⊆N.

**Proof: **By theorem 3.11, I is pseudo symmetric ternary Γ-ideal. If is I-dominant, then x cannot be semi simple. It is a contradiction. Therefore, M_{0} (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, M_{0} (I) ⊆ N.

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