STUDY ON THE STRUCTURE OF PERIODIC Γ-SEMIGROUP

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OF PERIODIC Γ-SEMIGROUP

KOSTAQ HILA and JANI DINE

In this paper, we introduce and study periodic Γ-semigroups extending and generalizing the results obtained for periodic semigroups. We define a natural (equivalence) relationKin periodic Γ-semigroup that mimics the relationKin periodic semigroups studied by Schwarz. We investigate and characterize the periodic Γ- semigroups using Green’s relations. The main purpose of this paper is to study the relationship betweenKand Green’s relations, especially with the relation D and also with the relationN.

AMS 2010 Subject Classification: 20M10.

Key words: Γ-semigroup, Green’s relation, weakly commutative, periodic.

1. INTRODUCTION AND PRELIMINARIES

In 1964, N. Nobusawa [14] introduced the notion of a Γ-ring, more general than a ring. In 1966, W.E. Barnes [1] weakened slightly the conditions in the definition of Γ-ring in the sense of Nobusawa. Many fundamentals results in ring theory have been extended to Γ-rings by different authors obtaining vari- ous generalization analogous to corresponding parts in ring theory. In 1981, M.K. Sen [16] and later in 1986, Sen and Saha [17] introduced the concept of the Γ-semigroup as a generalization of semigroup and ternary semigroup.

Many classical notions and results of the theory of semigroups have been extended and generalized to Γ-semigroups by a lot of mathematicians. Green’s relations for semigroups were introduced by J.A. Green in a paper from 1951 [23]. Greens relations for Γ-semigroups defined in [5, 18] and studied in a several papers [3, 4, 5, 6, 7, 10, 18], play an important role in studying of the structure of Γ-semigroups as well as in case of the plain semigroups and become a familiar tool among Γ-semigroups. Periodic semigroups has been treated occasionally in the literature, essential contribution having been made by Schwarz [22], Yamada [24], Sedlock [23], Miller [13]. Periodic Γ-semigroups were first introduced in [5] and a related result was obtained. In this paper we introduce and study periodic Γ-semigroups extending and generalizing the results obtained for periodic semigroups. We investigate and characterize the

MATH. REPORTS13(63),3 (2011), 271–284

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periodic Γ-semigroups using Green’s relations. We define a natural (equivalence) relation K in periodic Γ-semigroup that mimics the relationK in periodic semigroups studied by Schwarz. In this paper we study the relationship betweenKand Green’s relations, especially with the relationDand also, with the relationN. We introduce the notion of weakly commutative Γ-semigroup and we investigate a several properties in the class of periodic Γ-semigroups and weakly commutative periodic Γ-semigroups. One can extend this work, searching other necessary and sufficient conditions in order that the relation K coincide with any one of the other Green’s relations.

In 1986, Sen and Saha [17] defined Γ-semigroup as a generalization of semigroup and ternary semigroup as follows:

Definition 1.1. LetM and Γ be two nonempty sets. Denote by the letters of the English alphabet the elements of M and with the letters of the Greek alphabet the elements of Γ. Then M is called a Γ-semigroup if

1. aγb∈M for all γ ∈Γ.

2. If m1, m2, m3, m4 ∈ M,γ1, γ2 ∈Γ such that m1 =m3,γ1 =γ2 and m2=m4, thenm1γ1m2 =m3γ2m4.

3. (aαb)βc=aα(bβc) for all a, b, c∈M and for all α, β ∈Γ.

Example 1.2. Let M be a semigroup and Γ be any nonempty set. If we define aγb=abfor all a, b∈M andγ ∈Γ. Then M is a Γ-semigroup.

Example1.3. LetMbe a set of all negative rational numbers. Obviously, M is not a semigroup under usual product of rational numbers. Let Γ =

−¹_p : p is prime . Let a, b, c ∈ M and α ∈ Γ. Now, if aαb is equal to the usual product of rational numbers a, α, b, then aαb ∈ M and (aαb)βc = aα(bβc).

Hence M is a Γ-semigroup.

Example 1.4. Let M ={−i,0, i}and Γ =M. ThenM is a Γ-semigroup under the multiplication over complex numbers while M is not a semigroup under complex number multiplication.

These examples shows that every semigroup is a Γ-semigroup. Therefore, Γ-semigroups are a generalization of semigroups.

Example1.5. LetGbe a group,I,∧two index sets and Γ the collection of some∧×Imatrices overGô=G∪{0}, the group with zero. Letµôbe the set of all elements (a)_iλwherei∈I,λ∈ ∧and (a)_iλtheI×∧matrix overGôhaving ain theith row andλth column, its remaining entries being zero. The expres- sion (0)_iλwill be used to denote the zero matrix. For any (a)_iλ,(b)jµ,(c)_kν ∈µô and α = (p_λi), β = (q_λi) ∈ Γ we define (a)_iλα(b)_jµ = (ap_λjb)_iµ. Then it is easy verified that [(a)iλα(b)jµ]β(c)kν = (a)iλα[(b)jµβ(c)kν]. Thus µô is a Γ- semigroup. We call Γ the sandwich matrix set andµô the ReesI× ∧matrix Γ- semigroup overGôwith sandwich matrix set Γ and denote it byµô(G:I,∧,Γ).

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Other examples of Γ-semigroups can be found in [8, 9, 17, 18].

For nonempty subsetsAandB ofM and a nonempty subset Γ⁰ of Γ, let AΓ⁰B ={aγb:a∈A,b∈B andγ ∈Γ⁰}. IfA={a}, then we also writeaΓ⁰B instead of {a}Γ⁰B, and similarly if B ={b} or Γ⁰ ={γ}.

A Γ-semigroupM is called commutative Γ-semigroup if for all a, b∈M and γ ∈Γ, aγb=bγa. A nonempty subset K of a Γ-semigroup M is called a sub-Γ-semigroup ofM if for all a, b∈K andγ ∈Γ,aγb∈K.

Example 1.6. LetM = [0,1] and Γ =₁

n |nis a positive integer . Then M is a Γ-semigroup under usual multiplication. Let K = [0,1/2]. We have that K is a nonemtpy subset ofM and aγb∈ K for all a, b∈ K and γ ∈Γ.

Then K is a sub-Γ-semigroup of M.

Definition 1.7. Let M be a Γ-semigroup. A non-empty subset I ofM is called a left (resp. right) ideal of M if MΓI ⊆ I (resp. IΓM ⊆ I). A nonempty subset I of M is called an ideal of M if it is a left ideal as well as a right ideal of M.

An element aof an Γ-semigroup M is calledidempotent ifa=aγa, for some γ ∈Γ. An element aof an Γ-semigroup M is calledzero element of M ifaγb=bγa=a,∀b∈M and∀γ ∈Γ and it is denoted by 0. A Γ-semigroup M is called left (respectively, right) simple if it does not contain proper left (respectively, right) ideals or equivalently, if for every left (respectively, right) ideal Aof M, we haveA=M.

The elementaof a Γ-semigroupM is calledregular inM ifa∈aΓMΓa, where aΓMΓa ={(aαb)βa |a, b ∈M, α, β ∈Γ}. M is called regular if and only if every element of M is regular.

Definition 1.8. A Γ-semigroup M with zero element is called 0-simple (left 0-simple,right 0-simple) if

1. MΓM 6={0}, and

2. {0} is the only proper two sided (left, right)-ideal ofM.

Definition 1.9. A two sided (left, right) ideal I of a Γ-semigroup M is called 0-minimal if

1. I 6={0}, and

2. {0} is the only two sided (left, right) ideal ofM contained in I.

For each element a of a Γ-semigroup M, the left ideal MΓa∪ {a} containing a is the smallest left ideal of M containing a, for if A is any other left ideal containing athen MΓa∪ {a} ⊆A and this ideal is denoted by (a)_l and called the principal left ideal generated by the element a. Similarly, for each a∈M, the smallest right ideal containing a is aΓM ∪ {a} which is denoted by (a)r and called the principal right ideal generated by the elementa.

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The principal ideal of M generated by the element a is denoted by (a) and (a) ={a} ∪MΓ{a} ∪ {a}ΓM∪MΓ{a}ΓM.

LetM be a Γ-semigroup and x be a fixed element of Γ. We definea◦b in M by a◦b = axb, ∀a, b ∈ M. The authors [17] have shown that M is a semigroup and denoted this semigroup by M_x. They have shown that ifM_x is a group for some x∈ Γ, thenMx is a group for all x ∈Γ. A Γ-semigroup M is called a Γ-group ifM_x is a group for some (hence for all)x∈Γ.

In [5], the authors defined the Green’s equivalences on Γ-semigroup as follows:

LetM be a Γ-semigroup. Let a, b∈M,

aLb⇔(a)_l = (b)_l, aRb⇔(a)_r = (b)_r, aJb⇔(a) = (b), aHb⇔aLband aRb, aDb⇔aLcand cRbfor somec∈M.

It is clear that a Γ-semigroup M is left [right] simple if and only if it consists of a single L[R] class, and thatM is simple if and only if it consists of a single J-class. We say that a Γ-semigroup M isbisimple if it consists of a single D-class. Since D ⊆ J, every bisimple Γ-semigroup is also simple.

Definition 1.10. Let M a Γ-semigroup. If La and Lb are L-classes containing a and bof a Γ-semigroup M respectively, thenL_a ≤L_b if (a)_l ⊆(b)_l. Then “≤” is a partial order inM/L which is the set ofL-classes ofM. Simi- larly,R_a≤R_b and J_a≤J_b are defined in M/Rand M/J.

Definition 1.11. A Γ-semigroupM is said to satisfy min_L or min_R condition if every nonempty set of L-classes or of R-classes possess a minimal member respectively.

Definition 1.12. A Γ-semigroup M is calledcompletely 0-simple ifM is 0-simple and it satisfies the min_L and min_R conditions.

2. THE NATURAL EQUIVALENCE ANDD

A Γ-semigroup M is said to be a periodic Γ-semigroup [5] if for any a∈M and anyγ ∈Γ, there exist positive integersnandmsuch that (aγ)ⁿb= (aγ)^n+mband b(γa)ⁿ=b(γa)^n+m for all b∈M. Equivalently, a Γ-semigroup M is said to be aperiodicΓ-semigroupif each element ofM has a finite order, where the order of a ∈ M is the order of the cyclic sub-Γ-semigroup of M generated by a, that is, to each elementaofM,∀γ ∈Γ, there corresponds an idempotent e and a positive integernsuch that (aγ)ⁿ⁻¹a=e; the element a is then said to belong to e.

The fact that for each elementaof a periodic Γ-semigroupMsome power of ais idempotent leads to defining a natural (equivalence) relation K on M by: for a, b∈M,aKb if and only if for all γ ∈Γ, there exists an idempotent

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e and integers m, n such that (aγ)ⁿ⁻¹a= (bγ)^m−1b=e. The K-classes of M will be denoted by K^e,eidempotent.

Letebe an idempotent. A sub-Γ-group Gof a periodic Γ-semigroup M is called maximal sub-Γ-group to belong to e if a) e ∈ G; and b) it is not properly contained in any other sub-Γ-group of M. This will be denoted by G^e. For the set G^e we have obviously the following facts:

I) for each α-idempotent e, eγx = xγe for each x ∈ Kê, γ ∈ Γ, and eαKê = Kêαe = Gê = {x ∈ Kê | eαx = xαe = x} is the maximal sub-Γ- group of M containinge;

II) M is a union of sub-Γ-groups if and only if each element of M has index one if and only if for each idempotent e,K^e=G^e;

III) for each idempotente,H_e=G^e. In [5], the authors proved the following:

Theorem2.13 ([5], Theorem 3.4). IfM is a periodicΓ-semigroup, then D=J.

We will determine necessary and sufficient conditions on a periodic Γ- semigroup M in order thatK coincide with any one of the Green relations.

Theorem 2.14. For each idempotent e in a periodic Γ-semigroup M, K^e∩De=G^e.

Proof. Since Gê ⊆ Kê and Gê = H_e ⊆ D_e it is immediate that Gê ⊆ Kê∩De.

Conversely, letx∈K^e∩D_e. Then for anyγ ∈Γ,eγx=xγe∈G^e⊆D_e, so xDeγx and x = pαeγxβq for some p, q ∈ M¹ and α, β ∈ Γ. Since M si periodic, there exist γ1-idempotent f, γ2-idempotent h and positive integers m, n such that ((pαe)γ₁)^m(pαe) =f and (qγ₂)ⁿ⁻¹q=h; therefore,

x=pαeγxβq= (pαe)γ₁(pαe)γxβ(qγ₂q) =· · ·=

= ((pαe)γ1)^mn−1(pαe)γxβ(qγ2)^mn−1q=f γxβh.

Noting thatf γe= ((pαe)γ1)^m(pαe)γe=f, we have

M¹ΓeΓx⊆M¹Γx=M¹Γf γxβh=M¹Γf γeγxβh⊆M¹Γeγxβh.

But it is easily verified thatM¹Γeγxβh=M¹Γeγx, henceM¹Γeγx=M¹Γx= M¹Γeγxβh. This shows that xLeγx.

Dually,

xγeΓM¹⊆xΓM¹=f γxβhΓM¹⊆f γxΓM¹ =f γeγxΓM¹ =f γxγeΓM¹. Now sincef γx=f it follows that f γxγeΓM¹ =xγeΓM¹, so that xγeΓM¹= xΓM¹=f γxγeΓM¹, andxReγx. Therefore,x∈Leγx∩Reγx =Heγx=He= G^e.

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Corollary2.15. For each idempotentein a periodicΓ-semigroup,Kê∩ L_e=Gê and Kê∩R_e=Gê.

Proof. Again Gê ⊆ Kê and Gê = H_e ⊆ L_e, so it is immediate that Gê ⊆Kê∩L_e. From Theorem 2.2, Kê∩L_e ⊆Kê∩D_e=Gê.

Similarly, we can show that K^e∩Re=G^e.

Corollary 2.16. Periodic Γ-semigroup M is a union ofΓ-subgroups if and only if K ⊆ D.

Proof. If M is a union of Γ-subgroups, then for each idempotent e, Kê=Gê=H_e⊆D_e. On the other hand, ifK ⊆ D, then for each idempotent e,Kê=Kê∩De=Gê. Hence,M is a union of its maximal Γ-subgroups.

Definition 2.17. Γ-semigroupM isweakly comutativeif for eacha, b∈M andα, γ ∈Γ, there existx, y∈M and an integerksuch that ((aαb)γ)^k−1(aαb)

=xαa=bαy.

Definition2.18. Γ-semigroupMis asemilattice of Γ-semigroups of typeα ifM is a disjoint union of Γ-semigroups of typeα{M_i|i∈I,I index set}, and for eachi, j∈I there existsk∈I such that MiΓMj ⊆Mkand MjΓMi ⊆Mk. LetM be a Γ-semigroup and F a sub-Γ-semigroup. Then F is called a filter of M if a, b ∈ M, aγb ∈ F (γ ∈ Γ) ⇒ a ∈ F and b ∈ F [8, 19]. It is clear that for every a ∈ M there is a unique smallest filter of M containing the element a, denoted byN(a), which is called the principal filter generated by a. We denote by “N” the equivalence relation on M defined by N = {(a, b) ∈M² |N(a) =N(b)}. N is a semilattice congruence on M. For any a ∈ M, the N-class containing a is denoted by (a)N and it is clear that it is a sub-Γ-semigroup of M. On the set M/N = {(a)_N | a ∈ M} we define (a)Nγ(b)N = (aγb)N, for all (a)N,(b)N ∈M/N, γ∈Γ. It is clear that the set M/N is a Γ-semigroup.

Theorem 2.19. Let M be a Γ-semigroup. For every x∈M, N(x) ={y∈M | hxi ∩MΓy∩yΓM 6=∅}

if and only if M is weakly commutative.

Proof. We first prove the necessity. Let x, y∈M. Then, forα∈Γ x∈N(x)⊆N(xαy) ={z∈M | hxαyi ∩MΓz∩zΓM 6=∅}

and thus ((xαy)γ)^m−1(xαy) = aαx for some a ∈ M and some integer m;

similarly, ((xαy)γ)ⁿ⁻¹(xαy) = yαb for some b ∈ M and some integer n. If m > n, then

((xαy)γ)^m−1(xαy) =aαx=yα[bγ((xαy)γ)^m−n(xαy)];

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the other cases are similar. Hence M is weakly commutative.

We next prove the sufficiency. Letx∈M, and let T ={y∈M | hxi ∩MΓy6=∅}.

We first show that T is filter ofM; this together with the fact that x∈T will prove that N(x) ⊆ T. If y, z ∈T, then for all γ ∈ Γ, (xγ)^m−1x = aαy and (xγ)ⁿ⁻¹x=bβz for somea, b∈M,α, β ∈Γ and for some integerm, n. Since M is weakly commutative, (xγ)^nr−1x = ((bβz)γ)^r−1(bβz) =zδc for somec∈ M,δ ∈Γ and for some integerr. Consequently, (xγ)^(m+nr)−1x= [aα(yγz)]δc, whence, again by weak commutativity, (xγ)^(m+nr)k−1x=dρ[aα(yγz)] for some d∈M,ρ∈Γ and for some integerk. Thusyγz∈T for allγ ∈Γ. Conversely, suppose that yγz ∈ T for all γ ∈ Γ. Hence (xγ)^m−1x = (aαy)γz for some a ∈ M, α ∈ Γ and for some integer m, and thus z ∈ T. It follows by weak commutativity that (xγ)^mn−1x = ([(aαy)γz]γ)ⁿ⁻¹[(aαy)γz] = bβ(aαy) for some b ∈ M, β ∈ Γ and for some integer n; that is, y ∈ T. Hence T is a filter and N(x) ⊆T. Since the opposite inclusion is clearly satisfied, we have N(x) =T.

By symmetry we conclude that also

N(x) ={y∈M | hxi ∩yΓM 6=∅}.

It is easily to see that the set

{y∈M | hxi ∩yΓM 6=∅} ∩ {y∈M | hxi ∩MΓy6=∅}.

is the one in the statement of the theorem.

Theorem2.20. Let M be a periodic Γ-semigroup. Then every K^e is an N-class of M if and only if M is weakly commutative.

Proof. Suppose that every K^e is an N-class of M. If x, y ∈ M, then xαy, yαx ∈(xαy)N =K^f for allα ∈Γ and for some γ-idempotent f. Thus, ((xαy)γ)^m−1(xαy) = ((yαx)γ)ⁿ⁻¹(yαx) =f for somem and n. Hence

((xαy)γ)^m−1(xαy) = ((yαx)γ)²ⁿ⁻¹(yαx) =yα[xγ((yαx)γ)²ⁿ⁻¹] =

= [((yαx)γ)²ⁿ⁻¹(yαx)γy]αx.

Conversely, suppose M is weakly commutative. For any x ∈ M, (x)N is Γ- semigroup and thus contains an γ-idempotent. If e, f ∈ E_γ ∩(x)N, then e = aγ1f and f = eγ2b for some a, b ∈ M, γ1, γ2 ∈ Γ by Theorem 2.7.

Consequently,

e=aγ₁f = (aγ₁f)γf =eγf =eγ(eγ₂b) =eγ₂b=f.

Hence (x)N ⊆ K^e and since the opposite inclusion is obvious, we conclude that (x)N =K^e.

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Proposition2.21. If M is a weakly commutative periodicΓ-semigroup, then D ⊆ K=N and each maximal sub-Γ-group is aD-class of M.

Proof. It can be seen that for the minimal semilattice congruence N, we have J ⊆ N for an arbitrary Γ-semigroup. In particular, ifM is a weakly commutative periodic Γ-semigroup, then D = J ⊆ N = K. Hence, each K-class is a sub-Γ-semigroup of M and a union ofD-classes.

Moreover, by Theorem 2.2, sinceD ⊆ K, we haveG^e=K_e∩D_e=D_e.

Definition 2.22. A Γ-semigroup M is called unipotent if it contains e- xactly one idempotent.

Definition 2.23. An element u of a Γ-semigroup M is called a zeroid element of M if, for each elementaof M, there existx, y∈M,α, β ∈Γ such that aαx=yβa=u.

Definition 2.24. Γ-homogroup is called a Γ-semigroup having zeroid elements.

LetM be a periodic Γ-semigroup. We introduce the following three conditions which will play principal roles in sequel.

Condition A. For any elements a, b ∈ M and γ ∈ Γ, if (aγ)ⁿ⁻¹a = (bγ)^m−1bfor some integern, m, then for allα∈Γ, there exist three positive integerr, s and tsuch that ((aαb)γ)^r−1(aαb) = (aγ)^saα(bγ)^tb= (bγ)^tbα(aγ)^sa.

Condition B. For any elements a, b ∈ M, and for any positive integers n, m and α, γ ∈ Γ, then there exist two positive integer r, s such that ((aαb)γ)^r(aαb) = ((aγ)ⁿ⁻¹aα(bγ)^m−1b)γ)^s−1((aγ)ⁿ⁻¹aα(bγ)^m−1b)).

Condition C. For any elements a, b ∈ M, and for any positive integers n, m and α, γ ∈Γ, then there exist two positive integer r, ssuch that

((aαb)γ)^r−1(aαb) = (((aγ)ⁿ⁻¹aα(bγ)^m−1b)γ)^s−1((aγ)ⁿ⁻¹aα(bγ)^m−1b)) =

= (((bγ)^m−1bα(aγ)ⁿ⁻¹a)γ)^t−1(bγ)^m−1bα(aγ)ⁿ⁻¹a).

Lemma 2.25. For periodic Γ-semigroups, Condition A is a consequence of Condition B.

Proof. Let M satisfy ConditionB. Suppose that a, b are elements of M such that (aγ)ⁿ⁻¹a = (bγ)^m−1b for all γ ∈Γ. Then there exists an integer j and an idempotent e∈ M satisfying (aγ)^nj−1a= (bγ)^mj−1b= e. By Condi- tion B, there exist integers r, s such that for all α ∈ Γ, ((aαb)γ)^r−1(aαb) = ((aγ)^nj−1aα(bγ)^mj−1b)γ)^s−1 ((aγ)^nj−1aα(bγ)^mj−1b)) =e. Hence, we have

((aαb)γ)^r−1(aαb) = (aγ)^nj−1aα(bγ)^mj⁻¹b= (bγ)^mj⁻¹bα(aγ)^nj−1a.

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Lemma 2.26. For periodic Γ-semigroups, Condition B is a consequence of Condition C.

Proof. It is immediate from the definitions of ConditionsB and C.

Theorem2.27. A periodicΓ-semigroupM is decomposable into the class sum of mutually disjoint unipotentΓ-homogroups if and only if it satisfies Con- dition A. Further, in this case such a decomposition is uniquely determined.

Proof. LetM satisfy ConditionA. To prove the “if” part of this theorem, we need to show only that each K^e is a Γ-homogroup. Ifa, bare two elements of K^e, then there exist integers n, m such that (aγ)ⁿ⁻¹a = (bγ)^m−1b = e.

Since M satisfies Condition A, there exist three positive integer r, s and t such that ((aαb)γ)^r−1(aαb) = (aγ)^s−1aα(bγ)^t−1b= (bγ)^t−1bα(aγ)^s−1a. Hence, ((aαb)γ)^rnm−1(aαb) = (aγ)^snm−1aα(bγ)^tnm−1b = e. This implies that Kê is Γ-semigroup. Sinceeis clearly a zeroid element ofKê, the semigroupKê is a Γ-homogroup.

Conversely, assume thatM is decomposed into the class sum of mutually disjoint unipotent Γ-homogroups H_i and suppose that (aγ)ⁿ⁻¹a= (bγ)^m−1b.

Then bothaandbare contained in the same Γ-homogroup, sayHi, since there exists an idempotent eand a integers such that (aγ)^ns−1a= (bγ)^ms−1b=e.

Therefore, aαb∈H_i. Hence, we have

(aγ)^ns−1aα(bγ)^ms−1b= (bγ)^ms−1bα(aγ)^ns−1a=e= ((aαb)γ)^r−1(aαb).

for some integerr. Thus, the proof of the first half of this theorem is complete.

The latter half of this theorem is clear.

Aband is an idempotent Γ-semigroup. Let J be a band. A Γ-semigroup G is said to be a band J of Γ-semigroups of type T, if G is the class sum of a set {G_i |i∈J} of mutually disjoint sub-Γ-semigroup G_i each type T, such that for any i, j ∈ J, GiΓGj ⊂ Giγj, γ ∈ Γ. If J is a commutative band, that is, ifJ is a semilattice, then M is called a semilattice of Γ-semigroups of type T.

Theorem2.28. A periodic Γ-semigroup M is decomposable into a band of unipotent Γ-homogroups if and only if it satisfies Condition B.

Further, in this case such a decomposition is uniquely determined.

Proof. Assume that M is a band of unipotent Γ-homogroup Hi. Let a, b ∈ M. Let n, m be any integers. Then, there exist Hi and Hj which contain a and b respectively. Since, by the assumption on M, both aαb and (aγ)ⁿ⁻¹aα(bγ)^m−1b are contained in Hiαj, there exist integers r and s such that ((aαb)γ)^r−1(aαb) =eiαj whereeiαj is the idempotent of Hiαj.

Conversely, letM satisfy ConditionB. By Theorem 2.15,M is then the class sum of mutually disjoint unipotent Γ-homogroups Hi, since M satisfies

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also Condition A. Pick up anya1, a2∈Hi and b1, b2 ∈Hj respectively. There exist integers n1, n2, m1, m2 such that (a1γ)ⁿ¹⁻¹a1 = ei, (a2γ)ⁿ²⁻¹a2 = ei, (b₁γ)^m¹⁻¹b₁ =e_j, (b₂γ)^m²⁻¹b₂ =e_j wheree_i, e_j are idempotents ofH_i andH_j respectively. By Condition B, we have

((a1αb1)γ)^r¹⁻¹(a1αb1) = ((eiαej)γ)^s¹⁻¹(eiαej) and

((a₂αb₂)γ)^r²⁻¹(a₂αb₂) = ((e_iαe_j)γ)^s²⁻¹(e_iαe_j) for some integers r₁, s₁, r₂, s₂.

Ifeiαej ∈Ht, there exists an integernsuch that ((eiαej)γ)ⁿ⁻¹(eiαej) = et, where et is the idempotent of Ht. Therefore, we have

((a₁αb₁)γ)^r¹ⁿ⁻¹(a₁αb₁) = ((e_iαe_j)γ)^s¹ⁿ⁻¹(e_iαe_j) =e_t=

= ((eiαej)γ)^s²ⁿ⁻¹(eiαej) = ((a2αb2)γ)^r²ⁿ⁻¹(a2αb2).

This implies that both a1αb1 and a2αb2 are contained in Ht, that is, H_iΓH_j ⊂ H_t. Thus, the proof of the first half of this theorem is complete.

The latter half of the theroem follows from Theorem 2.15.

Theorem2.29. A periodicΓ-semigroupM is decomposable into a semilattice of unipotent Γ-homogroups if and only if it satisfies Condition C.

Further, in this case such a decomposition is uniquely determined.

Proof. Let M satisfy Condition C. By Lemma 2.14, M satisfies Condi- tion B. Therefore, by Theorem 2.16, M is decomposed uniquely into a band J of unipotent Γ-homogroupsH_i. Leta, b∈M. Then, by ConditionC, there exist two integers s, tsuch that ((aαb)γ)^s−1(aαb) = ((bαa)γ)^t−1(bαa). Hence, there existsH_i which contains bothaαbandbαa. This implies thatJis a semilattice.

Conversely, assume that M is decomposable into a semilattice of unipotent Γ-homogroupsHi. Leta, b∈M. Letn, mbe any integers. SinceM satisfies ConditionB, there exist integersr₁, r₂, r₃, r₄such that ((aαb)γ)^r¹⁻¹(aαb) = ((aγ)ⁿ⁻¹aα(bγ)^m−1b)γ)^r²⁻¹((aγ)ⁿ⁻¹aα(bγ)^m−1b)) and ((bαa)γ)^r³⁻¹(bαa) = ((bγ)^m−1bα(aγ)ⁿ⁻¹a)γ)^r⁴⁻¹((bγ)^m−1bα(aγ)ⁿ⁻¹a)).

On the other hand, it follows from our assumption onMthataαbandbαa are contained in the same unipotent Γ-homogroup, say Hi under the decomposition. Hence, there exist two integers t₁, t₂ such that ((aαb)γ)^t¹⁻¹(aαb) = ((bαa)γ)^t²⁻¹(bαa) = ei, where ei is the idempotent of Hi. Consequently, we have

((aαb)γ)^t¹^t²^r¹^r³⁻¹(aαb) =

= ((aγ)ⁿ⁻¹aα(bγ)^m−1b)γ)^t¹^t²^r²^r³⁻¹((aγ)ⁿ⁻¹aα(bγ)^m−1b)) =

= ((bγ)^m−1bα(aγ)ⁿ⁻¹a)γ)^t¹^t²^r¹^r⁴⁻¹((bγ)^m−1bα(aγ)ⁿ⁻¹a)).

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Lemma2.30. PeriodicΓ-semigroupM is weakly commutative if and only if it satifies Condition C.

Proof. IfM is weakly commutative, thenN =K andM is a semilattice of unipotent Γ-homogroups, that is the K-classes of M. By Theorem 2.17,M satisfies Condition C.

Conversely, letM satisfy ConditionC. Fora, b∈M theree exist integers r, t such that ((aαb)γ)^r−1(aαb) = ((bαa)γ)^t−1(bαa). Hence,

((aαb)γ)^r−1(aαb) =bα[aγ((bαa)γ)^t−2(bαa)] = [((bαa)γ)^t−2(bαa)γb]αa.

To prove the main theorem, we need some other results as follows. As an application of the results proved in Theorem 1, Theorem 2, Theorem 3 and Theorem 4 in [11, 12] and Proposition 2.21 in [10], we have the following:

Lemma 2.31. A Γ-semigroupM is left[right, intra-]regular if and only if every left [right, two-sided]ideal of M is semiprime.

Theorem 2.32. The following are equivalent conditions on a Γ-semigroup M.

1. M is left regular.

2.Every left ideal of M is semiprime.

3.Every L-class of M is a left simple sub-Γ-semigroup of M. 4.Every left L-class of M is a sub-Γ-semigroup of M.

5. M is a disjoint union of left simple sub-Γ-semigroups.

6. M is a union of left simple sub-Γ-semigroups.

Theorem 2.33. The following are equivalent conditions on a Γ-semigroup M.

1. M is a union ofΓ-groups.

2. M is both left and right regular.

3.Every left and every right ideal of M is semiprime.

4.Every H-class of M is a Γ-group.

5. M is a union of disjointΓ-groups.

Proof. If (1) holds, then M is clearly left regular, right regular and regular; for we may solve respectively xγaµa=a,aµaγy =a,aγzµa=a for all a∈M, for someγ, µ∈Γ and x, y, z within a Γ-group to which abelongs.

Thus (1) implies (2). Moreover, (2) is equivalent to (3) by Lemma 2.19.

By definition of left and right regular Γ-semigroup and by Greens’ Theo- rem for Γ-semigroups [15, Theorem 2.1], it follows that (4) holds. (4) implies (5) since H-classes are disjoint, and (5) implies (1) trivially. So, we have es- tablished the equivalence of (1), (2), (3), (4) and (5).

Theorem 2.34. The following four statements are equivalent.

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1. M is a union of simpleΓ-semigroups.

2. M is intra-regular.

3.Every ideal of M is semiprime.

4.The principal ideals ofM constitute a semilatticeY under intersection;

in fact J(a)∩J(b) =J(aγb) for everya, b∈M, γ∈Γ; furthermore, M is the union of the semilattice Y of simple Γ-semigroups Mi (i∈Y), eachMi being a J-class ofM.

Proof. By Lemma 2.19, Theorem 2.20 and Theorem 2.21 it follows that (1) implies (2) and (2) is equivalent to (3). Evidently (4) implies (1). The proof of the fact that (4) follows from (2) and (3) can be done by easily respectively modifications in the last part of the proof of Theorem 4.4 in [2].

Theorem 2.35. The following statements are mutually equivalent.

1. M is union of Γ-groups.

2. M is a union of completely simple Γ-semigroups.

3. M is a semilattice Y of completely simple Γ-semigroups Mi (i∈ Y), where Y is the semilattice of principal ideals of M, and each M_i is a J-class of M.

Proof. (3) implies (2) trivially and (2) implies (1) by Lemma 2.9 [20]

and Theorem 2.1 [15]. By Theorem 2.22, (1) implies (3) except for the complete simplicity of the simple sub-Γ-semigroup Mi. This is immediate from the Theorem 4.4 [20], since each J-classM_i is a union of the H-classes of M contained in it, while (1) implies that every H-classes of M is a Γ-group from Theorem 2.21.

Now we prove the main theorem of this section.

Theorem 2.36. Let M be a periodic Γ-semigroup. Then the following are equivalent:

1. M is a semilattice of Γ-subgroups.

2. M is a union ofΓ-subgroups and weakly commutative.

3.K =D.

Proof. (1) ⇒ (2). Since M is a semilattice of (periodic) Γ-subgroups, then it is trivially both a union of Γ-subgroups and a semilattice of unipotent Γ-homogroups. Due to Theorem 2.17, M satisfies Condition C, and by Lem- ma 2.18 it is weakly commutative.

(2) ⇒ (3). If M is a union of Γ-subgroups, then for each idempotent e, K^e = G^e = H_e. Thus K = H. Therefore combining this with all above mentioned results, we get D = J ⊆ N = K = H. However, it follows from the definitions of Green’s relations that H ⊆ L ⊆ D and H ⊆ R ⊆ D. So all of Green’s relations coincide with K.

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(3) ⇒ (1). From Corollary 2.4, K ⊆ D implies that M is a union Γ- subgroups, that is, each K-class is a maximal Γ-group. By Theorem 2.23, M is a semilattice of D=H-classes.

Acknowledgement.The first author wish to express his warmest thanks to Professor Apostolos Thoma and Department of Mathematics, University of Ioannina (Greece) for their hospitality during the author’s postdoctoral scholarship supported by The State Scholarships Foundation of Hellenic Republic.

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Received 1 March 2010 University of Gjirokastra Albania Faculty of Natural Sciences

Department of Mathematics and Computer Science kostaq hila@yahoo.com