Abstract

In this study, researchers have studied the Γ-algebraic structures and some characteristics of them. According to Sen and Saha, we defined algebraic structures: Γ-semigroup, Γ-regular semigroup, Γ-idempotent semigroup, Γ-invers semigroup and Γ-group. Theorem 2, 3 and 4 proves the existence of Γ-group and gives necessary and sufficient conditions where one Γ-semigroup is a Γ-group. Finally, theorem 5 shows necessary and sufficient conditions where one Γ-regular semigroup is a Γ-group. In addition, for every Γ- algebraic structure that we mentioned before we give an original example.

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INTRODUCTION

Idea for Γ-algebraic structures follows from Γ-ring which notion has been introduced by Nobusawa (1964). Theory of Γ-semigroup is expanded in a natural way while as foundation we take in the theory of semigroups.

Definition 1: Let:

S = {a, b, c,...}

and

Γ = {α, β, γ,...}

two nonempty sets. A mapping f: SxΓxS→S or f: (a, α, b)→c; a, b, c ε S; α ε Γ called ternary operation in S and Γ. This operation we denote by (.)_Γ or by (+)_Γ. The element (a, α, b) we denote simple by aαb. Operation f is commutative if ∀a, b ε S, α ε Γ, satisfies condition:

aαb = bαa

Operation f is associative if it satisfies condition:

(aαb)βc = aα (bβc); ∀a, b, c ε S, α, β ε Γ

Definition 2: Let:

S = {a, b, c,...}

and

Γ = {α, β, γ,...}

two non-empty sets. Order pair (S, (.)_Γ) is called Γ-algebraic structure.

Definition 3: A Γ-algebraic structure (S, (.)_Γ) is called Γ-groupoid if it satisfies condition:

(i)	∀a, b ε S, α, β ε Γ⇒ααb ε S

Example 1: Let:

S = {a = 4z + 3, z ε Z} = {...-13, -9, -5, -1, 3, 7, 11, 15,...}

Γ = {α = 4z + 1, z ε Z} = {...-11, -7, -3, 1, 5, 9, 13,...}

two sets. If a = 4z + 3, b = 4z + 3 and α = 4z₃ + 1 where a, b ε S and α ε Γ. Then:

Therefore (S, (+)_Γ is Γ-groupoid where operation (+)_r or aαb is addition of integers.

MATERIALS AND METHODS

In definition 4, we defined Γ-semigroup by Γ-algebraic structure and examined few examples by using some of their characteristics. Next, briefly present definition of Γ-subsemigroup (Saha, 1987), also definition of ideal in Γ-semigroup (Saha, 1988).

Definition 4: A Γ-algebraic structure (S (.)_Γ is called Γ- semigroup if it satisfies condition:

(i)	∀a, b ε S, α ε Γ ⇒ aαb ε S

 

(ii)	∀a, b, c ε S; α, β ε Γ (aαb) βc = aα (bβc)

 

Γ-semigroup we can definite also in this way: A Γ-groupoid (S, (.)_Γ satisfying the associative law is a Γ-semigroup:

(aαb)βc = aα (bβc); (∀a, b, c ε S; α, β ε Γ

Example 2: Let S be the set of all integers of the form 6z + 1:

and Γ be the set of all integers of the form 6z+5:

Then order pair (S, (+)_Γ) is a Γ-semigroup where aαb denotes the addition of integers.

Solution
Closure property: If

and

then:

therefore, pair (S, (+)_Γ is a grupoid.

Associative property: If

and

then:

From the other side:

Consequently,

We now prove that the pair (S, (+)_Γ is a Γ-semigroup.

Example 3: Let S be the set of all matrices of type:

and Γ be the set of all matrices:

We can prove that the system (S, (.)_Γ is Γ-semigroup, where operation (.)_Γ is the product of the matrices.

Solution
Closure property: If

and

then:

where,

and

Associative property is true because the product of the matrices is associative and relation is evident:

since the algebraic system (S, (.)_Γ is Γ-semigroup.

Semigoup S can be considered always like Γ- semigroup if operation in S expands in S∪{1} where 11 = 1 and 1a = a1 = a, (1≠a), ∀a ε S. Hence, S∪{1} is semigroup with identity element. If we define ab = a1b and Γ = {1} in this case S is Γ-semigroup. Therefore, principal results of the theory of semigroups can be expanded in the theory of a Γ-semigroup.

Lemma 1: If S is a semigroup, Γ = {1} and ab = a1b then S is Γ-semigroup.

Proof: From definition 4, it is evident.

Definition 5: Let S be a Γ-semigroup. Subset M of S is Γ-subsemigroup of Γ-semigroup S if MΓM⊆M, where:

Other alternative way of defining Γ-subsemigroup M of semigroup S can be found by Saha (1987).

Example 4: Let S = [0. 1] and

Then S is Γ-semigroup. But subset:

is Γ- subsemigroup of Γ-semigroup S. We can see that:

Determination of ideal in Γ-semigroup S, although not according to the definition provided by Saha, the idea and some characteristics were taken from Saha (1988).

Definition 6: A left (right) ideal of a Γ-semigroup S is non-empty subset I of S (I⊂S, I≠φ) such that SΓI⊂I (IΓS⊂I). If I is both a left and right ideal, then we say that I is an ideal of S. Let Q non-empty set of S.Q is quasi-ideal of Γ-semigroup S if QΓS∩SΓQ⊆Q.

Definition 7: A Γ-semigroup S is left (right) simple if it has no proper left (right) ideal. A Γ-semigroup S is said to be simple if it has no proper ideal.

Definition 8: An element aεS is said to be a regular in the Γ-semigroup S if aεaΓSΓa, where:

A Γ-semigroup S is said to be regular if every element of S is regular.

Example 5: Let A = {1, 2, 3} and B = {4, 5} be two non-empty sets. Let S = {f, g, h} Γ = {α, β, γ, δ, θ, φ}, where f, g, h are maping from the set A to the set B and α, β, γ, δ, θ, φ are maping from the set B to the set A. They are defined by:

and:

Where, 1f = 4, 2f = 4, 3f = 4 and similarly others. We can see that:

For example: if 1εA∧fεS, we have 1f = 4. From the other side:

We proof fαfδf = f. Therefore, S is regular Γ- semigroup. We can give another definition about regular Γ-semigroups.

Definition 9: A Γ-semigroup S is called regular Γ- semigroup if for any aεS exists bεS, α, βεΓ such that a = (aαb) βa.

Example 6: Let S be the set of all 2x3 matrices over the field and Γ be the set of all 3x2 matrices over the same field:

then S is regular Γ-semigroup where AαB (A, BεS, αεΓ) denote the product of the matrices. Indeed for AεS, we can chose αεΓ such that:

Definition 10: An element eεS is said to be an idempotent of Γ-semigroup S, if eαe = e for some αεΓ. In this case, we call e an α-idempotent. S is a idempotent Γ-semigroup if and only if every element of S is idempotent.

Example 7: Let

Then S is a idempotent Γ-semigroup.

Definition 11: Let S be a Γ-semigroup and aεS. If for bεS exists α, βεΓ such that a = (aαb)βa and b = (bβa) αb then b is called an (α, β) invers of a. In this case researcher write:

Proposition 1: If S is a regular Γ-semigroup and aεS, then for some α, βεΓ.

Proof: If S is a regular Γ-semigroup, then for some aεS exists bεS, α, βεΓ such that a = (aαb)βa. For element x = bβaαbεS, we can prove that . Indeed:

and

Then:

Definition 12: A Γ-semigroup S is called a inverse Γ- semigroup if every element a of S has a unique (α, β)- inverse, whenever (α, β)-inverse of a exists.

Proposition 2: A regular Γ-semigroup S is a inverse Γ- semigroup if for all aεS and for all α, βεΓ.

Proof: It is evident from definition 12.

Theorem 1: Let S be a inverse Γ-semigroup and aεS, α, βεΓ. If then aαa^-1 is β-idempotent and a^-1βa is α-idempotent of S.

Proof: If S is a inverse Γ-semigroup, then for aεS and α, βεΓ exists unique inverse element such that a = aαa^-1 βa and a^-1 = a^-1 βaαa^-1. Element (aαa^-1)εS is a β-idempotent:

and element a^-1βa is a α-idempotent i S:

RESULTS AND DISCUSSION

Initially we create semigroup S_α (Chinram and Siammai, 2009). In Theorem 3 and 4, researchers proof necessary and sufficient condition where semigroup G_α is a Γ-group. Furthermore, we have determined necessary and sufficient condition where regular and inverse semigroup G_α is a Γ-group. Now we will create the semigroup S_α.

Let S be a Γ-semigroup and α be a fixed element of Γ. If a, bεS, define operation B in S by, aBb = aαb, ∀a, bεS. Then, S is a semigroup.

Proof:

(i)	∀a, bεS; αεΓ ⇒ aBb = aαbεS, α-fixed element of Γ

(ii)	∀a, b, cεS, αεΓ we have a

On the other hand:

Since:

We denote this semigroup by S_α.

Example 8: If

is fixed element of Γ. Then S_α is semigroup, where operation is the product of matrices. We took definition of Γ-group from Sen and Saha (1986) which in a free form appears as:

Definition 13: A Γ-semigroup G is called a Γ-group if G_α is group for some αεΓ (Sen and Saha, 1986).

Example 9: If S = {6z + 1: zε} and Γ = {5} from example 2. We can see that S₅ is semigroup:

(i)

 

(ii)

 

 

 

Consequently, S₅ is Γ-semigroup and denoted by G₅. We can see that G₅ is Γ-group. Identity element in G₅ is -5εG₅:

Consequently, G₅ is Γ-group. Similarly, we can see that G_α is Γ-group for all αεΓ.

Theorem 2: G_α is a Γ-group if and only if G is simple Γ- semigroup .

Proof: Suppose G is simple, then G is right simple and left simple. Let aεG, consider the set aαG. We can show that aαG is a right ideal in G. Since G is right simple we have aαG = G. Similarly, we can show that Gαa = G. Hence, aBG = G and GBa = G for any aεG. Then from the known (Ljapin, 1960) result it follows that G_α is a Γ-group.

Conversely assume that G_α is a group and e_α be the identity element in G_α. Let I be a left ideal in the Γ- semigroup G and aεI. Then there exists bεG such that bBa = e_α. Hence, e_α = bBa = bαa = εI. Let cεG. Then c = cBe_α = cαe_αεI. This shows that G = I. Similarly, we can show that G has no proper right ideal. Hence G both left simple and right simple Γ-semigroup. Then G is simple Γ-semigroup.

Corollary 1: G_α is a Γ-group if and only if G is a Γ- semigroup and if it has not proper ideal.

Proof : From Theorem 2 and definition 6 and 7 is evident.

Theorem 3: Let G a Γ-semigroup and αεΓ. G_α is Γ-group if and only if G does not have proper quasi-ideal.

Proof: Suppose that G does not have proper quasi-ideal. Let aεG, consider the set aαG. Then:

Therefore, aαG is quasi-ideal. But G does not have proper quasi-ideal, then aαG = G. Similarly we can show that Gαa = G. Hence, for every aεG we have aBG = GBa = G. It shows that G_αis a Γ-group.

Conversely, suppose that G_α is a group and Q is quasi-ideal of G. Let aεQ, then:

Consequently, G = Q. This shows that G does not have proper quasi-ideal.

Theorem 4: Let G be a Γ-semigroup if G_α is Γ-group for some αεΓ then G_α is Γ-group for all αεΓ.

Proof: Let G_α be a group. Consider the sets aβG and Gβa; aεG, βεΓ. Now (aβG) αG = aβ (GαG) ⊂ aβG and Gα (Gβa) = (GαG) βa ⊂ Gβa. Hence, aβG is a right ideal and Gβa is a left ideal in G_α. Since G_αis a group we have aβG = G and Gβa = G. Then aBG_β = G and G_βBa = G for all aεG. Hence, from known result it follows that G_β is a group.

Theorem 5: A regular Γ-semigroup G will be a Γ-group if and only if eαf = fαe = f and eβf = fβe = e for any two idempotents e = eαe and f = fβf of G.

Proof: Suppose G is a Γ-group. Let e = eαe and f = fβf of G be two idempotents in G. Then e is the identity element of the group G_α and f is dhe identity element of the group G_β. Now fεG_α. Hence eBf = fBe (e-identity of G_α) and eBf = eαf, fBe = fαe. This shows that eαf = fαe = f. Similarly eβf = fβe = e. Conversely, suppose that the given condition holds in a regular Γ-semigroup G. Let aεG. Then there exist α, βεΓ and bεG such that a = aα (bβa) = (aαb)βa. Let e = bβa and f = aαb. Hence eαe = e and fβf = f are two idempotents in G . We shall show that G_α is a group. Let c = (cγd)δc be an element in G where γ, δεΓ and dεG. Then:

are idempotents. Now:

and

Hence, eBc = eεc = cBe = cαe = c for all. Again fβe = e. Hence, (aαb)βe = e. Then aα(bβe) = e. Also

Hence, for a there exists bβe in G_α such that:

Hence, G_α is a group.

Example 10: If

then S is Γ-semigroup. For fixed element:

We can see that G_α is Γ-group. Identity element is:

For:

Theorem 6: An invers Γ-semigroup G will be a Γ-group if and only if eαf = fαe = f and eβf = fβe = e for any two idempotents e = eαe and f = fβf of G.

Proof: Inverse Γ-semigroup G always is a regular Γ- semigroup. Hence, the remaining part of proof of Theorem 6 follows from Theorem 5.

CONCLUSION

A Γ-semigroup G is a Γ-group, if G is simple or G does not have proper quasi-ideal or ideal. A regular or invers Γ- semigroup G will be a Γ-group if G satisfies condition in theorem 5 or 6.

How to cite this article:

Sabri Sadiku. Necessary and Sufficient Conditions Where One Γ-Semigroup is a Γ-Group.
DOI: https://doi.org/10.36478/jmmstat.2010.44.49
URL: https://www.makhillpublications.co/view-article/1994-5388/jmmstat.2010.44.49