British Journal of Research Open Access

Keywords

Prime, Strongly prime, Semi prime, Irreducible and strongly irreducible bi-ideals in ternary semi rings.

Introduction

Algebraic structures play a very outstanding role in mathematics with classification in multifarious disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, topological areas etc. Ternary generalizations of algebraic structures are the very natural ways for further development and in depth comprehension of their basic traits.

Cayley for the first time pioneered and launched first ternary algebraic operations in the way back in the 19th century. Cayley’s ideas expounded and developed nary generalizations of matrices and their determinants [9, 14] and general theory of n-ary algebras [3, 10] and ternary rings [11] (For physical apllications in Nambu mechanics, super symmetry, Yang-Baxter equations etc). Ternary structures and their generalizations create some hopes because of their possibility of applications in physics. A few important physical applications are listed in [1,2,6,7]. In pursuance of Lister’s generalizations of ternary rings introduced in 1971, T. K. Dutta and S. Kar came up with the notion of ternary semirings.

T. K. Dutta and S. Kar initiated prime ideals and prime radical of ternary semirings in [4]. The same researchers launched semiprime ideals and irreducible ideals of ternary semirings in [5]. Furthermore S. Kar in [8] came up with the notion of quasiideals and bi-ideals in ternary semi rings. Similarly, M. Shabir and M. Bano floated prime bi-ideals in ternary semi groups in [13]. In the opening section of this paper, we assemble requisite material on prime, strongly prime and semi prime bi-ternary Γ- ideals in ternary Γ-semi rings. Deploying the taxonomic order, we define irreducible and strongly irreducible bi-ternary Γ-ideals in ternary Γ-semi rings and specify some classes of ternary Γ-semi rings by the characteristics of these ternary Γ-ideals.

Preliminaries

Definition 1

Let T and Γ be two additive commutative semi groups. T is said to be a if there exist a mapping from T ×Γ× T ×Γ× T to T which maps satisfying the conditions :

Definition 2

An element 0of a ternary Γ-semiring T is said to be an absorbing zero of T provided 0 + x = x = x + 0 and

Note 3

Throughout this paper, T will always denote a ternary Γ-semiring with zero and unless otherwise stated a ternary Γ-semiring means a ternary Γ-semiring with zero.

Definition 4

A ternary Γ-semiring T is said to be

Note 5

For the convenience we write

Note 6

Let T be a ternary semiring. If A, B and C are three subsets of T, we shall denote the set AΓBΓC =

Note 7

Let T be a ternary semiring. If A, B are two subsets of T, we shall denote the set

Definition 8

Let T be ternary Γ-semiring. A non empty subset ‘S’ is said to be a of T if S is an additive subsemigroup of T and for all

Note 9

A non empty subset S of a ternary Γ- semiring T is a ternary sub Γ-semiring if and only if

Definition 10

A nonempty subset A of a ternary Γ- semiring T is said to be

Note 11

A nonempty subset A of a ternary Γ- semiring T is a left ternary Γ-ideal of T if and only if A is additive subsemi group of T and

Definition 12

A nonempty subset of a ternary Γ- semiring T is said to be a

Note 13

A nonempty subset of A of a ternary Γ-semiring T is a lateral ternary Γ-ideal of T if and only if A is additive subsemi group of

Definition 14

A nonempty subset A of a ternary Γ- semiring T is a

Note 15

A nonempty subset A of a ternary Γ- semiring T is a right ternary Γ-ideal of T if and only if A is additive subsemi group of T

Definition 16

A nonempty subset A of a ternary Γ- semiring T is said to be ternary

Note 17

A nonempty subset A of a ternary Γ- semiring T is a ternary Γ-ideal of T if and only if it is left ternary Γ-ideal, lateral ternary Γ-ideal and right ternary Γ-ideal of T.

Definition 18

An element a of a ternary Γ- semiring. T is said to be regular if there

Definition 19

An element a of a ternary Γ-semiring T is said to be an α-idempotent element provided

Definition 20

A ternary Γ-semiring T is called α- idempotent ternary Γ-semiring if every element of T is α-idempotent.

Definition 21

An element a of a ternary Γ-semiring T is said to be an element provided

Definition 22

A ternary Γ-subsemiring B of a ternary Γ-semiring T is called a bi-ternary

Theorem 22

The intersection of any family of

Proof

Let be any family of prime bi-ternary Γ-ideals of a ternary Γ-semiring T. Let I.As each B_i is a bi-ternary Γ-ideal of T, we have x + y∈ B_i for all i∈ I implies x + z∈ B_i for all i∈ I.As each B_i is a bi-ternary Γ-ideal of T, we have implies

Theorem 23

Every left (right, lateral) ternary Γ-ideal of a ternary Γ-semiring A is a biternary Γ-ideal of A.

Theorem 24

If B is a bi-ternary Γ-ideal of a regular ternary Γ-semiring T and X, Y, be any non-empty subsets of T, then B

Corollary 25

If B₁,B₂,B₃ are any bi-ternary Γ- ideals of a regular ternary Γ-semiring T, then is a bi-ternary Γ-ideal of T.

PRIME, STRONGLY PRIME AND SEMI PRIME BI-TERNARY

Definition 1

A bi-ternary Γ-ideal B of a ternary Γ- semiring T is said to be a prime bi-ternary if B₁ΓB₂ΓB₃⊆ B implies B₁⊆ B or B₂⊆ B or B₃⊆ B for any bi-ternary Γ-ideals B₁,B₂,B₃ of T.

Definition 2

A bi-ternary Γ-ideal B of a ternary Γ- semiring T is said to be a strongly prime bi of T if B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆ B implies B₁⊆ B or B₂⊆ B or B₃⊆B for any bi-ternary Γ- ideals B₁,B₂,B₃ of T.

Definition 3

A bi-ternary Γ-ideal B of a ternary Γ- semiring T is said to be a semi prime bi ternaryΓ-ideal of T if implies B₁⊆B for any bi-ternary Γ-ideal B₁ of T.

Proposition 4

Every strongly prime bi-ternary Γ- ideal of a ternary Γ-semiring T is a prime bi-ternary Γ-ideal of T.

Proof

Let B be a strongly prime bi-ternary Γ-ideal of a ternary Γ-semiring A. Let B₁,B₂,B₃ be three bi-ternary Γ-ideals of T such that B₁ΓB₂ΓB₃⊆ B. This implies B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆ B. Thus B₁⊆ B or B₂⊆ B or B₃⊆ B. Hence B is a prime bi-ternary Γ-ideal of T.

Theorem 5

Every prime bi-ternary Γ-ideal of a ternary Γ-semiring T is a semi prime biternary Γ-ideal of T.

Proof

Let B be a prime bi-ternary Γ-ideal of a ternary Γ-semiring A. Now let B₁ be any bi-ternary Γ-ideal of T such that Then B₁⊆B.Thus B is a semi prime bi-ternary Γ-ideal of A.

Note 6

A prime bi-ternary Γ-ideal of a ternary Γ-semiring is not necessarily a strongly prime bi-ternary Γ-ideal and a semi prime bi-ternary Γ-ideal of a ternary Γ- semiring is not necessarily a prime biternary Γ-ideal. This fact is clear from the following examples:

Example 7

Let A Define addition and ternary multiplication on T as X+Y = XΔY = (X ∪Y)−(X ∩Y) and (XΓY)ΓZ = X∩Y∩Z for all X,Y,Z ∈ T. Then T isaternaryΓ- semiring. Bi-ideals of Tare {Φ}, {Φ,{a}}, {Φ,{b}} and {Φ,{a},{b},{a,b}}. Since (XΓ)²X = X for all X ∈ T, so each bi-ternary Γ-ideal of T is semi prime. In particular {Φ} is a semi prime bi-ternary Γ-ideal of T but not a prime bi-ternary Γ-ideal of T, because {Φ,{a}}Γ{Φ,{b}}Γ{Φ,{a},{b},{a,b}} = {Φ,{a}}∩{Φ,{b}}∩{Φ,{a},{b},{a, b}} = {Φ}⊆{Φ}.

But none of {Φ,{a}}, {Φ,{b}} and {Φ,{a},{b},{a,b}} is contained in {Φ}.

Theorem 8

The intersection of any family of prime bi-ternary Γ-ideals of a ternary Γ- semiring T is a semi prime bi-ternary Γ- ideal of T.

Proof

Let{B_i :i∈I}be any family of prime bi-ternary Γ-ideals of a ternary Γ-semiring T. We have to show that is a semi prime bi-ternary Γ-ideal of T. By theorem 2.22, is a bi-ternary Γ-ideal of T. Now let B be any bi-ternary Γ-ideal of T such that implies BΓBΓB = (BΓ)²B⊆B_i for all i∈I. Thus B ⊆Bi for all i∈I, because each Bi is a prime bi-ternary Γ- ideal of T. This implies Hence is a semi prime bi-ternary Γ-ideal of T.

IRREDUCIBLE AND STRONGLY IRREDUCIBLE BI-TERNARY

Definition 1

A bi-ternary Γ-ideal B of a ternary Γ- semiring T is said to be an irreducible bi ternaryΓ-ideal of T if B₁∩B₂∩B₃ = B implies either B₁ = B or B₂ = B or B₃ = B for any bi-ternary Γ-ideals B₁,B₂,B₃ of T.

Definition 2

bi-ternary Γ-ideal B of a ternary Γ- semiring T is said to be a strongly irreducible bi-ternary Γ-ideal of T if B₁∩B₂∩B₃⊆ B implies either B₁⊆ B or B₂ ⊆ B or B₃ ⊆ B for any bi-ternary Γ- idealsB₁,B₂,B₃ of T.

Theorem 3

Every strongly irreducible semi prime bi-ternary of a ternary T is a strongly prime biternary

Proof

Let B be a strongly irreducible semi prime bi-ternary Γ-ideal of a ternary Γ- semiring T. Let B₁,B₂,B₃ be three bi-ternary Γ-ideals of T such that

B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆ B →(i)

Then we have to show that either B₁⊆B or B₂⊆B or B₃⊆B. As B₁∩B₂∩B₃⊆ B₁,

B₁∩B₂∩B₃⊆ B₂ and B₁∩B₂∩B₃⊆ B₃ implies [(B₁∩B₂∩B₃)Γ]²(B₁∩B₂∩B₃)⊆B₁Γ B₂ΓB₃, [(B₁∩B₂∩B₃)Γ]²(B₁∩B₂∩B₃)⊆B₂Γ B₃ΓB₁ and [(B₁∩B₂∩B₃)Γ]²(B₁∩B₂∩B₃)⊆ B₃ΓB₁ΓB₂.

Thus [(B₁∩B₂∩B₃)Γ]²(B₁∩B₂∩B₃) ⊆B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆ B (by using (i)).

This implies B₁∩B₂∩B₃⊆B, because B is a semi prime bi-ternary Γ-ideal of T.

Thus B₁⊆B or B₂⊆B or B₃⊆B, because B is a strongly irreducible bi-ternary Γ-ideal of T. Hence B is a strongly prime biternary Γ-ideal of A.

Theorem 4

Let B be a bi-ternary Γ-ideal of a ternary Γ-semiring T and a∈ T such that a B. Then there exists an irreducible biternary Γ-ideal I of T such that B ⊆ I and aI.

Proof

Let X be the collection of all biternary Γ-ideals of T which contain B but do not contain a, that is X = {Y_i: Y_i is a biternary Γ-ideal of T, B ⊆Y_i and a Y_i}. Then X is non-empty as B ∈ X. The collection X is a partially ordered set under inclusion. If {Y_i: i∈I} is any totally ordered subset (chain) of X, then is also a biternary Γ-ideal of T containing B and a Y. So Y is an upper bound of {Y_i: i∈I}. Thus every chain in X has an upper bound in X. Hence by Zorn’s lemma, there exists a maximal element I (say) in X. This implies B ⊆I and a I. Now we show that I is an irreducible bi-ternary Γ-ideal of T. For this let C, D and E be three bi-ternary Γ-ideals of T such that I = C∩D∩E. If C, D and E properly contain I, then a∈C, a∈D and a∈E. Thus a∈C∩D∩E = I. Which is a contradiction to the fact that a I. So either I = C or I = D or I = E. Hence I is an irreducible bi-ternary Γ-ideal of T.

Theorem 5

For a ternary T, the following assertions are equivalent:

(1) Every bi-ternary of T is idempotent.

(2) B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂ = B₁∩B₂∩B₃ for any bi-ternary

(3) Each bi-ternary of T is semiprime.

(4) Each proper bi-ternary of T is the intersection of irreducible semiprime bi-ternary

Proof

(1) ⇒ (2) Let B₁,B₂,B₃ be any three bi-ternary Γ-ideals of T. Then B₁∩B₂∩B₃ is also a bi-ternary Γ-ideal of T, by theorem 22. By hypothesis, we have

B₁∩ B₂∩B₃= [(B₁∩B₂∩B₃)Γ]²(B₁∩B₂∩B₃) = (B₁∩B₂∩B₃)Γ(B₁∩B₂∩B₃)Γ(B₁∩B₂∩B₃)⊆B ₁ΓB₂ΓB₃.

Similarly

B₁∩B₂∩B₃⊆ B₂ΓB₃ΓB₁ and B₁∩B₂ ∩B₃⊆ B₃ΓB₁ΓB₂.

So B₁∩B₂∩B₃⊆ B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂.Now B₁ΓB₂ΓB₃,B₂ΓB₃ΓB₁ andB₃ΓB₁ΓB₂, being the products of three bi-ternary Γ-ideals of T, are bi-ternary Γ- ideals of T, by Corollary 2.25. Also B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂ is a biternary Γ-ideal of T, by theorem 2.22. Then by hypothesis

B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂ = [(B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂)Γ]² (B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂) ⊆(B₁Γ B₂ΓB₃)Γ(B₃ΓB₁ΓB₂)Γ(B₂ΓB₃ΓB₁)⊆(B₁ΓTΓ T)Γ(TΓB₁ΓT)Γ(TΓTΓB₁) = B₁Γ(TΓTΓT) ΓB₁Γ(TΓTΓT)ΓB₁ = B₁ΓTΓB₁ΓTΓB₁⊆ B₁

Similarly

B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂⊆B₂ and B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂⊆B₃.Thus B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂⊆B₁∩B₂ ∩B₃.

Hence

B₁ΓB₂ΓB₃∩B₂ΓB₃ΓB₁∩B₃ΓB₁ΓB₂= B₁∩B₂ ∩B₃.

(2) ⇒ (1) Let B be a bi-ternary Γ- ideal of T.

Then by hypothesis B = B∩B∩B =BΓBΓB ∩BΓBΓB∩BΓBΓB = BΓBΓB.

(1) ⇒ (3) Let B be a bi-ternary Γ- ideal of T such that for any biternary Γ-ideal B₁ of T. Then by hypothesis, we have B₁ = Hence every biternary Γ-ideal of T is a semiprime biternary Γ-ideal of T

(3) ⇒ (4) Let each bi-ternary Γ-ideal of T is semiprime. Now let B be a proper biternary Γ-ideal of T. If is the intersection of all bi-ternary Γ-ideals of T containing B, then B ⊆. If this inclusion is proper, then there exists such that

This implies then by Theorem 4.4, there exists an irreducible bi-ternary Γ-ideal I (say) of T such that B ⊆ I and Which is a contradiction to the fact that for all α. So By hypothesis, each bi-ternary Γ-ideal of T is semiprime. Thus each proper bi-ternary Γ-ideal of T is the intersection of irreducible semiprime bi-ternary Γ-ideals of T which contain it.

(4)⇒(1) Let each proper bi-ternary Γ-ideal of T is the intersection of irreducible semi prime bi-ternary Γ-ideals of T which contain it. Now if B is a bi-ternary Γ-ideal of T, then is also a bi-ternary Γ-ideal of T, by Corollary 2.25. If = T (improper bi-ternary Γ-ideal), then A This implies

So for each bi-ternary Γ-ideal B of T. Now if is a proper bi-ternary Γ-ideal of T, that is then is an irreducible semiprime bi-ternary Γ-ideal of T such that for all α}. This implies B ⊆ Bα for all α, because each Bα is a semi prime biternary Γ-ideal of T. Thus B ⊆∩Bα=

Also as B is closed under multiplication. Hence for each bi-ternary Γ-ideal B of T.

Theorem 6

If each bi-ternary Γ-ideal of a ternary Γ-semiring T is idempotent then a bi-ternary Γ-ideal B of T is strongly irreducible if and only if B is strongly prime.

Proof

Let B be a strongly irreducible biternary Γ-ideal of T. Then we have to show that B is a strongly prime bi-ternary Γ-ideal of T. For this let B₁, B₂, B₃ be any three biternary Γ-ideals of T such that B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆B. By Theorem 4.5, we have B₁ ∩B₂ ∩B₃ = B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆B. But B is a strongly irreducible bi-ternary Γ-ideal of T. Thus we have B₁⊆ B or B₂⊆ B or B₃⊆ B. Hence B is a strongly prime bi-ternary Γ- ideal of T.

Conversely suppose that B is a strongly prime bi-ternary Γ-ideal of T. To show that B is strongly irreducible bi-ternary Γ-ideal of T, let B₁, B₂, B₃ be any bi-ternary Γ-ideals of T such that B₁ ∩ B₂ ∩ B₃⊆ B. By Theorem 4.5, we have B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂ = B₁ ∩ B₂ ∩ B₃⊆ B. But B is a strongly prime bi-ternary Γ- ideal of T, we have B₁⊆ B or B₂⊆ B or B₃⊆ B. Hence B is a strongly irreducible bi-ternary Γ-ideal of T.

Next we characterize those ternary Γ-semi rings in which each bi-ternary Γ- ideal is strongly prime and also those ternary Γ-semi rings in which each bi-ternary Γ- ideal is strongly irreducible.

Theorem 7

Each bi-ternary Γ-ideal of a ternary Γ-semiring T is strongly prime if and only if each bi-ternary Γ-ideal of T is Γ-idempotent and the set of bi-ternary Γ-ideal of T is totally ordered by inclusion.

Proof

Suppose that each bi-ternary Γ-ideal of T is strongly prime. This implies that each bi-ternary Γ-ideal of T is semiprime. Thus by Theorem 4.5, each bi- ternary Γ- ideal of T is idempotent. Now we show that the set of bi-ternary Γ-ideals of T is totally ordered by inclusion. For this, let B₁, B₂ be two bi-ternary Γ-ideals of T. Then by Theorem 4.5, we have

B₁∩B₂ = B₁∩B₂∩T = B₁ΓB₂ΓT∩B₂Γ TΓB₁∩AΓB₁ΓB₂, ⇒ B₁ΓB₂ΓT∩B₂ΓTΓB₁∩ AΓB₁ΓB₂⊆B₁∩B₂.

By hypothesis, B₁ and B₂ are strongly prime bi-ternary Γ-ideals of T, so is B₁∩B₂. Then B₁⊆B₁∩B₂ or B₂⊆B₁∩B₂ or T⊆B₁∩B₂.Thus B₁⊆B₂ or B₂⊆B₁.Hence the set of bi-ternary Γ-ideals of T is totally ordered by inclusion.

Conversely, assume that each biternary Γ-ideal of T is Γ-idempotent and the set of bi-ternary Γ-ideals of T is totally ordered by inclusion. We have to show that each bi-ternary Γ-ideal of T is strongly prime. For this, let B be an arbitrary biternary Γ-ideal of T and B₁, B₂, B₃ be any bi-ternary Γ-ideals of T such that B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆B. By Theorem 4.5, we have B₁∩B₂∩B₃ = B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆B →(i).

Since the set of bi-ternary Γ-ideals of T is totally ordered by inclusion, so for B₁, B₂, B₃ we have the following six possibilities:

(ii) B₁⊆B₂⊆B₃ (iii) B₁⊆B₃⊆B₂

(iv) B₂⊆B₃⊆B₁ (v) B₂⊆B₁⊆B₃

(vi) B₃⊆B₁⊆B₂ (vii) B₃⊆B₂⊆B₁.

In these cases we have

(ii) B₁∩B₂∩B₃ = B₁ (iii) B₁∩B₂∩B₃ = B₁

(iv) B₁∩B₂∩B₃ = B₂ (v) B₁∩B₂∩B₃ = B₂

(vi) B₁∩B₂∩B₃ = B₃ (vii) B₁∩B₂∩B₃ = B₃.

Thus (i) gives, either B₁⊆B or B₂⊆B or B₃⊆B. Hence B is strongly prime.

Theorem 8

If the set of bi-ternary Γ-ideal of a ternary Γ-semiring T is totally ordered, then each bi-ternary Γ-ideal of T is Γ-idempotent if and only if each bi-ternary Γ-ideal of T is prime.

Proof

Suppose each bi-ternary Γ-ideal of T is Γ-idempotent and B is an arbitrary biternary Γ-ideal of T and B₁, B₂, B₃ be any bi-ternary Γ-ideals of T such that B₁ΓB₂ΓB₃⊆B. As the set of bi-ternary Γ- ideals of T is totally ordered, then for B₁, B₂, B₃ we have the following six possibilities:

(i) B₁⊆B₂⊆B₃ (ii) B₁⊆B₃⊆B₂

(iii) B₂⊆B₃⊆B₁ (iv) B₂⊆B₁⊆B₃

(v) B₃⊆B₁⊆B₂ (vi) B₃⊆B₂⊆B₁.

For (i) and (ii), we have B₁ΓB₁ΓB₁⊆ B₁ΓB₂ΓB₃⊆B, implies B₁⊆B, as B is Γ-idempotent. Similarly for other possibilities we have B₂⊆B or B₃⊆B.

Conversely, suppose that each biternary Γ-ideal of T is prime, so is semiprime, by Theorem 3.5. Thus by Theorem 4.5, each bi-ternary Γ-ideal of T is Γ-idempotent.

Theorem 9

If the set of bi-ternary Γ-ideal of a ternary Γ-semiring T is totally ordered, then the concepts of primeness and strongly primeness coincide.

Proof

Let B be a prime bi-ternary Γ-ideal of T. To show that B is a strongly prime biternary Γ-ideal of T, let B₁, B₂, B₃ be any biternary Γ-ideals of T such that

B₁ΓB₂ΓB₃ ∩B₂ΓB₃ΓB₁ ∩B₃ΓB₁ΓB₂⊆B.

As the set of bi-ternary Γ-ideals of ternary Γ-semiring T is totally ordered, then for B₁, B₂, B₃ we have the following six possibilities:

(i) B₁⊆B₂⊆B₃ (ii) B₁⊆B₃⊆B₂

(iii) B₂⊆B₃⊆B₁ (iv) B₂⊆B₁⊆B₃

(v) B₃⊆B₁⊆B₂ (vi) B₃⊆B₂⊆B₁.

For (i) and (ii), we have

Implies B₁⊆ B, as B is a prime biternary Γ-ideal of T. Similarly for other possibilities we have B₂⊆ B or B₃⊆ B. This shows that B is a strongly prime bi-ternary Γ-ideal of T. Thus every prime bi-ternary Γ- ideal of T is a strongly prime bi-ternary Γ- ideal of T. Now let B be a strongly prime biternary Γ-ideal of T. To show that B is a prime bi-ternary Γ-ideal of T, let B₁, B₂, B₃ be any bi-ternary Γ-ideals of T such that B₁ΓB₂ΓB₃⊆B.Implies B₁ΓB₂ΓB₃ ∩B₂ΓB₃ ΓB₁ ∩B₃ΓB₁ΓB₂⊆B. Implies either B₁⊆ B or B₂⊆ B or B₃⊆ B, as B is a strongly prime bi-ternary Γ-ideal of T. This shows that B is a prime bi-ternary Γ-ideal of T. Thus every strongly prime bi-ternary Γ-ideal of T is a prime bi-ternary Γ-ideal of T.

Theorem 10

For a ternary Γ-semiring T, the following assertions are equivalent:

(1) The set of bi-ternary Γ-ideals of T is totally ordered by set inclusion.

(2) Each bi-ternary Γ-ideal of T is strongly irreducible.

(3) Each bi-ternary Γ-ideal of T is irreducible.

Proof

(1) ⇒(2) Let the set of bi-ternary Γ- ideals of Tis totally ordered by set inclusion. To show that each bi-ternary Γ-ideal of T is strongly irreducible, let B be an arbitrary biternary Γ-ideal of T and B₁, B₂, B₃ be any biternary Γ-ideals of T such that Since the set of bi-ideals of T is totally ordered by set inclusion, then B₁ or B₂ or B₃. Thus either B₁⊆ B or B₂⊆ B or B₃⊆ B. So B is strongly irreducible. Hence each bi-ternary Γ-ideal of T is strongly irreducible.

(2) ⇒ (3) Let each bi-ternary Γ-ideal of T is strongly irreducible. To show that each bi-ternary Γ-ideal of T is irreducible, let B be an arbitrary bi-ternary Γ-ideal of T and B₁, B₂, B₃ be any bi-ternary Γ-ideals of T such that . This implies B ⊆ B₁, B ⊆ B₂ and B ⊆ B₃. On the other hand, by hypothesis we have B₁⊆ B or B₂⊆ B or B₃⊆ B. Hence eitherB₁ = B or B₂ = B or B₃ = B. Thus B is an irreducible bi-ternary Γ-ideal of T. Hence each bi-ternary Γ-ideal of T is irreducible.

(3) ⇒ (1) Let each bi-ternary Γ-ideal of Tis irreducible. To show that the set of biternary Γ-ideals of T is totally ordered by set inclusion, let B₁, B₂ be any two bi-ternary Γ- ideals of T. Then by Theorem 22, B₁ B₂ is also a bi-ternary Γ-ideal of Tand so is irreducible bi-ternary Γ-ideal of T. Since B₁B₂T= B₁B₂, implies B₁ = B₁B₂ or B₂ = B₁B₂ or T= B₁B₂, implies either B₁⊆ B₂ or B₂⊆ B₁ or B₁ = B₂ = T. Hence the set of bi-ternary Γ-ideals of T is totally ordered by set inclusion.

Conclusion

In this paper mainly we start the study of prime Bi-ternary Γ-ideals, irreducible bi-ternary Γ-ideals in ternary Γ- semi rings. We characterize them and results in this paper may apply to many algebraic structures for further research.

Acknowledgements

This research is supported by the Department of Mathematics, VSR & NVR College, Tenali, Guntur (Dt), Andhra Pradesh, India.

The first and third authors express their warmest thanks to the University Grants Commission (UGC), India, for doing this research under Faculty Development Programme.

The authors would like to thank the experts who have contributed towards preparation and development of the paper and the authors also wish to express their sincere thanks to the referees for the valuable suggestions which lead to an improvement of this paper.

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