ON THE STRUCTURE OF GREEN S RELATIONS IN BQ Γ-SEMIGROUPS

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I. CUZA DIN IAŞI (S.N.) MATEMATICĂ, Tomul LX, 2014, f.1 DOI: /aicu ON THE STRUCTURE OF GREEN S RELATIONS IN BQ Γ-SEMIGROUPS BY KOSTAQ HILA and JANI DINE Abstract. In this paper, we study and characterize the structure of Green s relations in BQ Γ-semigroups, i.e., Γ-semigroups in which the bi-ideals and quasi-ideals coincide. Mathematics Subject Classification 2010: 20M12, 20M10. Key words: Γ-semigroup, left ideal, quasi-ideal, bi-ideal, Green s relation. 1. Introduction and preliminaries In 1981, Sen [18] introduced Γ-semigroups as a generalization of semigroups and ternary semigroups. Many classical notions and results of the theory of semigroups have been extended and generalized to Γ-semigroups. In 1956, Steinfeld [20, 22] introduced the notion of quasi-ideal for semigroups as a generalization of the one-sided ideal. Several results on quasiideals have been obtained, see for example [1, 2, 13, 14, 15, 21]. Some of them were extended to Γ-semigroups in [11]. In fact, the notion of biideal was given earlier by Good and Hughes [6]. It was actually introduced in Green s relations for semigroups were introduced by Green in a paper from 1951 [7] (cf. [4]). Green s relations for Γ-semigroups [3, 5, 16, 12, 17], play an important role in studying the structure of Γ- semigroups as well as in case of the plain semigroups. In this paper, we study and characterize the structure of Green s relations in BQ Γ-semigroups, i.e., Γ-semigroups in which the bi-ideals and quasi-ideals coincide. We will show that an H-class contains an irregular element only when it consists of exactly that element. Also, we will show that in a BQ Γ-semigroup

2 202 KOSTAQ HILA and JANI DINE 2 M, an element a M is regular if and only if it is quasiregular. We will also show that if M is a BQ Γ-semigroup M and a, b M with adb and R a < R b and L a < L b, then a and b are regular. Finally, we show that in a BQ Γ-semigroup M any irregular D-class is either an L-class or an R-class. We introduce below necessary notions and present a few auxiliary results that will be used throughout the paper. In 1986, Sen and Saha [19] defined Γ-semigroup as a generalization of semigroup and ternary semigroup as follows: Definition 1.1. Let M and Γ be two non-empty 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 there exists a mapping M Γ M M, written as (a, γ, b) aγb satisfying the following identity (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 non-empty set. Define a mapping M Γ M M by aγb = ab for all a, b M and γ Γ. Then M is a Γ-semigroup. Example 1.3. Let M be a set of all negative rational numbers. Obviously M is not a semigroup under the 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 the 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. Then M is a Γ- semigroup under the multiplication over complex numbers while M is not a semigroup under complex number multiplication. These examples show that every semigroup is a Γ-semigroup and Γ- semigroups are a generalization of semigroups. A Γ-semigroup M is called commutative Γ-semigroup if for all a, b M and γ Γ, aγb = bγa. A non-empty subset K of a Γ-semigroup M is called a sub-γ-semigroup of M if for all a, b K and γ Γ, aγb K. Example 1.5. Let M = [0, 1] and Γ = { 1 n n is 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 of M and aγb K for all a, b K and γ Γ. Then K is a sub-γ-semigroup of M.

3 3 GREEN S RELATIONS IN BQ Γ-SEMIGROUPS 203 Other examples of Γ-semigroups can be found in [8, 9, 10, 19, 17]. For non-empty subsets A and B of M and a non-empty subset Γ of Γ, let AΓ B = {aγb : a A, b B and γ Γ }. If A = {a}, then we also write {a}γ B as aγ B, and similarly if B = {b} or Γ = {γ}. Let M be a Γ-semigroup and A be a non-empty subset of M. Then A is called a right (resp. left) ideal of M if AΓM A (resp. MΓA A). A is called an ideal of M if it is right and left ideal of M. A right, left or ideal A of a Γ-semigroup M is called proper if A M. An element a of an Γ-semigroup M is called idempotent if a = aγa, for some γ Γ. An element a of a Γ-semigroup M with at least two elements is called zero element of M if aγb = bγa = a, b M and γ Γ and it is denoted by 0. 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 a then 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 element a. The principal ideal of M generated by the element a is denoted by (a) and (a) = {a} MΓ{a} {a}γm MΓ{a}ΓM. Definition 1.6. Let M be a Γ-semigroup and Q a non-empty subset of M. Then Q is called quasi-ideal of M if QΓM MΓQ Q Example 1.7. Let M be a semigroup and Γ be any non-empty set. Define a mapping M Γ M M by aγb = ab, a, b M and γ Γ. Then M is a Γ-semigroup. Let Q be a quasi-ideal of M. Thus MQ QM Q. We have that MΓQ QΓM=MQ QM Q. Hence, Q is a quasi-ideal of M. This example implies that the class of quasi-ideals in Γ-semigroups is a generalization of quasi-ideals in semigroups. Definition 1.8. Let M be a Γ-semigroup and a M. Then the principal quasi-ideal Q(a) generated by a is the smallest quasi-ideal of M containing a. Clearly Q(a) = a (aγm 1 M 1 Γa). A quasi-ideal Q of a Γ-semigroup M is called a minimal quasi-ideal of M if there is no quasi-ideal A of M such that A Q. Equivalently, if for any quasi-ideal A of M such that A Q, we have A = Q. Definition 1.9. Let M be a Γ-semigroup and B a non-empty subset of M. Then B is called bi-ideal of M if B BΓMΓB B

4 204 KOSTAQ HILA and JANI DINE 4 Definition Let M be a Γ-semigroup and a M. Then the principal bi-ideal B(a) generated by a is the smallest bi-ideal of M containing a. Clearly B(a) = a aγm 1 Γa. In [5], the authors defined the Green s equivalences on Γ-semigroup as follows: Let M be a Γ-semigroup. Let a, b M, alb (a) l = (b) l, arb (a) r = (b) r, aj b (a) = (b), ahb alb and arb, adb alc and crb for some c M. 2. H-class structure of BQ Γ-semigroups Definition 2.1. Let M be a Γ-semigroup and a, b M. We define now the following equivalence relation B in Γ-semigroup M as follows: 1. a = b or 2. there exist u, v M, γ 1, γ 2, α 1, α 2 Γ such that aγ 1 uγ 2 a = b and bα 1 vα 2 b = a. It is clear that abb B(a) = B(b). In the same way we have aqb Q(a) = Q(b). We denote B a the B-class containing a. Definition 2.2. The class of BQ-Γ-semigroups will consist of those Γ- semigroups whose sets of bi-ideals and quasi-ideals coincide. Example Every regular Γ-semigroup M satisfies the condition B = Q and in this case we have B = BΓM MΓB = BΓB BΓMΓB for every bi-ideal B of M. 2. Every left (resp. right) simple Γ-semigroup M satisfies the condition B = Q and in this case we have BΓM MΓB = BΓB BΓM B for every bi-ideal B of M. It can be easily verified the following: Lemma 2.4. Let M be a Γ-semigroup. Then for a, b M, ahb if and only if Q(a) = Q(b). Proposition 2.5. The relation B is an equivalence relation, indeed, B H.

5 5 GREEN S RELATIONS IN BQ Γ-SEMIGROUPS 205 Lemma 2.6. Let M BQ, then B = H. Proof. By Proposition 2.5 we have B H. Let ahb. Since M is a BQ Γ-semigroup, it can be easily shown that B(a) = Q(a) for all a M. Using Lemma 2.4, we have B(a) = Q(a) = Q(b) = B(b). Thus abb since B(a) = B(b) if and only if abb, and this complete the proof. In general, although M BQ implies B = H, we may have B = H and M / BQ. For example, let M = {a, aγa, (aγ) 2 a, 0} where (aγ) 3 a = 0 and Γ = {γ}. In Γ-semigroup M, we have B = H = J, but B = {0, aγ} is a bi-ideal which is not a quasi-ideal, since {0, aγa}γm MΓ{0, aγa} = MΓ{0, aγa} = {0, (aγ) 2 a} B. It can be easily proved the following: Lemma 2.7. Let M be a Γ-semigroup and a M. Then either 1) a is irregular and B a = {a}, or 2) a is regular and B a = H a. An immediate consequence of of Lemma 2.6 and Lemma 2.7 is the following: Theorem 2.8. Let M BQ. If H a is an H-class of M and a is irregular, then H a = {a}. 3. D-class structure of BQ Γ-semigroups The following theorem extends the result proved by Calais [2]. Theorem 3.1. Let M be a Γ-semigroup. Let B(a, b) denote the minimal bi-ideal of M containing a, b M, and let Q(a, b) be the minimal quasi-ideal of M containing a and b. Then M BQ if and only if B(a, b) = Q(a, b) ( ). Proof. The condition (*) is evidently necessary. Let us show that it is also a sufficient condition. It is clear that the condition (*) implies the condition (**) : a M, B(a) = Q(a). It follows that if B is an arbitrary bi-ideal of M, we have a BΓB BΓMΓB Q(a) BΓB BΓMΓB and {a, b} BΓB BΓMΓB Q(a, b) BΓB BΓMΓB. Let c BΓM MΓB. We have 1. c BΓB BΓMΓB c B. 2. c / BΓB BΓMΓB there exist a, b B such that c aγm MΓb Q(a, b) = B(a, b). c / BΓB BΓMΓB {a, b} BΓB BΓMΓB. It follows that c = a or c = b, thus c B. Theorem 3.1 implies the following propositions.

6 206 KOSTAQ HILA and JANI DINE 6 Proposition 3.2. If a Γ-semigroup M satisfies the condition (*) of Theorem 3.1 and further, if for all a M, a aγm MΓa, then M satisfies B = Q and every bi-ideal of M can be written : B = BΓM MΓB = BΓB BΓMΓB. Proposition 3.3. If a Γ-semigroup M satisfies the condition (*) of Theorem 3.1 and further, if for all a M, a / MΓa (or a / aγm), then M satisfies B = Q and for every bi-ideal of M we have: a BΓM MΓB = BΓB BΓMΓB B. It is easily seen that B(a, b) = {a, b} aγm 1 Γa bγm 1 Γb aγm 1 Γb bγm 1 Γa, and that Q(a, b) = (aγm 1 M 1 Γa) (bγm 1 M 1 Γb) (aγm 1 M 1 Γb) (bγm 1 M 1 Γb). Definition 3.4. Let M be a Γ-semigroup. A non-zero element a of M is said to be quasi-regular if there exist elements b, c, d, e M, γ 1, γ 2, γ 3, α 1, α 2, α 3 Γ such that a = bγ 1 aγ 2 cγ 3 a = aα 1 dα 2 aα 3 e. A Γ-semigroup is said to be quasi-regular if each of its element is quasi-regular. Proposition 3.5. Let M be a Γ-semigroup. Then if a M is a quasiregular element of M, every element of D a is quasiregular. Proof. Let a M be a quasi-regular. We will show that every element of L a is quasi-regular. Dually, every element of R a will be quasi-regular, and the result will then follow for D a. Assume that a is quasi-regular, then a = aα 1 uα 2 aα 3 v = sγ 1 aγ 2 rγ 3 a for some u, v, s, r M and γ 1, γ 2, γ 3, α 1, α 2, α 3 Γ. Let x L a. If x a, then there are t 1, t 2 M such that a = t 1 β 1 x and x = t 2 β 2 a. We then have x = t 2 β 2 a = t 2 β 2 sγ 1 aγ 2 rγ 3 a = (t 2 β 2 sγ 1 t 1 )β 1 xγ 2 (rγ 3 t 1 )β 1 x and x = t 2 β 2 a = (t 2 β 2 a)α 1 uα 2 aα 3 v = xα 1 uα 2 (t 1 β 1 x)α 3 v = xα 1 (uα 2 t 1 )β 1 xα 3 v, hence x is quasi-regular. The result now follows. Lemma 3.6. Let M BQ. An element a M is regular if and only if it is quasi-regular. Proof. If a is regular, then there exist a M, γ 1, γ 2 Γ such that a = aγ 1 a γ 2 a. Then a = aγ 1 a γ 2 aγ 1 (a γ 2 a) = (aγ 1 a )γ 2 aγ 1 a γ 2 a so that a is quasi-regular. If a is quasi-regular, a MΓaΓMΓa and a aγmγaγm. But aγmγa is a bi-ideal and since M BQ, aγmγa is a quasi-ideal. Therefore, a (aγmγa)γm MΓ(aΓMΓa) aγmγa. Whence a is regular. By Lemma 3.6 it follows the following:

7 7 GREEN S RELATIONS IN BQ Γ-SEMIGROUPS 207 Proposition 3.7. Let M BQ. Then M is regular if and only if M is quasi-regular. Definition 3.8 ([5]). Let M a Γ-semigroup. If L a and L b are L-classes containing a and b of a Γ-semigroup M respectively, then L a L b if (a) l (b) l (or equivalently: M 1 Γa M 1 Γb). Then is a partial order in M/L which is the set of L-classes of M. Similarly R a R b and J a J b are defined in M/R and M/J. Theorem 3.9. Let M BQ. If a, b M with adb and both L a < L b and R a < R b, then b is regular (i.e., both a and b are regular). Proof. Since adb and R a R b and L a L b there exists t, s M such that t R a L b and s R b L a, where t a, b, s a, b. Since R a < R b, t R a aγm 1 bγm 1 and t L b M 1 Γb, it follows that t bγm 1 M 1 Γb = b bγm 1 Γb. Every quasi-ideal is a bi-ideal, thus b bγm 1 Γb bγm 1 M 1 Γb, since b bγm 1 Γb is the smallest bi-ideal containing b. M BQ, thus b bγm 1 Γb is a quasi-ideal containing b, but bγm 1 M 1 Γb is the smallest quasi-ideal containing b, so that b bγm 1 Γb bγm 1 M 1 Γb. Since t b, t bγm 1 Γb. Similarly, s bγm 1 Γb. Hence there exists r 1, r 2 M 1, γ 1, γ 2, α 1, α 2 Γ, such that t = bγ 1 r 1 γ 2 b and s = bα 1 r 2 α 2 b. Since t L b \{b} and s R b \{b}, we have m 1, m 2 M sucht that b = m 1 β 1 t = sβ 2 m 2 for some β 1, β 2 Γ. If both r 1, r 2 M, we have b = m 1 β 1 t = m 1 β 1 bγ 1 r 1 γ 2 b and b = sβ 2 m 2 = bα 1 r 2 α 2 bβ 2 m 2, hence b is quasi-regular, therefore regular. If r 1 = 1, then t = bγ 1 r 1 γ 2 b = bγb for some γ Γ, b = m 1 β 1 t = m 1 β 1 bγb = m 1 β 1 bγ 1 m 1 β 1 bγb = m 1 β 1 bγ(m 1 β 1 b)γb and b = bα 1 r 2 α 2 bβ 2 m 2, therefore b is quasi-regular, hence regular. Similarly, if r 2 = 1 and r 1 M, b is regular. Since t s, we cannot have r 1 = r 2 = 1 otherwise t = bγb = s, and in every case, we have b is regular. Using the Theorem 3.9, we now discuss the restricted partial ordering of L- and R-classes in irregular D-classes. Proposition If M BQ and D a is an irregular D-class, then either aγm 1 Γa D a R a, or aγm 1 Γa D a L a. Proof. Suppose neither aγm 1 Γa D a R a, nor aγm 1 Γa D a L a. Then we have elements b and c such that b (aγm 1 Γa D a )\R a and c (aγm 1 Γa D a )\L a. Since bdc, there exists t R b L c, and R t = R b < R a for b aγm 1 Γa aγm 1. Furhtermore, L t = L c < L a since c aγm 1 Γa M 1 Γa. Thus by Theorem 3.9, a is regular which

8 208 KOSTAQ HILA and JANI DINE 8 is impossible. Therefore, we must have either aγm 1 Γa D a R a or aγm 1 Γa D a L a. Proposition If M BQ and D a is an irregular D-class, then aγm 1 Γa D a L a if and only if L a is minimal among the L-classes of M in D a. Proof. If L a is a minimal L-class of M in D a, suppose b aγm 1 Γa D a (if aγm 1 Γa D a = we are done), then L b L a and since L a is a minimal L-class in D a we have L b = L a and aγm 1 Γa D a L a. Assume that aγm 1 Γa D a L a. Let b D a with L b L a, then there exists r L b R a. hence r L b M 1 Γb M 1 Γa and r R a aγm 1. Thus r aγm 1 M 1 Γa = a aγm 1 Γa, for M BQ. If r = a we are done, for then L b = L r = L a. Otherwise r aγm 1 Γa D a L a, L r = L a and hence L a = L b. Thus, L a is a minimal L-class of M in D a. If aγm 1 Γa D a =, then R a and L a are both minimal among the R and L-classes of M in D a. Using Proposition 3.10 and Proposition 3.11 we get: Corollary Let M be a Γ-semigroup. If M BQ and D a ia an irregular D-class, then either L a or R a is minimal in the set of L- or R-classes of M in D a respectively. Lemma 3.13 ([15],Lemma 3.12). Let M be a Γ-semigroup. If M BQ and D is an irregular D-class, then for any two a, b D, either L a and L b are minimal in the set of L-classes of M in D, or R a and R b are minimal in the set of R-classes of M in D. Proof. Let x D, then either L x is minimal among the L-classes of D, or R x is minimal among the R-classes of D. Let a, b D, and suppose to the contrary that L a and R b are minimal while neither L b nor R a is minimal in the restricted partial ordering. Since L b is not minimal, there exists u D such that L u < L b, and similarly there exists v D such that R v < R a. Let t L u R v and r L b R a, then L t = L u < L b = L r and R t = R v < R a = R r. Therefore by Theorem 3.9, t is regular, which is impossible, since D is an irregular D-class. Thus either both L a and L b are minimal, or both R a and R b are minimal. Theorem Let M be a Γ-semigroup. If M BQ and D a is an irregular D-class of M. Then either D a = L a or D a = R a.

9 9 GREEN S RELATIONS IN BQ Γ-SEMIGROUPS 209 Proof. If D a L a and D a R a, then there is an element b D a such that L b L a and R b = R a. By Lemma 3.13 it follows that either both R a and R b are minimal among the R-classes of M in D a, or both L a and L b are minimal among the L-classes of M in D a. Assume R a and R b are minimal. Since M BQ we have: {a, b} aγm 1 Γa bγm 1 Γb aγm 1 Γb bγm 1 Γa = B(a, b) = Q(a, b) = (aγm 1 M 1 Γa) (bγm 1 M 1 Γb) (aγm 1 M 1 Γb) (bγm 1 M 1 Γa). ( ) Let u R a L b and r R b L a. It si clear that we must have u, r / {a, b}, and r u. Then u aγm 1 M 1 Γb, so u B(a, b). We consider (*). Since u is not regular, u / uγm 1 Γu = aγm 1 Γb. If u bγm 1 Γa or bγm 1 Γb, then R a = R u R b, and since R b is minimal, R a = R b, which is impossible. Thus u aγm 1 Γa and L b = L u L a. Similarly, r bγm 1 Γb and L a = L r L b. Thus L a = L b which is impossible. Hence if c D a, either c L a or c R a, and we have D a = R a L a. Assume that u R a \{a} and v L a \{a}, then let w R u L v D a = L a R a. Now either R v = R a or L u = L a, and thus either {v} = R v L v = R a L a = {a} or {u} = R u L u = R a L a = {a}, contrary to the hypothesis that u R a \{a} and v L a \{a}. Thus either R a \{a} = or L a \{a} =, and therefore either D a = L a or D a = R a. REFERENCES 1. Calais, J. Demi-groupes quasi-inversifs, C. R. Acad. Sci. Paris, 252 (1961), Calais, J. Demi-groups dans lesquels tout bi-ideal est un quasi-ideal, Symp. semigroups, Smolenice, June, Chinram, R.; Siammai, P. On Green s relations for Γ-semigroups and reductive Γ-semigroups, Int. J. Algebra, 2 (2008), Clifford, A.H.; Preston, G.B. The Algebraic Theory of Semigroups, Vol. I. Mathematical Surveys, No. 7 American Mathematical Society, Providence, R.I., Dutta, T.K.; Chatterjee, T.K. Green s equivalences on Γ-semigroup, Bull. Calcutta Math. Soc., 80 (1988), Good, R.A.; Hughes, D.R. Associated groups for a semigroup, Bull. Amer. Math. Soc., 58 (1952), Green, J.A. On the structure of semigroups, Ann. of Math., 54 (1951),

10 210 KOSTAQ HILA and JANI DINE Hila, K.; Pisha, E. On lattice-ordered Rees matrix Γ-semigroups, An. Ştiinţ. Univ. Al.I. Cuza Iaşi. Mat. (N.S.), 59 (2013), Hila, K. Filters in ordered Γ-semigroups, Rocky Mountain J. Math., 41 (2011), Hila, K. On quasi-prime, weakly quasi-prime left ideals in ordered-γ-semigroups, Math. Slovaca, 60 (2010), Hila, K. On regular, semiprime and quasi-reflexive Γ-semigroup and minimal quasi-ideals, Lobachevskii J. Math., 29 (2008), Hila, K.; Dine, J. On Green s relations, 20-regularity and quasi-ideals in Γ- semigroups, Acta Math. Sin. (Engl. Ser.), 29 (2013), Kapp, K.M. On bi-ideals and quasi-ideals in semigroups, Publ. Math. Debrecen, 16 (1969), Lajos, S. Generalized ideals in semigroups, Acta Sci. Math. Szeged, 22 (1961), Mielke, B.W. A note on bi-ideals and quasi-ideals in semigroups, Publ. Math. Debrecen, 18 (1971), (1972). 16. Petro, P.; Xhillari, T. Green s theorem and minimal quasi-ideals in Γ- semigroups, Int. J. Algebra, 5 (2011), Saha, N.K. On Γ-semigroup. II, Bull. Calcutta Math. Soc., 79 (1987), Sen, M.K. On Γ-semigroup, Algebra and its applications (New Delhi, 1981), , Lecture Notes in Pure and Appl. Math., 91, Dekker, New York, Sen, M.K.; Saha, N.K. On Γ-semigroup. I, Bull. Calcutta Math. Soc., 78 (1986), Steinfeld, O. Über die Quasiideale von Halbgruppen, Publ. Math. Debrecen, 4 (1956), Steinfeld, O. Quasi-Ideals in Rings and Semigroups, Disquisitiones Mathematicae Hungaricae 10, Akadémiai Kiadó, Budapest, Steinfeld, O. Über die Quasiideale von Ringe, (German) Acta Sci. Math. Szeged, 17 (1956), Received: 11.VIII.2011 Revised: 23.IV.2012 Accepted: 10.V.2012 Department of Mathematics & Computer Science, Faculty of Natural Sciences, University of Gjirokastra, ALBANIA kostaq hila@yahoo.com jani dine@yahoo.com