In this paper we will prove some theorems that discover the structure ofminimal quasi-ideals in -semigroups without zero. The main result is these are -subgroups. We will prove also that this structure is not true for respective quasi ideal of -semigroup without zero, living opened the problem of conditions thatmust satisfy a -semigroup without zero in order to have, in this case, an analogous structure

Islam BRAJA

Epoka University

“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.510Minimal Quasi-Ideals In -SemigroupIslam BRAJADepartment of Mathematics, Faculty of Natural Sciences, University “Aleksander Xhuvani”, Elbasan,Albania.Email: braja_islam@yahoo.comABSTRACTIn this paper we will prove some theorems that discover the structure ofminimal quasi-ideals in -semigroups without zero. The main result is these are -subgroups. We will prove also that this structure is not true for respective quasi-ideal of -semigroup without zero, living opened the problem of conditions thatmust satisfy a -semigroup without zero in order to have, in this case, an analogousstructure.Keywords: -semigroup, right (left) principal ideal, quasi-ideal, idempotentelement, minimal quasi-ideal, principal quasi-ideal.INTRODUCTIONDefinition 1.1  [1] Let be M = {a, b, c, ..., } and Γ = {x, y, z, ..., } twononempty set. We call the set M a Γ-semigroup if:1. axb  M,2. (axb) yc = ax (byc) for a, b, c  M and x, y  Γ.In this definition so called multiplication, which is similar withmultiply of an ordinary semigroup is described intuitively, so let’s try toprecise it.Let M = {a, b, c, ..., } and Γ = {x, y, z, ..., } be two nonempty sets. Themultiplication in the set M by elements of  set, which are between, we callevery map of M  Γ  M in M. This multiplication we call also -multiplication in M and will denote by (). Result of - multiplication in Mfor every two elements a, b of M and every element x  , which is theimage of mapping () for triple (a, x, b), ()(a, x,b), will denote simply byaxb.Definition 1.1’. [3] We call I proper ideal if it is a right (resp. left, two-sided) ideal of -semigroup M and a proper subset ofM.“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.511Now, in a Γ-semigroup M we may define analog relations ofGreen’s relations in a semigroup [4].Let M be a Γ-semigroup and a, b from M. We define binaryrelations in the following way:(1) a L b  (a) l = (b) l.(2) a R b  (a) r = (b) r.(3) a J b  (a) = (b).(4) a H b  a L b  a R b(5) a D b   c  M, a L c  c R b.Note that relation L, R, H, J are equivalence relations in M. Theclasses of equivalence of an element a from -semigroup M by theseequivalence relations we denote respectively by:La, Ra, Ha, Ja.Theorem 1.2. (Green’s Theorem) [6]. If H is a H –class of Γ-semigroupM, then either for all x from Γ, HxH ∩ H = Ф or H isa subgroup of Mx.Definition 1.3. [4] An element e  M is called idempotent in a Γ-semigroup M if there isx  Γ such that exe = e.Definition 1.4. The quasi-ideal of Γ-semigroup M we call anonempty subset Q of M such thatQ ΓM ∩ M ΓQ Q.It is clear that every left (resp. right, two-sided) ideal of a Γ- semigroupM is quasi-ideal of M. It is clear also that every quasi-ideal Q of M is Γ-subsemigroup. A Γ-semigroup may have not a proper quasi-ideal. Asresult of the absence of the proper quasi-ideals of a Γ-semigroup wehave an important property. So, is true the following theorem:Theorem 1.5. Let M be a Γ-semigroup and x  M be a fixed whateverelement. Mx is group if and only if M have not properquasi-ideals.Theorem 1.6. If e = exe (e  M, x  Γ) is idempotent element of Γ-semigroup of M and L, R are left, right ideals of M,respectively, then:Rxe = R ∩ Mxe,exL = L ∩ exM,are quasi-ideals of M.“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.512Definition 1.7. Let A be a nonempty subset of a Γ-semigroup M. Thequasi-ideal generated by A will call intersection of allquasi-ideals (A)q of M that contain A.Definition 1.8. If A = {a}, then quasi-ideal ({a}q), which denoted by(a)q, we will call principal quasi-ideal generated byelement a of M.The following proposition give the structure of quasi-idealgenerated by a nonempty subset A ≠ Φ of a Γ-semigroup M.Theorem 1.9. If A is a nonempty subset of a Γ-semigroup M, then (A M ΓA) ∩(A  M ΓA) is quasi-ideal (A)q of M, generated by A.2. Main ResultsThe minimal quasi-ideals in -semigroups without zero are discussedbefore, but without discover their -groupoid structure, which will beobtained through Greens relations.Definition 2.1. A quasi-ideal Q of a -semigroup without zero M is calledminimal if Q not hold in proper quasi-ideals of this -semigroup.The following theorem characterizes the minimal quasi-ideal:Theorem 2.2. A quasi-ideal Q of a -semigroup M is minimal if and onlyif it is a interception of the minimal left ideal L andminimal right ideal R, i.e. Q = L R.Theorem 2.3. A quasi-ideal Q of a -semigroup M is minimal if and onlyif Q is a H -class.Proof. Let suppose Q is a minimal quasi-ideal and a  Q. Since(a)q = a  (M a  a M)we have:(a)q Q  (M Q  Q M) Qand since Q is minimal, (a)q  Q.Conversely, let have“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.513 a Q, (a)q  Q.If Q is a quasi-ideal of M, such that Q Q, for an a Q we have Q =(a)q  Q , hence Q = Q . Thus, Q is minimal. So, we have prove theequivalence:Q is minimal if and only if for every a Q, (a)q  Q.From this we get Q is a H -class. Theorem 2.4. A quasi-ideal Q of a -semigroup without zero M isminimal if and only if Q is -subgroup of M. Moreover, every minimalquasi-ideal Q of M, for all x   holds equalities:Q = exM  Mxe = exMxe,where e = exe is the unity element of subgroup Qx.Proof. Let suppose Q is a minimal quasi-ideal of -semigroup M, then fromTheorem 2.3 Q is a H –class. If H-class Q we denote by H, then for x  , Qx= H is a -subgroup because have not proper quasi-ideals of M.Conversely, let suppose for x the quasi-ideal Q is a -subgroup. If Q isa quasi-ideal of -semigroup M such Q  Q, then:Q Q  Q Q Q M  M Q  Q ,which shows Q is a quasi-ideal of Q.We know Q is a -subgroup, then it not contain proper quasi-ideals,thus Q = Q, which imply Q is minimal quasi-ideal.Let Q be a minimal quasi-ideal of -semigroup M and e = exe, where e Q is unity of -subsemigroup Qx, for every x  . From Theorem 2.2 Qis the intersection of a right minimal ideal R and a left minimal ideal L andsince e  Q = R  L imply e R and e L. Since:exM  R   R,then from minimality of R we haveR = exM.Thus, analogously, we show L = Mxe. Now we have:“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.514Q = R L = exM  Mxe.Finally, we proveexM  Mxe = exMxe.Let a Q = exM Mxe. Then we have a = exm = nxe where n, m  andthus:a = (exe)xm = ex(exm) = ex(nxe) exMxe.SoexM  Mxe exMxe.Conversely, let suppose a exMxe, then:a = exbxe = ex(bxe) exManda = exbxe = (exb)xe Mxe,then a  exM  Mxe, thus we findexMxe  exM  Mxe.From this inclusion and above conversely inclusion we have equality. Naturally, rise question: Are hold true the above theorem if wesubstitute the -semigroup without zero by a -semigroup with zero,certainly demanding that quasi-ideal Q of -semigroup with zero M notcontain quasi-ideals other than zero and M quasi-ideal as well as by substitutethe requirement -group have zero?The response of this question is negative, as demonstrate the followingcounterexample:Counterexample 1Let M be the set of complex numbers C and  = C. Let define the -multiplication in C by common multiplication axb for every two complexnumbers a, b and for every x  . It is clear (M, ()) is -semigroup withzero. The set M is a quasi-ideal. The quasi-ideal M does not contain anyone“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.515quasi-ideal other than zero and M. Indeed, if Q is a quasi-ideal different fromzero, then for a element a not equal zero and for every element b  M wehave element 1ba  M and 1   for which holds true equalities:b = 1ba 1a = a1 1a b .These equalities show b  Q, since Q is quasi-ideal of M. So, Q = M.The minimal quasi-ideal M is not -group with zero because for x = 0  ,we have MxM = 0 and thus M is not a group with zero of semigroup withzero M0.But, there are -semigroups with zero for which hold true the analogous theorem for -semigroup with zero of the Theorem 2.4. For this, in the counterexample 1 lets replace  by the set of complex number diverse from zero C and by analogous reasoning will take a such -semigroups with zero. Now, -semigroup withzero (C, () * ) is -group because for every complex number x diverse fromzero, the set CxC përputhet me x and so it is group with zero of thesemigroup with zero 0 .Considering situation for -semigroup with zero we let opened thisproblem:What conditions must satisfy a -semigroup with zero to hold true theanalogous theorem of Theorem 2.4 for -semigroup with zero?The minimal quasi-ideals in -semigroupd without zero are discussedbefore by other authors, but fail to show their groupoid structure, which willbe obtained through Greens relations.Theorem 2.5. If e = exe, x , is a idempotent element of a -semigroupM contained in a minimal left ideal L, then exL is -group, and moreover is a minimal quasi-ideal of M.Proof. Let e be an idempotent, e = exe, x , contained in left ideal L. Letconsider exL. We will show G = exL is a subgroup of Mx. Take elements exm,exn exL, where m, n  L. For every y   are true the following equalities:(exm) y (exn) = ex[my(exn) ] = ex[(mye) xn]showing G is -subsemigroup of -semigroup M. It is clear that e = exe isleft unity element of semigrup (G, ○), where ○ is the induced operationfrom operation of semigroup Mx. If g = exl is an arbitrary element ofsemigroup (G, ○), then Lxg is the left ideal of M, because:M (Lxg)  (M L) xg  LxgSince:“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.516Lxg = Lx (exl) = (Lxe) xl  M L  Land L is a minimal left ideal of M we have Lxg = L. From this we find(ex)Lxg = exL,or, same(exL) xg = exL.So, the element g = exl of semigroup (G, ○) has as left inverse element anelement exl1 of exL. Thus, semigroup (G, ○) is group. -semigroup (G,()) is -group, due to a theorem, and therefore is -subgroup of -semigroup (M, ()) if with -subgroup of the -semigroup (M, ()) wemean every -semigroup such that -semigroup generated by it is -group. Now, basing on Theorem 2.4 we get out exL is a minimal quasi-ideal of M. Theorem 2.6. If e = exe, x  is an idempotent contained in a rightminimal ideal R of a -semigroup M, then Rxe is a group and,moreover, a minimal quasi-ideal of M.Proof. Let e = exe be an idempotent element of -semigroup M, wich is anelement of right ideal R. Let show that G = Rex is a subgroup of Mx. Let takeelements mxe, nxe of Rxe. For every y   are true the following equalities:(mxe) y (nxe) = mx [(eyn)xe ] = [mx(eyn)]ex,which show that G is a -subsemigroup of -semigroup M. It is clear that e =exe is the left unity element of semigroup (G, ○), where ○ is the inducedoperation from operation of semigroup Mx. If g = rxe is an arbitraryelement of G, then gxR is a right ideal of M, because:(gxR) M  gx(R M) gxRwhiles:gxR = (rxe)xR = rx(exR)  R M  M.So, while R is minimal right ideal of M we have gxR = R. From this we get“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.517(gxR)xe = xeR,or, similarlygx(Rex) = Rxe.Thus, the element g = rxe of semigroup (G, ○) has as left inverse elementan element r1xe of Rxe. So, semigroup (G, ○) is group. From this we get-semigroup (G, ()) is -group and hence G is -subgroup. From this,basing on Theorem 2.4 we have Rxe is also a minimal quasi-ideal of M.Theorem 2.7. If -semigroup without zero M has a reducible element,contained in a minimal quasi-ideal Q of M, then M is a -group.Proof. Basing on Theorem 2.4 Q is a minimal quasi-ideal of -semigroup and thus it is a -subgroup. Let e = exe be the inverse element of Qx and a a reducible element in Q for the element a we have exa = a. If m isan arbitrary element, we find that:mx (exa) = mxa ose  (mxe) xa = mxa.Its imply mxe = m. In the similar manner we show that exm = m. So, e is theidentity element for semigroup (M, ○). Since e Q, if m we have:M = mxe = exm MQ  Q MQ.It imply M = Q. Thus, M is -group. RemarkIt seems that removing of idempotent element in Theorem 2.3 is replaced byinserting a reducible element but we may prove every idempotent in aminimal quasi-ideal Q is a reciprocal reducible element. For this is suffice toprove:exa = exb  a = b,axe = bxe  a = b.Since e is an idempotent in minimal quasi-ideal Q, then (Q x, ) is group,where e is its unity and then we will have:a = exa = (exe) xa = ex (exa) = ex (exb) = (exe) xb = exb = b.“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.518This prove e is the left reducible. Similarly, we prove also that e is rightreducible.We emphasize that existence of idempotent element in a minimal quasi-idealof a-semigroup is a necessary condition for it to be a group.Theorem 2.8. Let Q be a quasi-ideal of -semigroup without zero M. Q isminimal if and only if for q Qwe have:Qq ∩ qQ = Q = (a)q .Proof. Let suppose Q is a quasi-ideal of M, which satisfy the equality. LetQ be a quasi-ideal of -semigroup and Q Q.On the other hand, let q be an element of Q . Then:Q = q Q Qq QMM Q  Q .So, Q = Q and consequently Q is a minimal quasi-ideal.Conversely, let suppose Q is a minimal quasi-ideal of M and qQ is anarbitrary element of Q, then, firstly we will prove Q q  q Q is a quasi-ideal of M.We know Q = (a)q. Since q q  Q q  q Q then it is nonempty set.Moreover:Q Q q  q Q)Q q  q Q) Q Q Q q) q Q)Q  (Q Q) q  q QQ)  Q q  q Q.So, Q q  q Q is a quasi-ideal of Q and since it is minimal thenQ q  qQ = Q = (a)q.CONCLUSIONIn this paper we are proved some theorems that help to find out the structureof minimal quasi-ideals in -semigroups without zero. The our aim was to discovertheir structures, which, as we proved, is the same with -subgroups. But thisstructure was not true for respective quasi-ideal of -semigroup without zero, livingopened the problem of conditions that must satisfy a -semigroup without zero in“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.519order to have, in this case, an analogous structure.REFERENCES[1] Sen M.K. (1981) On Γ-semigroups. Proceeding of International Conference onAlgebra and it’s Applications. Decker Publication, New York, 301.[2] Sen M.K. and Saha N.K. (1986) On Γ-semigroup I. Bull. Cal., Math. Soc. 78,180-6.[3] Saha, N.K. (1987) On Γ-semigroup II. Bull. Cal. Math. Soc., 79, 331-5.[4] Steinfeld, O. (1978) Quasi-ideals in rings and semigroups. Akademia Kiado,Budapest.[5] Dutta, T.K. and Chatterjee, T.K. (1987) Green’s equivalences on Γ-semigroup. Bull. Col. Soc. 80, 30-35.

Minimal Quasi-Ideals In F-Semigroup

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Minimal Quasi-Ideals In F-Semigroup

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