Kngweon-Kyungki Mth. Jour. 10 (2002), No. 2, pp. 117 122 ON LEFT(RIGHT) SEMI-REGULAR AND g-reguar po-semigroups Sng Keun Lee Astrt. In this pper, we give some properties of left(right) semi-regulr nd g-regulr po-semigroups. 1. Introdution Lee introdued the onepts of the left(right) semi-regulrity([8]) nd the g-regulrity ([9]) in po-semigroup S whih re the generlized regulrities. The uthor investigtes some hrteriztions of the left(right) semi-regulrity in terms of some types of idels([8], [10], [11]). In this pper, we give some properties of left(right) semi-regulr nd g-regulr po-semigroups. A po-semigroup(: ordered semigroup) is n ordered set (S, ) t the sme time semigroup suh tht = nd for ll S. An element of po-semigroup S is regulr if x for some x S nd S is regulr if every element of S is regulr. An element of po-semigroup S is left(resp. right) regulr if x 2 (resp. 2 x) for some x S nd S is left(resp. regulr) if every element of S is left(resp. right) regulr([3], [4]). An element of po-semigroup S is left(resp. right) semi-regulr if xy(resp. x y ) for some x, y, x, y S nd S is left(resp. right) semi-regulr if every element of S is left(resp. right) semi-regulr([8]). An element of po-semigroup S is g-regulr if xy for some x, y S nd S is g-regulr if every element of S is g-regulr([9]). Reeived Mrh 28, 2002. 2000 Mthemtis Sujet Clssifition: 06F05. Key words nd phrses: po-semigroup, poe-semigroup, left(right) semi-regulr, left(right) regulr, g-regulr, left(right) idel, idel.
118 Sng Keun Lee Remrk. (1) A regulr(left regulr) po-semigroup S is left semiregulr. (2) A regulr(right regulr) po-semigroup S is right semi-regulr. (3) A left(right) semi-regulr po-semigroup S is g-regulr. The onverses of (1), (2) nd (3) re not true, in generl(f. Exmple 1, 2). We denote (H] := {t S t h for some h H} for suset H of po-semigroup S. A non-empty suset A of po-semigroup S is lled left(resp. right) idel of S if (1) SA A (resp. AS A), (2) A, for S imply A. A non-empty suset A of po-semigroup S is n idel of S if it is oth left nd right idel of S([5]). We note tht the left, right idel nd the idel of po-semigroup S generted y S re respetively: L() = ( S], R() = ( S], I() = ( S S SS]. For po-semigroup S, the Green s reltions L, R re defined s follows: R := {(x, y) R(x) = R(y)}, L := {(x, y) L(x) = L(y)}. Then L nd R re equivlene reltions of po-semigroup S. 2. Min Theorems Theorem 1. If n element of po-semigroup S is left(resp. right) semi-regulr nd L(resp. R,)( S), then is left(resp. right) semi-regulr. Proof. If L, then ( S] = ( S]. Thus ( or u) nd ( or v) for some u, v S. Thus we hve four ses: (1) =, (2), v, (3) u, or (4) u, v for some u, v S. Cse (1) If =, then = xy = xy for some x, y S. Cse (2) If, nd v, then xy x(v)y(v) = (xv)(yv) for some x, y, v S.
On left(right) semi-regulr nd g-regur po-semigroups 119 Cse (3) If u, nd, then u uxy (ux)y for some x, y, u S. Cse (4) If u, nd v, then u uxy ux(v)y(v) = (uxv)y(v) for some x, y, u, v S. For ny ses, (SS]. Therefore is left semi-regulr. If R, then we n show tht is right semi-regulr y the similr method. By the Exmple 2 in the next setion, we hve the following theorem. Theorem 2. If n element of po-semigroup S is left(resp. right) semi-regulr, then L()(resp. R()) need not e left(resp. right) semiregulr. Theorem 3. If n element of po-semigroup S is g-regulr nd L(R)( S), then is g-regulr. Proof. If L, then ( S] = ( S]. Thus ( or u) nd ( or v) for some u, v S. Thus we hve four ses: (1) =, (2), v, (3) u, or (4) u, v for some u, v S. Cse (1) If =, then = xy = xy for some x, y S. Cse (2) If, v : xy x(v)y = (xv)y for some x, y, v S. Cse (3) If u, : u uxy (ux)y for some x, y, u S. Cse (4) If u, v : u uxy ux(v)y = (uxv)y for some x, y, u, v S. For ny ses, (SS]. Therefore is g-regulr. If R, then we n show tht is g-regulr y the similr method. By the Exmple 3 in the next setion, we hve the following theorem. Theorem 4. If n element x of po-semigroup S is regulr nd ylx(yrx)(y S), then y need not to e regulr element. 3. Exmples Exmple 1([1]). Let S := {,,, d, f, g} e po-semigroup with Cyley tle (Tle 1) nd Hsse digrm (Figure 1) s follows:
120 Sng Keun Lee d f g d d d d d d d d d d d f. f g d d f g g Tle 1 Figure 1 S is g-regulr. Indeed: = = nd f = = f. And for other elements, it is trivil. S is not left(right) semi-regulr. Indeed: (SS] = (SS] = {, d, g} for S. Thus there does not exist x, y S suh tht xy nd xy for S. Hene the g-regulrity is generlised onept thn the left(right) semi-regulrity in po-semigroups. Exmple 2([2]). Let S := {,,, d, e} e po-semigroup with Cyley tle (Tle 2) nd Hsse digrm (Figure 2) s follows: e d e d e d d d d d d d e d e Tle 2. Figure 2 d S is left semi-regulr ut not right semi-regulr. Indeed: If x S is idempotent, then x = x 2 = x 4 SxSx (SxSx] nd x = x 2 = x 4 SxSx (xsxs]. Thus x is left(right) semi-regulr. Sine ll elements of S exept re idempotent, it is suffiient to show tht is left semi-
On left(right) semi-regulr nd g-regur po-semigroups 121 regulr. Sine (SS] = ({, d}s] = ({, d}] = ({, d}] = {,, d}, (SS]. Thus xy for some x, y S, nd so S is left semiregulr. But, sine (SS] = ({}S] = (S] = (] = {}, / (SS]. Thus is not right semi-regulr, nd so S is not right semi-regulr. Also S is not regulr. Left idels generted y n element of S re L() = L() = L(d) = {,, d} nd L() = L(e) = S. Right idels generted y n element of S re R() = {}, R() = {, }, R() = {,, }, R(d) = {,, d} nd R(e) = S. Moreover d is right semi-regulr. But R(d) is not right semiregulr euse R(d) ontins whih is not right semi-regulr. Exmple 3([6]). Let S := {,,, d, e} e po-semigroup with Cyley tle (Tle 3) nd Hsse digrm (Figure 3) s follows: d e d e d d d e Tle 3 d e. Figure 3 Left idels generted y n element of S re L() = L() = L() = {,, }, L(d) = {,,, d} nd L(e) = S. Right idels generted y n element of S re R() = {, }, R() = {}, R() = {,, }, R(d) = {,,, d} nd R(e) = S. Sine (S] = (] = {}, / (S]. Thus is not regulr element. But,, d, e re regulr elements. Sine L() = {,, } = L(), L. is regulr element of S, ut is not regulr element. Referenes 1. N. Kehyopulu nd M. Tsingelis, Remrk on ordered semigroups, Soveremen-
122 Sng Keun Lee nj Alger, St. Petersurg Gos. Ped. Herzen Inst., [In: Deompositions nd Homomorphi Mppings of Semigroups.] 4, (1992), 50-55. 2. N. Kehyopulu, On regulr, intr-regulr ordered semigroups, Pure Mthemtis nd Applitions, 4, (1993), 477-461. 3. N. Kehyopulu, On regulr, regulr duo ordered semigroups, Pure Mthemtis nd Applitions, 5(2), (1994), 161-176. 4. N. Kehyopulu, Note on i-idels in ordered semigroups, Pure Mthemtis nd Applitions, 6(4) (1995), 333-344. 5. N. Kehyopulu, On regulr ordered semigroups, Mthemie Jponie, 45(3) (1997), 549-553. 6. N. Kehyopulu, On norml ordered semigroups, Pure Mthemtis nd Applitions, 8(2-3-4) (1997), 281-191. 7. N. Kehyopulu, Note on interior idels, idel elements in ordered semigroups, Sinetie Mthemtie, 2(3) (1999), 407-409. 8. S. K. Lee nd Y. I. Kwon, On hrteriztions of right(left) semi-regulr posemigroups, Comm. Koren Mth. So., 9(2) (1997), 507-511. 9. S. K. Lee, On kehyopulu s theorems in po-semigroup, Sientie Mthemtie, 3(3) (2000), 367-369. 10. S. K. Lee, On left(right) semi-regulr po-semigroups, Kngweon-Kyungi Mth. J., 9(2) (2001), 99-104. 11. S. K. Lee nd C. H. H, Right(left) semi-regulrity on po-semigroup, Sientie Mthemtie Jponie, 55(2) (2002), 271-274. Deprtment of Mthemtis Edutions Gyeongsng Ntionl University Jinju 660-701, Kore E-mil: sklee@nonge.gsnu..kr