Vol. 19 (2018), No. 1, pp. 141–147 DOI: 10.18514/MMN.2018.1974
GENERALIZED SUPPLEMENTED LATTICES
C¸ I ˘GDEM BIC¸ ER, CELIL NEBIYEV, AND ALI PANCAR Received 19 April, 2016
Abstract. In this work, we define (amply) generalized supplemented lattices and investigate some properties of these lattices. In this paper, all lattices are complete modular lattices with the smallest element0and the greatest element1. LetLbe a lattice,1Da1_a2_: : :_anand the quotient sublatticesa1=0,a2=0,. . . ,an=0be generalized supplemented, thenLis general- ized supplemented. IfLis an amply generalized supplemented lattice, then for everya2L, the quotient sublattice1=ais amply generalized supplemented.
2010Mathematics Subject Classification: 06C05; 06C15
Keywords: lattices, supplemented lattices, amply supplemented lattices, hollow lattices
1. INTRODUCTION
Throughout this paper, all lattices are complete modular lattices with the smallest element 0 and the greatest element 1. Let L be a lattice, a; b2L and ab. A sublattice fx2Ljaxbg is called a quotient sublattice, denoted by b=a. An elementa0of a latticeLis called acomplementofaifa^a0D0anda_a0D1. A latticeLis said to becomplementedif each element inLhas at least one complement.
An elementcofLis said to becompactif for every subsetXofLsuch thatc _X there is a finiteFXsuch thatc _F. A latticeLis said to becompactly generated if each of its elements is a join of compact elements. An elementaofLis said to be smallorsuperfluousifa_b¤1 holds for everyb¤1and denoted byaL.
The meet of all the maximal elements.¤1/of a lattice Lis called the radicalof Land denoted byr.L/. An elementc ofLis called asupplementofb inLif it is minimal forb_c D1. ais a supplement ofb in latticeLif and only ifa_bD1 and a^ba=0. L is called asupplemented lattice if every element ofL has a supplement in L. We say that an element b ofL lies above an elementa ofL if ab andb1=a. Lis said to be hollowif every element.¤1/ is superfluous inL, andLis said to belocalifLhas a greatest element.¤1/. An elementaof Lis called aweak supplement ofb inLifa_b D1 anda^bL. Lis called a weakly supplemented lattice, if every element ofLhas a weak supplement inL. An elementa2Lhasample supplementsinLif for everyb2Lwitha_bD1,ahas a supplementb0inLwithb0b.Lis called anamply supplemented lattice, if every
c 2018 Miskolc University Press
element ofLhas ample supplements inL. It is clear that every supplemented lattice is weakly supplemented and every amply supplemented lattice is supplemented.
More information about (amply) supplemented lattices are in [1,2] and [3]. More results about (amply) supplemented modules are in [4,6] and [7]. The definitions of (amply) generalized supplemented modules and some properties of them are in [5].
We generalize some properties of (amply) generalized supplemented modules.
In this paper, we constitute relationships between generalized supplemented quo- tient sublattices and generalized supplemented lattices by Corollary1, Theorem3and Corollary2. We also constitute relationships between amply generalized supplemen- ted quotient sublattices and amply generalized supplemented lattices by Proposition 3and Proposition5.
2. GENERALIZED SUPPLEMENTED LATTICES
In this part, generalized supplement elements and generalized supplemented lat- tices are defined and some properties of them are given.
Definition 1. LetLbe a lattice anda2L. An elementb2Lis called ageneral- ized supplementorRad-supplementofainLifa_bD1anda^br.b=0/. In this fact,ahas a generalized supplement.or Rad-supplement/inL. Lis said to begen- eralized supplementedorRad-supplementedif every element ofLhas a generalized supplement or Rad-supplement inL.
It is clear that every supplement element is generalized supplement in a lattice, also supplemented lattices are generalized supplemented but a generalized supplemented lattice need not to be supplemented. We give an example about this in the next part.
Lemma 1([3], Theorem 1.5). LetLbe a lattice anda; b2L. Then the quotient sublattices.a_b/=b anda=.a^b/are isomorphic.
Lemma 2. LetLbe a lattice andcbe a generalized supplement ofb inL.
.a/Ifabanda_cD1thencis a generalized supplement ofainL.
.b/ r.c=0/Dc^r.L/.
Proof. .a/Sincec is a generalized supplement ofb inL,b_cD1andb^c r.c=0/. Sinceab,a^cb^cr.c=0/. Thusc is a generalized supplement of ainL.
.b/ r.c=0/c^r.L/is clear. Letx be a maximal element.¤c/inc=0. Since
1
b_x Dbc__bx Dc_b_b_xx Šc^.bc_x/ D.c^b/c_x Dcx,b_xis a maximal element.¤1/
ofL. Hencer .L/b_xandc^r .L/c^.b_x/D.c^b/_xDx. Sincec^ r .L/xfor every maximal element.¤c/inc=0,c^r .L/r .c=0/andr.c=0/D
c^r.L/.
Theorem 1. LetLbe a lattice and b be a maximal element.¤1/ inL. Ifc is a generalized supplement of b in Lthen b^c Dr.c=0/ andr.c=0/is the unique maximal element.¤c/ofc=0.
Proof. Sinceb is maximal.¤1/inL, by Lemma1,b^c is a maximal element .¤c/ in c=0. Because of definition of the radical, r.c=0/ b^c, and c being generalized supplement of b in L, b^c r.c=0/. Hence b^c Dr.c=0/ is the
unique maximal element inc=0.
Theorem 2. LetLbe a lattice and1¤b2L. Ifcis a generalized supplement of b inLandr.L/L, then there exists a maximal elementm .¤1/inLsuch that bm.
Proof. Ifbr.L/, then it is clear. Letb—r.L/. Thenr.c=0/Dc^r.L/¤c by Lemma 2 .b/. Hence there exists an elementx c such that x is a maximal element.¤c/ inc=0. Let mDb_x. By Lemma1 and modularity m1 D b_1x D
b_c
b_x Šc^.bc_x/D.c^b/c _x Dxc and somis a maximal element.¤1/inLsuch that
bm.
Lemma 3([3], Lemma 7.8 (i)). LetLbe a lattice. Then a_r.L/r.1=a/ for everya2L.
Lemma 4([3], Exercise 7.3). LetLbe a lattice anda2L. Thenr.a=0/r.L/.
Lemma 5. LetLbe a lattice andc be a generalized supplement ofb inL. Then forab,a_c is a generalized supplement ofbin1=a.
Proof. Sincecis a generalized supplement ofbinL,b_cD1andb^cr.c=0/.
Thenb_.a_c/D1and by Lemma3and Lemma4,.a_c/^bDa_.b^c/a_ r.c=0/a_r..a_c/ =0/r..a_c/ =a/. Hencea_cis a generalized supplement
ofbin1=a.
Corollary 1. LetLbe a lattice. IfLis generalized supplemented anda2L, then 1=ais also generalized supplemented.
Proof. Clear from Lemma5.
Lemma 6 ([3], Lemma 12.3). Let L be a lattice. Then for every a; b; c 2L, Œ.b_c/^aŒb^.a_c/_Œc^.a_b/.
Lemma 7. LetLbe a lattice anda; b; c2L. Ifa=0is generalized supplemented anda_bhas a generalized supplement inL, thenbhas a generalized supplement in L.
Proof. Letc be a generalized supplement of a_b in Landd be a generalized supplement of.b_c/^aina=0. Clearly,1Da_b_cDb_c_d,.b_c/^d r.d=0/,b^.c_d /Œc^.b_d /_Œd^.b_c/r.c=0/_r.d=0/r..c_d / =0/.
Hencec_d is a generalized supplement ofbinL.
Theorem 3. LetLbe a latticea; b2Landa_bD1. Ifa=0andb=0are gener- alized supplemented, thenLis generalized supplemented.
Proof. Letc be arbitrary inL. a_.b_c/D1 has a generalized supplement0 inL. Since a=0is generalized supplemented, by Lemma7,b_c has a generalized supplement in L. And sinceb=0 is generalized supplemented,c has a generalized
supplement inL. So,Lis generalized supplemented.
Corollary 2. If1Da1_a2_: : :_anand the quotient sublatticesa1=0; a2=0; : : : ; an=0be generalized supplemented, thenLis generalized supplemented.
Ifa < b andac < bimplies thatcDa, thenais said to becovered byb. If0 is covered byafor some elementaofL, thenais called anatom. A latticeLis said to besemiatomicif1is a join of atoms inL. [3]
Lemma 8([3] , Theorem 6.7). Every modular compactly generated complemented lattice is semiatomic.
Proposition 1. LetLbe a modular compactly generated lattice. IfLis general- ized supplemented, then the quotient sublattice1=r.L/is semiatomic.
Proof. Clearly,1=r.L/is a modular compactly generated lattice. Leta21=r.L/.
SinceLis generalized supplemented,ahas a generalized supplementb inL. Then we have a_.b_r.L//D1anda^.b_r.L//Dr.L/. Hence1=r.L/is comple-
mented and by Lemma8,1=r.L/is semiatomic.
Proposition 2. LetLbe a lattice. IfLis generalized supplemented andr.L/L, thenLis weakly supplemented.
Proof. Clear.
3. AMPLY GENERALIZED SUPPLEMENTED LATTICES
In this section, we define amply generalized supplemented or amply Rad-supple- mented lattices and we give some properties of them.
Definition 2. LetLbe a lattice. We say an elementaofLhasample generalized supplementsinLif for everybofLwitha_bD1,ahas a generalized supplement b0 in L with b0b. L is called amply generalized supplemented or amply Rad- supplementedif every element ofLhas ample generalized supplements inL.
It is clear that every amply generalized supplemented lattice is generalized supple- mented. But the converse is not always true (see Example1). Every amply supple- mented modular lattice is amply generalized supplemented.
Proposition 3. LetLbe a lattice. IfLis amply generalized supplemented, then the quotient sublattice1=ais also amply generalized supplemented for every element aofL.
Proof. Leta2Landx; y21=awith x_y D1. SinceLis amply generalized supplemented, there exists an elementy0ofLsuch thaty0y; x_y0D1andx^
y0r.y0=0/. Thenx^.a_y0/Da_.x^y0/a_r.y0=0/r..a_y0/=a/. Hence a_y0is a generalized supplement ofxin1=awitha_y0y.
Proposition 4. LetLbe a lattice,a; b2Landa_bD1. Ifaandb have ample generalized supplements inL, thena^bhas ample generalized supplements inL.
Proof. Let.a^b/_c D1with c2L. We clearly see thatb_.a^c/D1and a_.b^c/D1. Sincea_.b^c/D1andahas ample generalized supplements inL, then there exists an elementxofLsuch thatxb^c,a_xD1anda^xr.x=0/.
Since b_.a^c/D1 and b has ample generalized supplements in L, then there exists an elementy ofLsuch thatya^c,b_yD1andb^y r.y=0/. Since xb anda_xD1,bDb^1Db^.a_x/D.a^b/_x. Similarly, we see that aD.a^b/_y. Then1Da_b DŒ.a^b/_y_Œ.a^b/_xD.a^b/_x_y.
Sincea^xr.x=0/andb^yr.y=0/, then.a^b/^.x_y/D.a^x/_.b^y/
r.x=0/_r.y=0/r..x_y/=0/. Hencex_y is a generalized supplement ofa^b
inLwithx_yc.
Proposition 5. LetL be a lattice andabe a supplement element in L. If Lis amply generalized supplemented, then the quotient sublatticea=0is amply general- ized supplemented.
Proof. Clear.
Theorem 4. Let L be a lattice. If L is amply generalized supplemented, then for every a2L, there exist x; y2Lsuch thatx is a generalized supplement inL, aDx_yandyr .L/.
Proof. Letbbe a generalized supplement ofainLandxbe a generalized supple- ment ofbinLwithxa. Sincebis a generalized supplement ofainL,a_bD1 anda^br.b=0/. Sincex is a generalized supplement ofb inL, x_bD1and x^br.x=0/. LetyDa^b. This caseyDa^br.b=0/r.L/. Sincex_bD1 andxa,aDa^1Da^.x_b/Dx_.a^b/Dx_y. Lemma 9. LetLbe a lattice anda; b2Lsuch thatab. Then the following assertions are equivalent.
.a/ blies aboveainL.
.b/ a_xD1for every elementx2Lwithb_xD1.
Proof. .a/).b/ Letb lies abovea inL. Thenb 1=a. Let b_xD1 with x2L. Thenb_a_xD1, and sincea_x21=aandb1=a, thena_xD1.
.b/).a/Letb_xD1withx21=a. By hypothesis,a_xD1and sincexa,
xDa_xD1. Thusblies abovea.
Lemma 10. LetLbe a lattice. If every element ofLlies above an elementxinL such thatx=0is generalized supplemented, thenLis amply generalized supplemen- ted.
Proof. Leta_b D1. By hypothesis, alies above an element x inL such that x=0 is generalized supplemented. Since a1=x, by Lemma 9, b_xD1. Let y be a generalized supplement of b^x inx=0. This case 1Db_xDb_y and b^yr.y=0/. Henceyis a generalized supplement ofbinLwithya.
Corollary 3. Let L be a lattice. If a=0 is generalized supplemented for every a2L, thenLis amply generalized supplemented.
Proof. Clear from Lemma10, because every elementaofLlies aboveainL.
We can prove this Corollary directly as follows:
Leta; b2Lwitha_bD1. Sincea^b2b=0, by assumption, there is a gener- alized supplementc ofa^binb=0. That is, c2b=0,.a^b/_cDb anda^cD a^b^cr .c=0/. Sincea_bD1and.a^b/_cDb,1Da_bDa_.a^b/_cD a_c. Hencec is a generalized supplement ofainLwithcb. ThusLis amply generalized supplemented.
Example1. Let˝ be family of all the submodules ofZ-moduleQ. ˝ is a lat- tice with . This case, for K; L2˝, K_LDsupfK; Lg DKCL, K^LD i nffK; Lg DK\L. ˝ is generalized supplemented but not supplemented [7] . SinceZis a Dedekind domain,Qis a quotient field ofZandZis not local,˝is not amply generalized supplemented.
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Authors’ addresses
C¸ i˘gdem Bic¸er
Ondokuz Mayıs University, Department of Mathematics, Kurupelit, Atakum Samsun, Turkey E-mail address:cigdem bicer184@hotmail.com
Celil Nebiyev
Ondokuz Mayıs University, Department of Mathematics, Kurupelit, Atakum Samsun, Turkey E-mail address:cnebiyev@omu.edu.tr
Ali Pancar
Ondokuz Mayıs University, Department of Mathematics, Kurupelit, Atakum Samsun, Turkey E-mail address:apancar@omu.edu.tr
