ON RIGHT(LEFT) DUO PO-SEMIGROUPS. S. K. Lee and K. Y. Park

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1 Kangwon-Kyungki Math. Jour. 11 (2003), No. 2, pp ON RIGHT(LEFT) DUO PO-SEMIGROUPS S. K. L and K. Y. Park Abstract. W invstigat som proprtis on right(rsp. lft) duo po-smigroups. 1. Introduction Khayopulu([6]) prov that vry idal of an N -class of an ordrd smigroup dos not contain propr prim idals. As a consqunc, ach prim idal of an ordrd smigroup is dcomposabl into its N - classs. In this papr, w giv th rlation btwn th lft(rsp. right) filtrs and th prim lft(rsp. right) idals. W dfin a smilattic congrunc N l (rsp. N r ) gnratd by th lft(rsp. right) filtr on a right(rsp. lft) duo po-smigroup and invstigat som proprtis on th right(rsp. lft) duo po-smigroups. Also w prov that vry lft(rsp. right) idal of N l -classrsp. N r -class) of a right(rsp. lft) duo po-smigroup dos not contain th propr prim lft(rsp. right) idals. As a consqunc, ach prim lft(rsp. right) idal of a right(rsp. lft) duo po-smigroup is dcomposabl into its N l - classs(rsp. N r -classs). A po-smigroup(: ordrd smigroup) is an ordrd st S at th sam tim a smigroup such that a b = xa xb and ax bx for all x S. Lt S b a po-smigroup. A nonmpty subst A of S is calld a lft(rsp. right) idal of S if (1) SA A(rsp. SA A), (2) a A and b a for b S = b A([3,5]). A is calld an idal of S if it is a lft and right idal of S. Rcivd July 14, Mathmatics Subjct Classification: 03G25, 06F35. Ky words and phrass: po-smigroup, idal, lft(right) idal, right(lft) duo, prim, prim lft(right) idal, lft(right) filtr, lft(right) congrunc, congrunc, smilattic congrunc.

2 148 S. K. L and K. Y. Park A po-smigroup S is said to b right(rsp. lft) duo if vry right(rsp. lft) idal is a lft(rsp. right) idal([4,5]). A non-mpty subst T of a po-smigroup S is said to b prim if AB T = A T or B T for substs A, B of S([8]). Equivalnt Dfinition: For lmnts a, b in a subst T ab T a T or b T. T is calld a prim lft(rsp. right) idal if T is prim as a lft(rsp. right) idal([2]). A non-mpty subsmigroup F of a po-smigroup S is calld a lft(rsp. right) filtr of S if (1) ab F for a, b S = b F (rsp. a F ), (2) a F, a c for c S = c F ([9]). A subsmigroup F of S is calld a filtr of S if F is a lft and right filtr ([2,4,5]). An quivalnc rlation σ on S is calld a lft congrunc(rsp. right congrunc) on S if (a, b) σ = (ac, bc) σ(rsp. (ca, cb) σ) for all c S. An quivalnc rlation σ on S is calld a congrunc it is a lft and right congrunc. A rlation σ is calld a smilattic congrunc on S if σ is a congrunc such that (x 2, x) σ and (xy, yx) σ([1,2,4]). Notation. For a smilattic congrunc σ, (z) σ is a class of th smilattic congrunc σ containing an lmnt z in a po-smigroup S. 2. Main Rsults. Lmma ([9]). Lt S b a po-smigroup and F a nonmpty subst of S. Th following ar quivalnt: 1) F is a lft(rsp. right) filtr of S. 2) S \ F = or S \ F is a prim lft(rsp. right) idal of S. From Lmma, w gt th following corollary. Corollary 1([2]). Lt S b a po-smigroup and F a nonmpty subst of S. Th following ar quivalnt: 1) F is a filtr of S. 2) S \ F = or S \ F is a prim idal of S. Proposition 1. A po-smigroup S dos not contain propr lft(rsp. right) filtrs if and only if S dos not contain propr prim lft(rsp. right) idals. Proof.. Assum that S contains a propr prim lft idal L of S. Thn = S \ L S. Sinc S \ (S \ L) = L, w not that S \ (S \ L)

3 On right(lft) duo po-smigroups 149 is a prim lft idal of S. By Lmma 1, S \ L is a propr lft filtr of S. It is impossibl. Hnc S dos not contain propr prim lft idals.. Suppos that F is a propr lft filtr of S. Thn S \ F. By Lmma 1, S \ F is a propr prim lft idal of S. It is impossibl. Hnc S dos not contain propr prim lft filtrs. By Proposition 1, w hav th following corollary. Corollary 2([6, Rmark 2]). A po-smigroup S dos not contain propr filtrs if and only if S dos not contain propr prim idals. Now w dfin a rlation N l on a po-smigroup S as follows: N l := {(x, y) N l (x) = N l (y)}, N r := {(x, y) N r (x) = N r (y)} whr N l (x) (rsp. N r (x)) is th lft (rsp. right) filtr of S gnratd by x S. Proposition 2. N l (rsp. N r ) is a smilattic congrunc on a right(rsp. lft) duo po-smigroup S. Proof. It is asy to chck that N l is an guivalnc rlation on S. Lt (x, y) N l. Thn N l (x) = N l (y). Sinc xz N l (xz) for all z S and N l (xz) is a lft filtr, w gt x N l (xz) and z N l (xz). Thus N l (x) N l (xz) and so y N l (y) = N l (x) N l (xz). Sinc y, z N l (xz) and N l (xz) is a subsmigroup of S, w gt yz N l (xz). Thrfor N l (yz) N l (xz). By symmtry, w gt N l (xz) N l (yz). Hnc N l (xz) = N l (yz). Thrfor N l is a right congrunc. Now w shall show that (x 2, x) N l. Lt x S. Sinc x 2 N l (x 2 ) and N l (x 2 ) is a lft filtr, w gt x N l (x 2 ). Thus N l (x) N l (x 2 ). Sinc x N l (x) and N l (x) is a subsmigroup of S, w gt x 2 N l (x). Hnc N l (x 2 ) N l (x). Thrfor N l (x 2 ) = N l (x), and so (x 2, x) N l. Nxt w shall show that (xy, yx) N l. Lt x, y S. Sinc xy N l (xy) and N l (xy) is a lft filtr, w hav x N l (xy). Suppos that y / N l (xy). Thn y S \ N l (xy). Sinc S \ N l (xy) is a prim right idal and S is a right duo, xy S(S \ N l (xy)) S \ N l (xy). It is impossibl. Thus y N l (xy). Sinc N l (xy) is a filtr, yx N l (xy). Thus N l (yx) N l (xy). By symmtry, N l (xy) N l (yx). Thrfor N l (xy) = N l (yx) and so (xy, yx) N l.

4 150 S. K. L and K. Y. Park Finally, w shall show that N l is a lft congrunc. Lt (x, y) N l, and z S. Thn N l (zx) = N l (xz) = N l (yz) = N l (zy). Thrfor N l is a lft congrunc. It follows that N l is a smilattic congrunc. Proposition 3. Lt S b a po-smigroup. If F is a lft filtr of S and F (z) Nl for z S, thn (z) Nl F. Proof. Assum that F is a lft filtr of S and a F (z) Nl for z S. If y (z) Nl thn (y) Nl = (z) Nl = (a) Nl. Thus (y, a) N l, and so N l (y) = N l (a). Sinc F is a lft filtr of S and a F, w hav N l (a) F. Thus y N l (y) = N l (a) F. Hnc (z) Nl F. Proposition 4. and (a, ab) N r. For a po-smigroup S, a b implis (a, ba) N l Proof. Suppos that a b. Sinc a N l (a) and N l (a) is a lft filtr, w gt b N l (a). Thus ba N l (a), and so N l (ba) N l (a). Sinc ba N l (ba) and N l (ba) is a lft filtr, w hav a N l (ba). Thus N l (a) N l (ba). Hnc N l (a) = N l (ba), and so (a, ba) N l. By symmtry, w can prov that (a, ab) N r. Proposition 5. Lt S b a right duo po-smigroup. If L is a lft idal of (z) Nl for z S thn L dos not contain propr prim lft idals. Proof. From Proposition 1, it is sufficint to prov that L dos not contain propr lft filtrs (of L). Lt F b a lft filtr of L and a F. Now w dfin T := {x S a 2 x F }. Thn T is a nonmpty st, sinc a 2 a = a 3 F. Now w show that F = T L. If y F, thn a 2 y F. Thus y T. Sinc F is a lft filtr of L, F L. Hnc y T L, and so F T L. Convrsly, if y T L, thn a 2 y F. Sinc F is a lft filtr of L, w gt y F. Thrfor F = T L. Nxt w show that T is a lft filtr of L. If x T and y T, thn a 2 x, a 2 y F. Sinc F is a lft filtr, w hav x, y F. Sinc a F, a 2 xy F. Thus xy T. If xy T for x, y L, thn (a 2 x)y = a 2 (xy) F. Sinc F is a lft filtr of L, w gt y F. If x T and x y for y L, thn a 2 x F. Sinc x y, w gt a 2 x a 2 y. Sinc F is a lft filtr, a 2 y F. Thus y T. Thrfor T is a lft filtr of L.

5 On right(lft) duo po-smigroups 151 W not that a F = T L L (z) Nl, and so T (z) Nl. Sinc T is a lft filtr of L, w hav (z) Nl T by Proposition 3. Thus L = (z) Nl L T L = F L, and so F = L. Hnc L dos not contain propr lft filtrs (of L). Thrfor by Proposition 1, L dos not contain propr prim right idals. Proposition 6. Lt S b a right duo po-smigroup and L a prim lft idal of S. Thn L = {(x) Nl x L}. Proof. Lt t (x) Nl for som x L. Sinc (x) Nl is a lft idal of (x) Nl, (x) Nl dos not contain propr prim lft idals by Proposition 5. If w prov that (x) Nl L is a prim lft idal of (x) Nl thn (x) Nl L = (x) Nl. W first show that (x) Nl L is a lft idal of (x) Nl. W not that (x) Nl L sinc x (x) Nl L. And (x) Nl ((x) Nl L) = (x) 2 Nl (x) Nl L (x) Nl SL (x) Nl L. Lt a (x) Nl L and b a for b (x) Nl. Sinc L is a lft idal of S, b is containd in L. Thus b (x) Nl L. Hnc (x) Nl L is a lft idal of (x) Nl. Finally, w show that (x) Nl L is prim in (x) Nl. Lt yz (x) Nl L for y, z (x) Nl. Sinc yz L and L is a prim lft idal of S, y is containd in L or z is containd in L. Hnc y (x) Nl L or z (x) Nl L. Thrfor (x) Nl L is a prim lft idal of (x) Nl. It follows that L {(x) Nl x L = {(x) Nl L x L} L. Thrfor L = {(x) Nl x L}. B. y similar mthods of Proposition 3, 5 and 6, w hav th followings: (1) If F is a right filtr of a po-smigroup S and F (z) Nr for z S, thn (z) Nr F. (2) If R is a right idal of (z) Nr of lft duo po-smigroups thn R dos not contain propr prim right idals. (3) If R is a prim right idal of lft duo po-smigroups, thn R = {(x) Nr x R}.

6 152 S. K. L and K. Y. Park 3. Exampls Now w giv an xampl of a lft filtr which is not a right filtr in po-smigroups and an xampl of a lft and right filtr in a posmigroup. Exampl 1([7]). Lt S := {a, b, c, d,, f} b a po-smigroup with Cayly tabl and Hass diagram on S as follows: a b c d f a b c d f b c d d d d c d d d d d d d d d d d d d d d d d f f f f f f f. d a b c Th st A := {, f} is a lft filtr, but not a right filtr of S. Thus A is not a filtr of S. Exampl 2([8]). Lt S := {a, b, c, d,, f} b a po-smigroup with Cayly tabl (Tabl 2) and Hass diagram (Figur 2) on S as follows: a b c d f a b c d f a b b d f b b b b b b b b b b b b d b b d f f f f f f f f f f d f. a b c Tabl 2 Figur 2 Th st B := {a, d, } is a lft and right filtr of S, and so B is a filtr of S.

7 On right(lft) duo po-smigroups 153 Rfrncs 1. N. Khayopulu, On filtrs gnralizd in po-smigroups, Math. Japon. 35(4) (1990), N. Khayoupulu, Rmark on ordrd smigroups, Math. Japon. 35(4) (1990), N.Khayoupulu, On lft rgular ordrd smigroups, Math. Japon. 35(6) (1990), N. Khayopulu, Right rgular and right duo ordrd smigroups, Math. Japon. 36(2) (1991), N. Khayopulu, Rgular duo ordrd smigroups, Math. Japon. 37(3) (1992), N. Khayopulu, On th dcomposition of prim idals of ordrd smigroups into thir N -class, Smigroup Forum 47 (1993), N. Khayopulu, A not on strongly rgular ordrd smigroups, Sci. Math. 1(1) (1993), S. K. L and Y. I. Kwon, A gnralization of th Thorm of Giri and Wazalwar, Kyungpook Math. J. 37(1) (1997), S. K. L and S. S. L, Lft(Right) Filtrs on po-smigroups, Kangwon- Kyungi Math. J. 8(1) (200), Dpartmnt of Mathmatics Education Gyongsang National Univrsity Chinju , Kora skl@nonga.gsnu.ac.kr