半群的L-模糊理想 L-Fuzzy Ideals of Semigroups

doi:10.12677/PM.2021.116124

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PureMathematicsnØêÆ,2021,11(6),1103-1111

PublishedOnlineJune2021inHans.http://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2021.116124

Œ+L-nŽ

oooÊÊÊÊÊÊ

=²nóŒÆnÆ§[‹=²

ÂvFÏµ2021c58F¶¹^FÏµ2021c69F¶uÙFÏµ2021c616F

Á‡

S´Œ+, L´‚, ùŸ©ÙïÄŒ+SL-nŽ, 3dÄ:þ, ·‚ïÄL-nŽ

3ÊÏŒ+±9AaA ÏŒ+þ5Ÿ.½Âü‡L-f8“◦”$Ž.‰ÑL-nŽÚ

AaAÏL-nŽVg±9¦‚d½Â.y²Œ+Sz‡L-V>nŽÑ´L-

SnŽ,z‡L-†£m¤nŽÑ´L-[nŽ,z‡L-VnŽÑ´L-2ÂVn

Ž,z‡L-[nŽÑ´L-VnŽ.

'…c

Œ+§L-f8§L-nŽ

L-FuzzyIdealsofSemigroups

QianqianLi

Scho olofScience,LanzhouUniversityofTechnology,LanzhouGansu

Received:May8

,2021;accepted:Jun.9

,2021;published:Jun.16

,2021

©ÙÚ^:oÊÊ.Œ+L-nŽ[J].nØêÆ,2021,11(6):1103-1111.

DOI:10.12677/pm.2021.116124

oÊÊ

Abstract

LetSbeasemigroupandLbeacompletelattice.Inthispaper,westudyL-fuzzy

ideals onasemigroup.On this basis,we studyproperties ofL-fuzzyidealsonordinary

semigroups andseveraltypes specialsemigroups arediscussed.Thisarticledeﬁnes the

“◦”oftwoL-fuzzysubsets.TheconceptsofL-fuzzyidealsandseveral specialL-fuzzy

idealsandtheirequivalentdeﬁnitionsaregiven.ItisprovedthateveryL-fuzzyideal

ofthesemigroupSisaL-fuzzyinteriorideal,andeveryL-fuzzyleft(right)idealisa

L-fuzzyquasi-ideal.Every L-fuzzybi-idealisaL-fuzzygeneralizedbi-ideal,andevery

L-fuzzyquasi-idealisaL-fuzzybi-ideal.

Keywords

Semigroup,L-FuzzySubset,L-FuzzyIdeal

2021byauthor(s)andHansPublishersInc.

ThisworkislicensedundertheCreativeCommonsAttributionInternationalLicense(CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

1.0

Œ+nŽ•Ð´1971cRosenfeld[1]Ú\,ƒKuroki[2–4]k?˜ÚïÄ¤J.

©z[2]¥Šö‰ÑŒ+nŽÚVnŽ˜5Ÿ,¿^nŽÚVnŽ•x

Œ+£†¤û!£†¤üÚfŒ+Œ‚.©z[4]¥Šö3Œ+¥Ú\Œƒ5Vg,§´

Œ+¥Œƒ5í2,¿^Œƒ5•xŒ+´üŒ+Œ‚.1993cKurokiqÚ\[

nŽVg[5].1992c,McleanÚKummer[6]5¿Œ+SnŽ†Œ+S‚'Xƒm

éX,Óž5¿S?˜nŽ´SnŽA¼ê·‚5|Ü%C.2002c,

Xie[7]Ú\Œ+SnŽ*Ü5Ÿ!rnŽ*Ü5Ÿ.©z[8–12]?˜ÚïÄn

Žƒ'(Ø.

ùÑ‰·‚JøéÐïÄg´,•·‚óŠ‰éÐÁ=.

S´Œ+,f´Sf8,XJé?¿x,y∈S,kf(xy)≥f(y)(f(x)),@o¡f´S

†£m¤nŽ.XJfQ´S†nŽq´SmnŽ,@o¡f´SV>nŽ,{

DOI:10.12677/pm.2021.1161241104nØêÆ

oÊÊ

¡nŽ.XJé?¿x,a,y∈Skf(xay)≥f(a),@o¡f´SSnŽ.XJé?¿

x,y,z∈S,kf(xyz)≥min{f(x),f(z)},@o¡f´SVnŽ.

2.Œ+L-nŽ

[0,1]´S8,´AÏ‚.b(X,≤)´ S8,…éX¥?¿˜‡š˜f8Yþ•3þ(

.Úe(.,@o¡(X,≤)•‚.í2“”VgÒ´‡r[0,1]«mC¤‚.©Ñy

Lþ•‚,0,1©O•L•Ú•Œ.

½Â2.1 X´š˜8Ü,¡Nf:X→L•XL-f8.

f,g´Œ+SþL-f8,½Âf,gƒm$ŽXe:

(f∩g)(x)=f(x)∧g(x)(∀x∈S);

(f∪g)(x)=f(x)∨g(x)(∀x∈S);

f⊆g⇔f(x)≤g(x)(∀x∈S).

(f◦g)(x)=











x=y z

{f(y)∧g(z)}(∃y,z∈S)x=yz;

0Ù¦.

½Â2.2S´Œ+,f´SL-f8.XJ∀x,y∈S,kf(xy)≥f(x)∧f(y),@o¡f´SL-

fŒ+.

½Â2.3S´Œ+,f´SL-f8.XJ(∀x,y∈S)f(xy)≥f(y)(f(x)),@o¡f´S

L-†£m¤nŽ.XJfQ´SL-†nŽq´SL-mn Ž,@o¡f´SL-

V>nŽ,{¡L-nŽ.

·‚5¿Œ+SŒ±Š•§gL-f8,=?¿x∈SÑkS(x)=1.u´·‚

±e(Ø.

½n2.4 S´Œ+,f´SþL-f8.Ke(Ø¤áµ

(1)f´SL-fŒ+⇔f◦f⊆f;

(2)f´SL-†nŽ⇔S◦f⊆f;

(3)f´SL-mnŽ⇔f◦S⊆f;

(4)f´SL-nŽ⇔S◦f⊆f…f◦S⊆f.

y²µ (1)7‡5,bf´SL-fŒ+,a∈S.e(f◦f)(a)=0,K(Øw,¤á.ÄK,

•3x,y∈S¦a=xy.u´

(f◦f)(a)=

a=xy

(f(x)∧f(y))≤

a=xy

f(xy)=f(a).

=f◦f⊆f.

DOI:10.12677/pm.2021.1161241105nØêÆ

oÊÊ

¿©5,x,y,a∈S,a=xy.e(f◦f)(x)≤f(x),K

f(xy)=f(a)≥(f◦f)(a)=

a=bc

(f(b)∧f(c))≥f(x)∧f(y)

¤±f´SL-fŒ+.

(2)7‡5,bf´SL-†nŽ,?¿a∈S.e(S◦f)(a)=0K(Jw,¤á.ÄK•

3x,y∈S¦a=xy.u´

(S◦f)(a)=

a=xy

(S(x)∧f(y))≤

a=xy

(1∧f(xy))=f(a).

=S◦f⊆f.

¿©5,x,y,a∈S,a=xy.e(S◦f)(x)≤f(x),K

f(xy)=f(a)≥(S◦f)(a)=

a=bc

(S(b)∧f(c))≥S(x)∧f(y)=f(y).

¤±f´SL-†nŽ.Šâ(2)y²,(3)Ú(4)aqŒy.

½n2.5 S´Œ+.Ke(Ø¤áµ

(1)ef,g´SL-fŒ+,Kf∩g•´SL-fŒ+;

(2)f,g´SL-†£m!V>¤nŽ,Kf∩g•´SL-†£m!V>¤nŽ.

y²µ(1)bf,g´SL-fŒ+,?¿a,b∈S.K(f∩g)(ab)=f(ab)∧g(ab)≥

(f(a)∧f(b))∧(g(a)∧g(b)),…

(f(a)∧f(b))∧(g(a)∧g(b))=(f(a)∧g(a))∧(f(b)∧g(b))

≥(f∩g)(a)∧(f∩g)(b).

nþ¤ã,(f∩g)(ab)≥(f∩g)(a)∧(f∩g)(b),=f∩g´SL-fŒ+.(2)Œ±aqy²,

ùpØ2Kã.

·K2.6 S´Œ+,f´SL-m£†¤nŽ.Kf∪(S◦f)(f∪(f◦S))´SL-V>

nŽ.

y²µbf´SL-mnŽ,K

S◦(f∪(S◦f))=(S◦f)∪(S◦(S◦f))=(S◦f)∪(S◦S)◦f

⊆(S◦f)∪(S◦f)=S◦f⊆f∪(S◦f).

Ïdf∪(S◦f)´SL-†nŽ.(f∪(S◦f))◦S=(f◦S)∪(S◦f◦S)⊆(f◦S)∪(S◦f)⊆

f∪(S◦f).¤±f∪(S◦f)´SL-V>nŽ.

DOI:10.12677/pm.2021.1161241106nØêÆ

oÊÊ

3.Œ+L-VnŽ

½Â3.1 S´Œ+,f´SL-fŒ+.XJ?¿x,y,z∈Skf(xyz)≥f(x)∧f(z),@o

¡f´SL-VnŽ.

½n3.2 S´Œ+,f´SL-fŒ+,Kf´SL-VnŽ…=f◦S◦f⊆f.

y²µ7‡5,bf´SL-VnŽ,?¿a∈S.e(f◦S◦f)(a)=0K(Øw,¤á,

ÄK∃x,y,p,q∈S¦a=xy,x=pq.duf´SL-fŒ+,¤±f(pqy)≥f(p)∧f(q).Ïd,

(f◦S◦f)(a)=

a=xy

((f◦S)(x)∧f(y))=

a=xy

(

x=pq

((f(p)∧S(q))∧f(y)))

a=xy

(

x=pq

((f(p)∧1)∧f(y)))=

a=pq y

(f(p)∧f(y))

≤

a=pq y

f(pqy)=f(a).

=f◦S◦f⊆f.

¿©5,x,y,z∈S,a=xyz.e(f◦S◦f)(x)≤f(x)Kk

f(xyz)=f(a)≥(f◦S◦f)(a)=

a=bc

((f◦S)(b)∧f(c))

≥

x=pq

((f(p)∧S(q))∧f(z)))

≥f(x)∧(y)∧f(z)=f(x)∧1∧f(z)=f(x)∧f(z).

=f´SL-VnŽ.y..

·K3.3S´Œ+,f´SL-f8,g´SL-VnŽ.Kf◦gÚg◦fÑ´SL-

VnŽ.

y²µdug´SL-VnŽ,Šâþ¡½n,Œ±í

(f◦g)◦(g◦f)=f◦g◦(f◦g)

⊆f◦g◦S◦g⊆f◦g.

¤±f◦g´SL-fŒ+.Ïd

(f◦g)◦S◦(f◦g)=f◦g◦(S◦f)◦g

⊆f◦(g◦S◦g)⊆f◦g

f◦g´SL-VnŽ.ÓnŒyg◦f´SL-VnŽ.

·K3.4 S´Œ+,f,g´SL-VnŽ.Kf∩g´SL-VnŽ.

DOI:10.12677/pm.2021.1161241107nØêÆ

oÊÊ

y²µw,,f∩g´L-fŒ+.∀a,b,x∈S,K(f∩g)(axb)=f(axb)∧g(axb).f(axb)∧

g(axb)≥(f(a)∧f(b))∧(g(a)∧g(b)).qÏ•

(f(a)∧f(b))∧(g(a)∧g(b))=f(a)∧g(a)∧f(b)∧g(b)

=(f∩g)(a)∧(f∩g)(b)

¤±(f∩g)(axb)≥(f∩g)(a)∧(f∩g)(b),=f∩g´SL-VnŽ.

4.L-SnŽ

½Â4.1S´Œ+,f´SL-fŒ+.XJ?¿x,y,z∈S,(xyz)≥f(y),@o¡f´S

L-SnŽ.

½n4.2 S´Œ+,f´SL-fŒ+.Kf´SL-SnŽ…=S◦f◦S⊆f.

y²µ7‡5,bf´SL-SnŽ,?¿x∈S.e(S◦f◦S)(x)=0≤f(x),(

Øw,¤á.ÄK,•3y,z,u,v∈S¦x=yz,y=uv.duf´S2ÂSnŽ,¤

±f(uvz)≥f(v).u´

(S◦f◦S)(x)=

x=y z

{(S◦f)(y)∧S(z)}

x=y z

{

y=uv

(S(u)∧f(v))∧S(z)}

≤

x=y z

{

y=uv

(1∧f(v))∧1}≤f(x).

¤±S◦f◦S⊆f.

¿©5,S◦f◦S⊆f,x,a,y∈S.=f(xay)≥(S◦f◦S)(xay).

(S◦f◦S)(x)=

xay =pq

{(S◦f)(p)∧S(q)}

≥(S◦f(xa))∧S(y)=(S◦f)(xa)∧1

xa=pq

{S(p)∧f(q)}≥S(x)∧f(a)=f(a).

¤±,f(xay)≥f(a),=f´SL-SnŽ.

w,Œ+Sz‡L-V>nŽÑ´SL-SnŽ.´SL-SnŽØ˜½

´SL-V>nŽ.e¡´L-SnŽØ´L-V>nŽ˜‡~f.

DOI:10.12677/pm.2021.1161241108nØêÆ

oÊÊ

~4.3µS={a,b,c,d}´Œ+.§¦{$ŽXe:

abcd

aaaaa

baaaa

caaba

daabb

f´SL-f8,f(a)=a,f(b)=b,f(c)=c,f(d)=d,…a,b,c,d∈Lka≥c≥b≥d.

¯¢þ∀x,y,z∈S,Ñkf(xyz)=f(a)=a≥f(y).Ïdf´SL-SnŽ.´,f(dc)=

f(b)=b≤c=f(c),¤±fØ´SL-†nŽ,=fØ´SL-V>nŽ.

½n4.4 S´KŒ+,f´SL-f8.KeQãd:

(1)f´SL-nŽ,

(2)f´SL-SnŽ.

y²µ(1)⇒(2)´w,,e¡y²(2)⇒(1).

bf´SL-SnŽ,∀a,b∈S.duS´KŒ+,¤±∃x,y∈S,¦a=axa,b=

byb,u´

f(ab)=f(axab)=f((ax)ab)≥f(a),

f(ab)=f(abyb)=f((ab(yb))≥f(b).

Ïd,f´SL-nŽ.

5.L-[nŽ

½Â5.1 S´Œ+,f´SL-f8.e(f◦S)∩(S◦f)⊆f,K¡f´SL-[nŽ.

w,,S?¿˜‡L-†nŽ£S◦f⊆f¤,L-mnŽ£f◦S⊆f¤Ñ´SL-[

nŽ.…,S?¿˜‡L-[nŽÑ´SL-VnŽ£f◦S◦f⊆f¤.´(Ø‡L5

™7¤á,e¡´ü‡äN~f.

~5.2µS={0,a,b,c}´Œ+.§¦{$ŽXe:

0abc

00000

a0ab0

b0000

c0c00

½ÂSL-f8Xe:f(0)=f(a)=a,f(b)=f(c)=b,…a,b∈Lka≥b..Kf´SL-

[nŽ,Ø´SL-†£m¤nŽ.

DOI:10.12677/pm.2021.1161241109nØêÆ

oÊÊ

~5.3µS={0,a,b,c}´Œ+.§¦{$ŽXe:

0abc

00000

a0000

b000a

c00ab

½ÂSL-f8Xe:f(0)=f(b)=b,f(a)=f(c)=a,…a,b∈Lkb≥a..Kf´SL-

VnŽ,Ø´SL-[nŽ.

·K5.4 Œ+Sz‡L-[nŽÑ´SL-fŒ+.

y²µf´SL-[nŽ.df⊆SŒf◦f⊆S◦f…f◦f⊆f◦S.Ïd

f◦f⊆(S◦f)∩(f◦S)⊆f.

=f´SL-fŒ+.

((f∩g)◦S)∩(S◦(f∩g))⊆(f◦S)∩(S◦g)

⊆f∩g.

¤±f∩g´SL-[nŽ.

·K5.6S´Œ+,f,g´SL-f8.ef,gÙ¥˜‡´L-[nŽ,Kf◦g´SL-

VnŽ.

y²µØ”f´SL-[nŽ,Kf◦S◦f⊆f.Ïd

(f◦g)◦(f◦g)=(f◦g◦f)◦g⊆(f◦S◦f)◦g⊆f◦g.

f◦g´SL-fŒ+.…

(f◦g)◦S◦(f◦g)=(f◦(g◦S)◦f)◦g⊆(f◦(S◦S)◦f)◦g

⊆(f◦S◦f)◦g⊆f◦g.

¤±f◦g´SL-VnŽ.

ë•©z

[1]Rosenfeld,A.(1971)FuzzyGroups.JournalofMathematicalAnalysisandApplications,35,

512-517.https://doi.org/10.1016/0022-247X(71)90199-5

[2]Kuroki,N.(1979)FuzzyBi-IdealsinSemigroups.CommentariiMathematiciUniversitatis

DOI:10.12677/pm.2021.1161241110nØêÆ

oÊÊ

SanctiPauli,28,17-21.

[3]Kuroki,N.(1981)OnFuzzyIdealsandFuzzyBi-IdealsinSemigroups.FuzzySetsandSystems,

5,203-215.https://doi.org/10.1016/0165-0114(81)90018-X

[4]Kuroki,N.(1982)FuzzySemiprimeIdealsinSemigroups.FuzzySetsandSystems,8,71-79.

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DOI:10.12677/pm.2021.1161241111nØêÆ