格的反软理想
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摘要
首先给出了格的反软理想新概念,证明2个反软理想分别在软集的限制并和"或"运算下仍然是反软理想.其次,利用软集的反对偶给出反软理想的等价刻画.再次,利用软集的反扩张原理给出反软理想在同态映射下反像与原像的性质.最后,在全体反软理想组成的集合H上,引入链条件并讨论H是阿丁的或诺特的充要条件.
Firstly,the definition of anti-soft ideals is given,and we show that the restricted union and‘OR'operation of two anti-soft lattices are still anti-soft ideal.Then,we discuss the equivalent characterizations of anti-soft ideals based on the anti-dual of soft set.In addition,based on the anti-extension principle,the properties of their anti-image and inverse image are derived under the homomorphic mapping.Finally,we introduce the condition of chain over H which is composed of all the anti-soft ideals,and discuss the necessary and sufficient condition for that His Artinian or Noetherian.
Firstly,the definition of anti-soft ideals is given,and we show that the restricted union and‘OR'operation of two anti-soft lattices are still anti-soft ideal.Then,we discuss the equivalent characterizations of anti-soft ideals based on the anti-dual of soft set.In addition,based on the anti-extension principle,the properties of their anti-image and inverse image are derived under the homomorphic mapping.Finally,we introduce the condition of chain over H which is composed of all the anti-soft ideals,and discuss the necessary and sufficient condition for that His Artinian or Noetherian.
引文
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[2]MAJI P K,BISWAS R,ROY A R.Soft set theory[J].Computers and Mathematics with Applications,2003,45:555-562.
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[5]BIRKHOFF G.Lattice Theory[M].Providence:American Mathematical Society,1961.
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[7]LI F.Soft lattices[J].Global Journal of Science Frontier Research,2010,10(4):157-159.
[8]吴广庆.格的L-fuzzy素理想[J].模糊系统与数学,2008,22(4):63-67.WU G Q.L-fuzzy prime ideals of lattice[J].Fuzzy Systems and Mathematics,2008,22(4):63-67.
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[11]李京效,辛小龙,贺鹏飞.模糊软格与模糊软理想[J].模糊系统与数学,2014,28(2):69-75.LI J X,XIN X L,HE P F.Fuzzy soft lattices and fuzzy soft ideals[J].Fuzzy Systems and Mathematics,2014,28(2):69-75.
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[15]郭二芳,李玉瑛,王绪柱.反模糊子群的运算[J].数学的实践与认识,2015,45(12):279-284.GUO E F,LI Y Y,WANG X Z.Operations of antifuzzy subgroups[J].Mathematics in Practice and Theory,2015,45(12):279-284.
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[19]童娟,廖祖华,路腾,等.反软子格[J].模糊系统与数学,2016,30(1):20-27.TONG J,LIAO Z H,LU T,et al.Anti-soft sublattices[J].Fuzzy Systems and Mathematics,2016,30(1):20-27.
[20]温永川.关于软集的研究[D].大连:辽宁师范大学,2008.WEN Y C.The Study of Soft Set[D].Dalian:Liaoning Normal University,2008.
[21]赵迪,廖祖华,朱晓英,等.半群的软理想[J].模糊系统与数学,2014,28(4):45-50.ZHAO D,LIAO Z H,ZHU X Y,et al.Soft ideals of semigroup[J].Fuzzy Systems and Mathematics,2014,28(4):45-50.
[22]殷霞,廖祖华,朱晓英,等.软集与新型软子群[J].计算机工程与运用,2012,48(33):40-43.YIN X,LIAO Z H,ZHU X Y,et al.Soft sets and the new soft subgroup[J].Computer Engineering and Applications,2012,48(33):40-43.
[23]叶灵军,廖祖华,朱晓英,等.软完全正则子半群[J].工程数学学报,2014,31(3):341-346.YE L J,LIAO Z H,ZHU X Y,et al.Soft completely regular sub-semigroup[J].Chinese Journal of Engineering Mathematics,2014,31(3):341-346.
[24]关贝贝,廖祖华,朱晓英,等.软子半群[J].模糊系统与数学,2014,28(4):39-44.GUAN B B,LIAO Z H,ZHU X Y,et al.Soft subsemigroups[J].Fuzzy Systems and Mathematics,2014,28(4):39-44.YE T,LIAO Z H,ZHU X Y,et al.Soft completely prime ideals of semigroup[J].Fuzzy Systems and Mathematics,2014,28(3):1-5.
[26]YIN X,LIAO Z H.Study on soft groups[J].Journal of Computers,2013(8):960-967.
[27]ZHENG G P,LIAO Z H,WANG N N,et al.Soft lattice implication subalgebra[J].Applied Mathematics and Information Sciences,2013,7(3):1181-1186.
[28]方捷.格论导引[M].北京:高等教育出版社,2014.FANG J.Lattice Theory Guidance[M].Beijing:Higher Education Press,2014.
[29]ALI M I,FENG F,LIU X Y,et.al.On some new operations in soft set theory[J].Computers and Mathematics with Applications,2009,57(9):1547-1553.
[2]MAJI P K,BISWAS R,ROY A R.Soft set theory[J].Computers and Mathematics with Applications,2003,45:555-562.
[3]MAJI P K,ROY A R.An application of soft sets in a decision making problem[J].Computers and Mathematics with Applications,2002,44:1077-1083.
[4]AKTAS H,CAGMAN N.Soft sets and soft groups[J].Information Sciences,2007,177:2726-2735.
[5]BIRKHOFF G.Lattice Theory[M].Providence:American Mathematical Society,1961.
[6]胡长流,宋振明.格论基础[M].开封:河南大学出版社,1990.HU C L,SONG Z M.Lattices Theory Foundation[M].Kaifeng:Henan University Press,1990.
[7]LI F.Soft lattices[J].Global Journal of Science Frontier Research,2010,10(4):157-159.
[8]吴广庆.格的L-fuzzy素理想[J].模糊系统与数学,2008,22(4):63-67.WU G Q.L-fuzzy prime ideals of lattice[J].Fuzzy Systems and Mathematics,2008,22(4):63-67.
[9]花秀娟,朱熙.L-fuzzy格及(λ,μ)-fuzzy凸子格[J].纺织高校基础科学学报,2009,22(4):529-531.HUA X J,ZHU X.L-fuzzy lattice and(λ,μ)-fuzzy convex sublattice[J].Basic Sciences Journal of Textile Universities,2009,22(4):529-531.
[10]花秀娟,辛小龙,刘庆.(∈,∈∨q)-模糊素理想与(λ,μ)-模糊子格(理想)[J].模糊系统与数学,2010,24(4):33-38.HUA X J,XIN X L,LIU Q.(∈,∈∨q)-Fuzzy prime ideal and(λ,μ)-fuzzy sublattice(ideal)[J].Fuzzy Systems and Mathematics,2010,24(4):33-38.
[11]李京效,辛小龙,贺鹏飞.模糊软格与模糊软理想[J].模糊系统与数学,2014,28(2):69-75.LI J X,XIN X L,HE P F.Fuzzy soft lattices and fuzzy soft ideals[J].Fuzzy Systems and Mathematics,2014,28(2):69-75.
[12]王德江,梁久祯,廖祖华.R-广义模糊子格的理[J].计算机科学,2014,41(1):271-274.WANG D J,LIANG J Z,LIAO Z H.Ideal of R-generalized fuzzy sublattice[J].Computer Science,2014,41(1):271-274.
[13]BISWAS R.Fuzzy subgroups and anti fuzzy subgroups[J].Fuzzy Sets and Systems,1990,35(1):121-124.
[14]FENG Y M,YAO B X.On(λ,μ)-anti-fuzzy subgroups[J].Journal of Inequalities and Applications,2012,78:1-5.
[15]郭二芳,李玉瑛,王绪柱.反模糊子群的运算[J].数学的实践与认识,2015,45(12):279-284.GUO E F,LI Y Y,WANG X Z.Operations of antifuzzy subgroups[J].Mathematics in Practice and Theory,2015,45(12):279-284.
[16]ATANASSOV K T.On Intuitionistic Fuzzy Sets Theory[M].Berlin/Heidelberg:Springer-Verlag,2012.
[17]BUSTINCE H,HERRERA F,MONTERO J.Fuzzy Sets and Their Extensions:Representation,Aggregation and Models[M].New York:Springer Publishing Company:Physica-Verlag,2007.
[18]吴修竹,廖祖华,刘维龙.反软子半群[J].模糊系统与数学,2015,29(5):44-50.WU X Z,LIAO Z H,LIU W L.Anti-soft subsemigroups[J].Fuzzy Systems and Mathematics,2015,29(5):44-50.
[19]童娟,廖祖华,路腾,等.反软子格[J].模糊系统与数学,2016,30(1):20-27.TONG J,LIAO Z H,LU T,et al.Anti-soft sublattices[J].Fuzzy Systems and Mathematics,2016,30(1):20-27.
[20]温永川.关于软集的研究[D].大连:辽宁师范大学,2008.WEN Y C.The Study of Soft Set[D].Dalian:Liaoning Normal University,2008.
[21]赵迪,廖祖华,朱晓英,等.半群的软理想[J].模糊系统与数学,2014,28(4):45-50.ZHAO D,LIAO Z H,ZHU X Y,et al.Soft ideals of semigroup[J].Fuzzy Systems and Mathematics,2014,28(4):45-50.
[22]殷霞,廖祖华,朱晓英,等.软集与新型软子群[J].计算机工程与运用,2012,48(33):40-43.YIN X,LIAO Z H,ZHU X Y,et al.Soft sets and the new soft subgroup[J].Computer Engineering and Applications,2012,48(33):40-43.
[23]叶灵军,廖祖华,朱晓英,等.软完全正则子半群[J].工程数学学报,2014,31(3):341-346.YE L J,LIAO Z H,ZHU X Y,et al.Soft completely regular sub-semigroup[J].Chinese Journal of Engineering Mathematics,2014,31(3):341-346.
[24]关贝贝,廖祖华,朱晓英,等.软子半群[J].模糊系统与数学,2014,28(4):39-44.GUAN B B,LIAO Z H,ZHU X Y,et al.Soft subsemigroups[J].Fuzzy Systems and Mathematics,2014,28(4):39-44.YE T,LIAO Z H,ZHU X Y,et al.Soft completely prime ideals of semigroup[J].Fuzzy Systems and Mathematics,2014,28(3):1-5.
[26]YIN X,LIAO Z H.Study on soft groups[J].Journal of Computers,2013(8):960-967.
[27]ZHENG G P,LIAO Z H,WANG N N,et al.Soft lattice implication subalgebra[J].Applied Mathematics and Information Sciences,2013,7(3):1181-1186.
[28]方捷.格论导引[M].北京:高等教育出版社,2014.FANG J.Lattice Theory Guidance[M].Beijing:Higher Education Press,2014.
[29]ALI M I,FENG F,LIU X Y,et.al.On some new operations in soft set theory[J].Computers and Mathematics with Applications,2009,57(9):1547-1553.