Regular Elements and BQ-Elements of the Semigroup (Z n, )

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1 Iteratioal Mathematical Forum, 5, 010, o. 51, Regular Elemets ad BQ-Elemets of the Semigroup (Z, Ng. Dapattaamogko ad Y. Kemprasit Departmet of Mathematics, Faculty of Sciece Chulalogkor Uiversity, Bagkok 10330, Thailad Abstract A semigroup S is called a BQ-semigroup if the sets of bi-ideals ad quasi-ideals of S coicide. By a BQ-elemet of S we mea a elemet x S such that the bi-ideal ad the quasi-ideal of S geerated by x coicide. It is kow that every regular semigroup is a BQ-semigroup. We also have that every regular elemet of S is a BQ-elemet. We have kow that (Z,, the multiplicative semigroup of itegers modulo, is a regular semigroup if ad oly if is a square-free, ad it is a BQsemigroup if ad oly if either =4or is square-free. I this paper, the regular elemets ad the BQ-elemets of (Z, are characterized ad we show that the above kow results become cosequeces of these characterizatios. Mathematics Subject Classificatio: 0M17, 0M99 Keywords: Regular elemet, BQ-elemet, the multiplicative semigroup of itegers modulo 1 Itroductio ad Prelimiaries A elemet x of a semigroup S is called a regular elemet if x = xyx for some y S ad S is called a regular semigroup if all elemets of S are regular. Let Reg(S be the set of all regular elemets of S. A subsemigroup Q of a semigroup S is called a quasi-ideal of S if SQ QS Q, ad a bi-ideal of S is a subsemigroup B of S such that BSB B. The quasi-ideals are a geeralizatio of oe-sided ideals ad bi-ideals are a geeralizatio of quasi-ideals. Notice that i a commutative semigroup, quasiideals are ideals. The otio of quasi-ideal for semigroups was itroduced by Steifeld [11] i 1956 while the otio of bi-ideal for semigroups was itroduced

2 534 Ng. Dapattaamogko ad Y. Kemprasit ealier by Good ad Hughes [5] i 195. Kapp [6] used BQ to deote the class of semigroups whose sets of bi-ideals ad quasi-ideals coicide. He showed that every left[right] simple semigroup ad left[right] 0-simple semigroup belogs to BQ. A semigroup i BQ is called a BQ-semigroup [10]. For a oempty subset X of a semigroup S, (X b ad (X q deote respectively the quasi-ideal ad the bi-ideal of S geerated by X, i.e., (X q is the itersectio of all quasi-ideals of S cotaiig X ad (X b is the itersectio of all bi-ideals of S cotaiig X (see [1], p.10 ad p.1. Notice that (X b (X q. It is clear that S is a BQ-semigroup if ad oly if (X b =(X q for every oempty subset X of S. Propositio 1.1. ([3], p For a oempty subset X of a semigroup S, (X q = X (SX XS ad (X b = X X XSX. The followig is kow. Propositio 1.. ([9] Every regular semigroup is a BQ-semigroup. The followig result geeralizes Propositio 1.. Propositio 1.3. ([7] If B is a bi-ideal of a semigroup S such that B Reg(S, the B is a quasi-ideal of S. I fact, Propositio 1.3 is a special case of the followig fact. Propositio 1.4. Let X be a oempty subset of a semigroup S. If X Reg(S, the (X b =(X q. Proof. We kow that (X b (X q. If a SX XS, the a = sx = yt for some x, y X ad s, t S. Sice x Reg(S, x = xx x for some x S. Thus a = sx = sxx x = ytx x = y(tx x XSX. From Propositio 1.1, (X q (X b ad hece the result follows. Calais [] itroduced the followig propositio. Propositio 1.5. ([] A semigroup S is a BQ-semigroup if ad oly if (x, y b =(x, y q for all x, y S. From the proof of Propositio 1.5 give i [1], p.76, oe ca see that the followig result holds. Propositio 1.6. A commutative semigroup S is a BQ-semigroup if ad oly if (x b =(x q for all x S.

3 Regular elemets ad BQ-elemets of the semigroup (Z, 535 By a BQ-elemet of a semigroup S we mea a elemet x S such that (x b =(x q. The every elemet of a BQ-semigroup S is a BQ-elemet of S. From Propositio 1.6, we have that i a commutative semigroup S, S is a BQ-semigroup if ad oly if every elemet of S is a BQ-elemet. Let BQ(S deote the set of all BQ-elemets of S. We have by Propositio 1.3 that Reg(S BQ(S. Notice that i a zero semigroup S, every ozero elemet is a BQ-elemet of S which is ot regular. Let Z be the set of all itegers. We shall cosider the semigroup (Z,, the multiplicative semigroup of itegers modulo. Recall that Z cotais elemets ad Z = {0, 1,..., 1} = {x x Z} where x is the equivalece class of x modulo. For a, b Z ot both zero, let (a, b deote the g.c.d. of a ad b. The (a, b = 1, that is a ad b are relatively prime if ad oly if xa + yb = 1 for some x, y Z. We also have that for k Z, kz =(k, Z = kz = { ( 0, (k,, (k,,..., (k, (k, 1 } (k, where X deotes the cardiality of a set X. For a, b Z ad a 0,a b meas that a divides b. We recall that is called square-free if there is o a Z such that a>1 ad a. The is square-free if ad oly if either =1or is a product of distict primes. Ehrlich [4] proved the followig fact. Theorem 1.7. ([4] The semigroup (Z, is a regular semigroup if ad oly is square-free. By makig use of Propositio 1. ad Theorem 1.7, Kemprasit ad Baupradist [8], gave the followig result. Theorem 1.8. ([8] The semigroup (Z, is a BQ-semigroup if ad oly if either =4or is square-free. I [1], the regular elemets of the semigroup (Z, were characterized i terms of the Euler s phi-fuctio ad the regular elemets of (Z, were couted. I this paper, we characterize the regular elemets of (Z, differetly ad we have Theorem 1.7 as a cosequece. The BQ-elemets of (Z, are also determied ad Theorem 1.8 is obtaied as a cosequece.

4 536 Ng. Dapattaamogko ad Y. Kemprasit Mai Results First, we characterize the regular elemets of the semigroup (Z,. Theorem.1. For x Z, x Reg(Z, if ad oly if x ad (x, are relatively prime. Proof. Assume that x Reg(Z,. The x = xȳ x for some y Z. The x(xy 1 = 0, so x(xy 1. Hece x (xy 1. But sice (x, (x, (x, ad x are relatively prime, it follows that k xy 1. The (x, (x, (x, = xy 1 ( for some k Z. Now we have xy + ( k = 1. This implies that x (x, ad xk + ad (x, are relatively prime. For the coverse, assume that x ad (x, l (x, = 1 for some k, l Z. Thusx = x k + xl are relatively prime. The (x, = x k + x (x, Z. This implies that x = x k ad thus x Reg(Z,. ( x (x, l Corollary.. The semigroup (Z, is a regular semigroup if ad oly is square-free. Proof. Assume that is ot square-free. The there is a Z such that a>1 ad a, soa a ad (, a = ( a. Thus a, = a>1. By Theorem (, a.1, a/ Reg(Z,. Hece (Z, is ot a regular semigroup. Coversely, assume that is a square-free. The either =1oris a product of distict primes. It clearly follows that for every x Z, x ad (x, are relatively prime. The we have by Theorem.1 that (Z, is a regular semigroup. Next, we characterize the BQ-elemets of (Z,. Sice (Z, is a commutative semigroup havig 1 as its idetity, we have that BQ(Z, ={x Z x Z ad (x b =(x q } = {x Z x Z ad {x} x Z = xz }. Theorem.3. For x Z, x BQ(Z, if ad oly if either (i x ad are relatively prime or (x, (ii x ad (x, =.

5 Regular elemets ad BQ-elemets of the semigroup (Z, 537 Proof. Assume that x BQ(Z,. The {x} x Z = xz. Case 1: x x Z. The x Reg(Z,, so by Theorem.1, x satisfies (i. Case : x / x Z. Sice x Z = (x, ad xz =, it follows (x, that 1 + (x, = {x} x Z = xz = (x,. But (x, (x,, thus. This implies that 1. The (x, (x, (x, (x, =1,so(x,=. Therefore x ad (x, =1+ = =. Hece x satisfies (ii. (x, Coversely, assume that x satisfies (i or (ii. If x satisfies (i, the by Theorem.1, x Reg(Z,. But Reg(Z, BQ(Z,, so x BQ(Z,. Next, let x satisfy (ii. Sice x, we have that x Z = {0}, so{x} x Z = {0, x}. We also have that xz = {x} x Z = xz. Hece x BQ(Z,. The theorem is thereby proved. (x, =. ThusxZ = {0, x}. Cosequetly, Corollary.4. The semigroup (Z, is a BQ-semigroup if ad oly if either =4or is a square-free. Proof. Assume that 4 ad is ot square-free. The there is a Z such that a>1 ad a. We claim that a does ot satisfy (i ad (ii of Theorem.3. Sice a, there is t Z such that = a t. The (a, = a t = at, so ( a a, = a>1. Hece a does ot satisfy (i. To show that a does ot (a, satisfy (ii, i.e., a or (a,, assume that a. But sice a, it follows that = a. Sice 4, we have that a>. Hece (a, = a a = a>. Thus a does ot satisfy (ii. By Theorem.3, a / BQ(Z,. Hece from Propositio 1.6, (Z, is ot a BQ-semigroup. For the coverse, assume that =4or is square-free. If is square-free, the by Corollary.(Theorem 1.7, (Z, is a regular semigroup ad hece (Z, isabq-semigroup by Propositio 1.. Next, assume that =4. If x {0, 1, 3}, the x satisfies (i of Theorem.3. If x =, the x satisfy (ii of Theorem.3. Hece by Theorem 0, 1,, 3 BQ(Z 4, ad therefore (Z 4, is a BQ-semigroup by Propositio 1.6. Therefore the proof is completed.

6 538 Ng. Dapattaamogko ad Y. Kemprasit Example.5. From Theorem.1 ad Theorem.3, we have Reg(Z 9, ={0, 1,, 4, 5, 7, 8} = BQ(Z 9,, Reg(Z 18, ={0, 1,, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17} = BQ(Z 18,, Reg(Z 8, ={0, 1, 3, 5, 7}, BQ(Z 8, =Reg(Z 8, {4}, Reg(Z 1, ={0, 1, 3, 4, 5, 7, 8, 9, 11}, BQ(Z 1, =Reg(Z, {6}. From Example.5, it is atural to ask whether { it is true that BQ(Z, = ( } ( Reg(Z, if4 ad BQ(Z, =Reg(Z, ad / Reg(Z, if 4. This is geerally true as the followig theorem shows : Theorem.6. The followig statemets holds. (i If 4, the BQ(Z, =Reg(Z,. { ( } ( (ii If 4, the BQ(Z, =Reg(Z, ad / Reg(Z,. Proof. (i Assume that there is x Z such that x ad =. The ( (x, x x ad. Sice (x, (x, (x, ad x are relatively prime, it (x, follows that x, so x. Hece (x,. Sice =(x, ad (x,, we (x, have that 4. This proves that if 4, the there is o x Z satisfyig (ii of Theorem.3. From Theorem.1 ad Theorem.3, we have that if 4, the BQ(Z, =Reg(Z,. (ii Assume that 4. The ad ( =, so we have by Theorem.1 that, / Reg(Z,. Sice = which is divided by, ( ( ( ( 4. Therefore by Theorem.3, BQ(Z,. It remais to show that { ( } BQ(Z, \ Reg(Z, =. Let x Z be such that x BQ(Z, ad x/ Reg(Z,. By Theorem.1 ad Theorem.3, x ad are ot relatively prime, x (x, ad =. The x 0 ad x. Let k Z be such (x, that x = k + r where 0 r<. Sice x, we have that 0 <r<. We also have that (x, r. Thus r. Cosequetly, r<,sor = + i for some i {0, 1,, } 1. Sice r, it follows that i = 0. The r = ad ( therefore x = r =. Hece (ii is proved.

7 Regular elemets ad BQ-elemets of the semigroup (Z, 539 Refereces [1] O. Alkam ad E. A. Usha, O regular elemets i Z, Turk J. Math. 3 (008, [] J. Calais, Demigroupes das lesquels tout bi-idéal est u quasi-idéal, Semigroup Symposium, Smoleice, [3] A. H. Clifford ad G. B. Presto, The Algebraic Theory of Semigroups, Vol. I, Amer. Math. Soc., Providece, [4] G. Ehrlich, Uit-regular rigs, Port. Math. 7 (1968, [5] R. A. Good ad D. R. Hughes, Associated groups for semigroups, Bull. Amer. Math. Soc. 58 (195, 64-65(Abstract. [6] K. M. Kapp, O bi-ideals ad quasi-ideals i semigroups, Publ. Math. Debrece 16 (1968, [7] K. M. Kapp, Bi-ideals i associative rigs ad semigroups, Acta Sci. Math. 33 (197, [8] Y. Kemprasit ad S. Baupradist, A ote o the multiplicative semigroups Z whose bi-ideals are quasi-ideals, Southeast Asia Bull. Math. 5 (001, [9] S. Lajos, Geeralized ideals i semigroups, Acta Sci. Math. (1961, 17-. [10] B. M. Mielke, A ote o Gree s relatios i BQ-semigroups, Czechoslovak Math. J. (197, 4-9. [11] O. Steifeld, Über die quasiideale vo halbgruppe, Publ. Math. Debrece, 4 (1956, [1] O. Steifeld, Quasi-ideals i Rigs ad Semigroups, Akadémiai Kiadó, Budapest, Received: April, 010