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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]

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 TernaryΓ-semigroupif there exist a mapping from T × Γ × T × Γ × T to T which maps (x₁,α,x₂,β,x₃) → [x₁αx₂βx₃] 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 primeΓ提供理想的T x, y, z∈T和xΓyΓz⊆implies either x∈ A or y∈A or z∈ A. and A is said to be aprimeΓ-idealof 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 semiprimeΓ-ideal providedfor some odd natural number n>1 impliesSimilarly, 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 andfor 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 impliesfor 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 ternaryΓ-ideal of a ternaryΓ-semigroup T. Then the following are equivalent.

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

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

3) 1₁^1I=最小完全semiprime ternaryΓ-ideal of T relative to containing I.

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

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

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

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

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

Theorem 2.4:If I is a ternaryΓ-ideal of a semi simple ternaryΓ-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 thatA 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.

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

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). Letbelongs to at least one I-dominant ternary Γ-idealsis an I-dominant element. Hence, x∈M₀(I). Therefore,Clearly M₂(I) is a ternary Γ-ideal of T. Since M₁(I) is the largest ternary Γ-ideal contained in M_o(I), we haveHence,

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) M_o(I)=The set of all I-dominant elements

(2) Suppose thatfor some odd natural number n. Suppose, if possibleM₁(I), < x > are the ternary Γ-ideals impliesis a ternary Γ-ideal. Since M₁(I) is the largest ternary Γ-ideal in M₀(I), We haveHence, there exists an element y such thatNowfor some odd natural numberIt 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 thatfor some oddTherefore,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 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-idealandButis a prime ternary Γ-ideal, let I, J, K are three ternary Γ-ideals of T such thatimplies all words containingor all words containingor all words containingorTherefore,is a prime ternary Γ-ideal. We havesoTherefore, we can remark that the inclusions inmay 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)=I₂and M₂(I)=I₄. Also by theorem 2.10, we have I2=I₄. Hence, M₀(I)=M₂(I). By the theorem 3.5,We haveNow letTherefore,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不能简单半(12,13].

Proof:Since x is a nontrivial I-dominant element, there exists an odd natural number n such thatSince I is semi pseudo symmetric ternary Γ-ideal, we haveIf x is semi simple, thenand 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 M₀(I)=I, then I is a completely semiprime ternary Γ-ideal and I is a pseudo symmetric ternary Γ-ideal.

Proof:LetandSinceThus, there exists an odd natural number n such thatTherefore, 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,LetIfthen 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 elementis said to be a left I-divisor (a lateral I-divisor, right I-divisor) provided there exist two elementssuch 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:LetSincewe havefor some odd natural number n. Let n be the least odd natural number such thatIf n=1 then x∈I and hence,

If n >1, thenwhere

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. Thusand

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

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. Thenwhere y, zI 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 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,

Letis an I-divisoris 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)=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:因为每一个伪对称三进制Γ理想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.

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

Since N is a maximal ternary Γ-ideal,

Sincewe haveLet n be the least positive odd integer such thatSince I is a pseudo symmetric ternary Γ-ideal then I is a semipseudo symmetric ternary Γ-ideal and hence,

Thereforeand 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 thatTherefore, there exists an odd natural number p such thatfor all

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

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

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

这是不正确的。在这两种情况下我们反对tradiction. 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 inThen 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. Ifis 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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