ON LEFT(RIGHT) SEMI-REGULAR AND g-reguar po-semigroups. Sang Keun Lee

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1 Kngweon-Kyungki Mth. Jour. 10 (2002), No. 2, pp 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, 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.

2 118 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.

3 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:

4 120 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-

5 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-

6 122 Sng Keun Lee nj Alger, St. Petersurg Gos. Ped. Herzen Inst., [In: Deompositions nd Homomorphi Mppings of Semigroups.] 4, (1992), N. Kehyopulu, On regulr, intr-regulr ordered semigroups, Pure Mthemtis nd Applitions, 4, (1993), N. Kehyopulu, On regulr, regulr duo ordered semigroups, Pure Mthemtis nd Applitions, 5(2), (1994), N. Kehyopulu, Note on i-idels in ordered semigroups, Pure Mthemtis nd Applitions, 6(4) (1995), N. Kehyopulu, On regulr ordered semigroups, Mthemie Jponie, 45(3) (1997), N. Kehyopulu, On norml ordered semigroups, Pure Mthemtis nd Applitions, 8(2-3-4) (1997), N. Kehyopulu, Note on interior idels, idel elements in ordered semigroups, Sinetie Mthemtie, 2(3) (1999), S. K. Lee nd Y. I. Kwon, On hrteriztions of right(left) semi-regulr posemigroups, Comm. Koren Mth. So., 9(2) (1997), S. K. Lee, On kehyopulu s theorems in po-semigroup, Sientie Mthemtie, 3(3) (2000), S. K. Lee, On left(right) semi-regulr po-semigroups, Kngweon-Kyungi Mth. J., 9(2) (2001), S. K. Lee nd C. H. H, Right(left) semi-regulrity on po-semigroup, Sientie Mthemtie Jponie, 55(2) (2002), Deprtment of Mthemtis Edutions Gyeongsng Ntionl University Jinju , Kore E-mil: sklee@nonge.gsnu..kr