格蕴涵代数的(∈,∈∨_(q(λ,μ)))-模糊LI-理想
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摘要
对格蕴涵代数的(∈,∈∨q(λ,μ))-模糊LI-理想进行了详细的研究,主要内容涉及以下几个方面。我们首先给出了(∈,∈∨q(λ,μ))-模糊LI-理想和广义模糊LI-理想的概念,讨论了两者之间的关系;其次给出了(∈,∈∨q(λ,μ))-模糊LI-理想的一些等价刻画,并研究了其相关性质;最后还讨论了(∈,∈∨q(λ,μ))-模糊LI-理想的同态像与同态原像的基本性质。
In this paper,we carry out a detailed investigation into the(∈,∈∨q(λ,μ))-fuzzy LI-ideals of the lattice implication algebra L.The main content involves the following several aspects:Firstly,the concepts of(∈,∈∨q(λ,μ))-fuzzy LI-ideals and generalized fuzzy LI-ideals are introduced,and the relationship between them is also discussed;Secondly,we present some equivalent descriptions and study the related properties of the(∈,∈∨q(λ,μ))-fuzzy LI-ideals;Finally,we discuss some properties of homomorphic image and homomorphic preimage of(∈,∈∨q(λ,μ))-fuzzy LI-ideals.
In this paper,we carry out a detailed investigation into the(∈,∈∨q(λ,μ))-fuzzy LI-ideals of the lattice implication algebra L.The main content involves the following several aspects:Firstly,the concepts of(∈,∈∨q(λ,μ))-fuzzy LI-ideals and generalized fuzzy LI-ideals are introduced,and the relationship between them is also discussed;Secondly,we present some equivalent descriptions and study the related properties of the(∈,∈∨q(λ,μ))-fuzzy LI-ideals;Finally,we discuss some properties of homomorphic image and homomorphic preimage of(∈,∈∨q(λ,μ))-fuzzy LI-ideals.
引文
[1]Zadeh L A.Fuzzy sets[J].Information and Control,1965,8:338~353.
[2]Rosenfeld A.Fuzzy groups[J].Journal of Mathematical Analysis and Applications,1971,35:512~517.
[3]Bhakat S K,Das P.On the definition of a fuzzy subgroup[J].Fuzzy Sets Systems,1992,51:235~241.
[4]Bhakat S K,Das P.fuzzy subgroups[J].Fuzzy Sets Systems,1996,80:359~368.
[5]Bhakat S K,Das P.Fuzzy subrings and ideals redefined[J].Fuzzy Sets Systems,1996,81:383~393.
[6]Liao Z H,Gu H.fuzzy normal subgroup[J].Fuzzy Systems and Mathematics,2006,20(5):38~43.
[7]Guo Z Y,Liao Z H,Guo J F.Fuzzy subgroups[C]//The Sixth International Conference on Information and Management Sciences.California Polytechnic State Univesity Press,2007:570~574.
[8]Chen M,Liao Z H.Fuzzy subsemigroups and fuzzy ideals of semigroups[C]//The Sixth International Conference on Information and Management Sciences.California Polytechnic State Univesity Press,2007:575~579.
[9]Du J,Liao Z H.Generalized fuzzy ideal of Algebra[C]//The Fourth International Conference on Fuzzy Systems and Knowledge Discovery.IEEE Press,2007:294~299.
[10]Yhou J,Liao Z H.Generalized(α,β)-convex fuzzy cones[C]//The 2008IEEE International Conference on Fuzzy Systems.IEEE Press,2008:439~442.
[11]Liao Z H,Hu M H,Chen M,Yi L H,Rui M l,Lu J H,Liu C Z.Generalized Fuzzy Interior Ideals in Semigroups[C]//2010Third International Joint Conference on Computational Science and Optimization.IEEE Press,2010:197~200.
[12]易丽华,廖祖华.半群的(∈,∈∨q(λ,μ))-模糊n伪理想与Drazin半群的刻画[J].模糊系统与数学,2009,23(5):48~55.
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[15]胡淼菡,廖祖华,易丽华,芮明力.半环的(∈,∈∨q(λ,μ))-模糊双理想[J].模糊系统与数学,2011,25(5):69~76.
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[19]Liao Z H,Yi L H,Hu M H.(∈,∈∨q(λ,μ))-fuzzy k-ideals of semigroups[J].Journal of Mathematics,2012,32(2):191~205.
[20]廖祖华,胡淼菡,芮明力,曹姝.链式条件下的半环的(∈,∈∨q(λ,μ))-模糊双理想[J].模糊系统与数学,2012,26(2):54~59.
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[22]Jun Y B.Generalization of(∈,∈∨q)-fuzzy subalgebras in algebras[J].Computers and Mathematics with Applications,2009,58:1383~1390.
[23]Jun Y B,Dudek W A,Shabir M,Kang M S.General types of(α,β)-fuzzy ideals of hemirings[J].Honam Mathematical Journal,2010,32(3):413~439.
[24]Shabir M,Jun Y B,Nawaz Y.Semigroups characterized by(∈,∈∨qk)-fuzzy ideals[J].Computers and Mathematics with Applications,2010,60:1473~1493.
[25]Ma X L,Zhan J M,Jun Y B.Some kinds of(∈γ,∈γ∨qσ)-fuzzy ideals ofBCI-algebras[J].Computers and Mathematics with Applications,2011,61:1005~1015.
[26]吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56~64.
[27]徐扬.格蕴涵代数[J].西南交通大学学报,1993,(1):20~27.
[28]Jun Y B,Roh E H,Xu Y.LI-ideals in lattice implication algebras[J].Bulletin Korean Mathematical Society,1998,5(1):13~24.
[29]Jun Y B,Xu Y.Fuzzy LI-ideals in lattice implication algebras[J].The Journal of Fuzzy Mathematics,1999,7(4):997~1003.
[30]Jun Y B,Xu Y,Liu J.Quotient lattice implieation algebra incluced by fuzzy LI-ideals[J].The Journal of Fuzzy Mathematics,1997,7:1013~1020.
[31]Zhao G F,Xu Y,Song Z M.Implicative and fuzzy implicative ideals of lattice implication algebras[J].Chinese Quarterly Journal of Mathematics,2001,16(4):105~110.
[32]Qin K Y,Xu Y,Si C,Jun Y B,Chin J.Ultra LI-ideals in lattice implication algebras[J].Czechoslovak Mathematical Journal,2002,52(127):463~468.
[33]Liu Y L,Liu S Y,Xu Y.ILI-ideals and prime LI-ideals in lattice implication algebra[J].Information Sciences,2003,(155):157~175.
[34]Xu Y,Ruan D,Qin K Y,Liu J.Lattice-Valued Logic[M].Berlin:Springer,2003.
[35]王学芳,刘培顺,徐扬.格蕴涵代数的素对偶理想[J].模糊系统与数学,2006,20(5):37~42.
[36]Chen S W,Jiang B Q,Yang X W.LI-ideal spaces of lattice implication algebras[J].Chinese Quarterly Journal of Mathematics,2007,22(4):504~511.
[37]潘小东,张青,徐扬.格蕴涵N序半群中的sl理想[J].模糊系统与数学,2008,22(2):65~69.
[38]Lai J J,Xu Y.On WLI-ideal space and the properties of WLI-ideals in lattice implication algebra(LIA)[J].Journal of Applied Mathematics and Computing,2009,31:113~127.
[39]Hou Z X.Measuring technology and mechatronics automation in electrical engineering[M].Berlin:Springer,2012.
[40]刘春辉,徐罗山.格蕴涵代数的fuzzy LI-理想[J].纯粹数学与应用数学,2011,27(3):363~376.
[41]刘春辉.格蕴涵代数的区间值fuzzy LI-理想[J].模糊系统与数学,2011,25(3):26~34.
[2]Rosenfeld A.Fuzzy groups[J].Journal of Mathematical Analysis and Applications,1971,35:512~517.
[3]Bhakat S K,Das P.On the definition of a fuzzy subgroup[J].Fuzzy Sets Systems,1992,51:235~241.
[4]Bhakat S K,Das P.fuzzy subgroups[J].Fuzzy Sets Systems,1996,80:359~368.
[5]Bhakat S K,Das P.Fuzzy subrings and ideals redefined[J].Fuzzy Sets Systems,1996,81:383~393.
[6]Liao Z H,Gu H.fuzzy normal subgroup[J].Fuzzy Systems and Mathematics,2006,20(5):38~43.
[7]Guo Z Y,Liao Z H,Guo J F.Fuzzy subgroups[C]//The Sixth International Conference on Information and Management Sciences.California Polytechnic State Univesity Press,2007:570~574.
[8]Chen M,Liao Z H.Fuzzy subsemigroups and fuzzy ideals of semigroups[C]//The Sixth International Conference on Information and Management Sciences.California Polytechnic State Univesity Press,2007:575~579.
[9]Du J,Liao Z H.Generalized fuzzy ideal of Algebra[C]//The Fourth International Conference on Fuzzy Systems and Knowledge Discovery.IEEE Press,2007:294~299.
[10]Yhou J,Liao Z H.Generalized(α,β)-convex fuzzy cones[C]//The 2008IEEE International Conference on Fuzzy Systems.IEEE Press,2008:439~442.
[11]Liao Z H,Hu M H,Chen M,Yi L H,Rui M l,Lu J H,Liu C Z.Generalized Fuzzy Interior Ideals in Semigroups[C]//2010Third International Joint Conference on Computational Science and Optimization.IEEE Press,2010:197~200.
[12]易丽华,廖祖华.半群的(∈,∈∨q(λ,μ))-模糊n伪理想与Drazin半群的刻画[J].模糊系统与数学,2009,23(5):48~55.
[13]廖祖华,陈敏.(∈,∈∨q(λ,μ))-模糊子近环和理想[J].模糊系统与数学,2009,23(6):12~16.
[14]陈敏,廖祖华.(,∨珔q(λ,μ))-模糊子近环及其理想[J].数学的实践与认识,2009,39(4):225~230.
[15]胡淼菡,廖祖华,易丽华,芮明力.半环的(∈,∈∨q(λ,μ))-模糊双理想[J].模糊系统与数学,2011,25(5):69~76.
[16]刘春芝,廖祖华,曹姝,胡淼菡,芮明力.(∈,∈∨q(λ,μ))-模糊子格[J].模糊系统与数学,2011,25(5):54~59.
[17]芮明力,廖祖华,胡淼菡,陆金花.广义模糊子坡与(∈,∈∨q(λ,μ))-模糊子坡[J].模糊系统与数学,2011,25(6):60~68.
[18]刘春芝,廖祖华,姜雪.布尔代数的(∈,∈∨q(λ,μ))-模糊理想[J].江南大学学报(自然科学版),2011,10(6):737~740.
[19]Liao Z H,Yi L H,Hu M H.(∈,∈∨q(λ,μ))-fuzzy k-ideals of semigroups[J].Journal of Mathematics,2012,32(2):191~205.
[20]廖祖华,胡淼菡,芮明力,曹姝.链式条件下的半环的(∈,∈∨q(λ,μ))-模糊双理想[J].模糊系统与数学,2012,26(2):54~59.
[21]王德江,廖祖华,曹姝,宋璐瑶,刘春芝.R-广义模糊子格[J].山东大学学报(理学版),2012,47(1):93~97.
[22]Jun Y B.Generalization of(∈,∈∨q)-fuzzy subalgebras in algebras[J].Computers and Mathematics with Applications,2009,58:1383~1390.
[23]Jun Y B,Dudek W A,Shabir M,Kang M S.General types of(α,β)-fuzzy ideals of hemirings[J].Honam Mathematical Journal,2010,32(3):413~439.
[24]Shabir M,Jun Y B,Nawaz Y.Semigroups characterized by(∈,∈∨qk)-fuzzy ideals[J].Computers and Mathematics with Applications,2010,60:1473~1493.
[25]Ma X L,Zhan J M,Jun Y B.Some kinds of(∈γ,∈γ∨qσ)-fuzzy ideals ofBCI-algebras[J].Computers and Mathematics with Applications,2011,61:1005~1015.
[26]吴望名.Fuzzy蕴涵代数[J].模糊系统与数学,1990,4(1):56~64.
[27]徐扬.格蕴涵代数[J].西南交通大学学报,1993,(1):20~27.
[28]Jun Y B,Roh E H,Xu Y.LI-ideals in lattice implication algebras[J].Bulletin Korean Mathematical Society,1998,5(1):13~24.
[29]Jun Y B,Xu Y.Fuzzy LI-ideals in lattice implication algebras[J].The Journal of Fuzzy Mathematics,1999,7(4):997~1003.
[30]Jun Y B,Xu Y,Liu J.Quotient lattice implieation algebra incluced by fuzzy LI-ideals[J].The Journal of Fuzzy Mathematics,1997,7:1013~1020.
[31]Zhao G F,Xu Y,Song Z M.Implicative and fuzzy implicative ideals of lattice implication algebras[J].Chinese Quarterly Journal of Mathematics,2001,16(4):105~110.
[32]Qin K Y,Xu Y,Si C,Jun Y B,Chin J.Ultra LI-ideals in lattice implication algebras[J].Czechoslovak Mathematical Journal,2002,52(127):463~468.
[33]Liu Y L,Liu S Y,Xu Y.ILI-ideals and prime LI-ideals in lattice implication algebra[J].Information Sciences,2003,(155):157~175.
[34]Xu Y,Ruan D,Qin K Y,Liu J.Lattice-Valued Logic[M].Berlin:Springer,2003.
[35]王学芳,刘培顺,徐扬.格蕴涵代数的素对偶理想[J].模糊系统与数学,2006,20(5):37~42.
[36]Chen S W,Jiang B Q,Yang X W.LI-ideal spaces of lattice implication algebras[J].Chinese Quarterly Journal of Mathematics,2007,22(4):504~511.
[37]潘小东,张青,徐扬.格蕴涵N序半群中的sl理想[J].模糊系统与数学,2008,22(2):65~69.
[38]Lai J J,Xu Y.On WLI-ideal space and the properties of WLI-ideals in lattice implication algebra(LIA)[J].Journal of Applied Mathematics and Computing,2009,31:113~127.
[39]Hou Z X.Measuring technology and mechatronics automation in electrical engineering[M].Berlin:Springer,2012.
[40]刘春辉,徐罗山.格蕴涵代数的fuzzy LI-理想[J].纯粹数学与应用数学,2011,27(3):363~376.
[41]刘春辉.格蕴涵代数的区间值fuzzy LI-理想[J].模糊系统与数学,2011,25(3):26~34.