BCI-代数的扰动模糊q-理想
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摘要
将扰动模糊集应用于BCI-代数中,引入了BCI-代数的扰动模糊q-理想的概念,研究了它相关性质,讨论了扰动模糊q-理想与扰动模糊理想、扰动模糊子代数之间的关系,给出了扰动模糊q-理想的几个等价刻画和扩展定理,指出了BCI-代数的扰动模糊q-理想的同态象与同态原象在一定条件下仍是扰动模糊q-理想。
Applying disturbing fuzzy sets to BCI-algebras, this paper introduces the concept of disturbing fuzzy qideals of BCI-algebras and studies their properties. The relations between disturbing fuzzy q-ideals and disturbing fuzzy ideals and disturbing fuzzy subalgebras are discussed. Some equivalent characterizations and the extensive theorem of disturbing fuzzy q-ideals are given. It is pointed out that the homomorphic image and homomorphic primitive of disturbing fuzzy q-ideal of BCI-algebras are still disturbing fuzzy q-ideals under certain conditions.
Applying disturbing fuzzy sets to BCI-algebras, this paper introduces the concept of disturbing fuzzy qideals of BCI-algebras and studies their properties. The relations between disturbing fuzzy q-ideals and disturbing fuzzy ideals and disturbing fuzzy subalgebras are discussed. Some equivalent characterizations and the extensive theorem of disturbing fuzzy q-ideals are given. It is pointed out that the homomorphic image and homomorphic primitive of disturbing fuzzy q-ideal of BCI-algebras are still disturbing fuzzy q-ideals under certain conditions.
引文
[1]Bunder W M.BCK and related algebras and their corresponding logics[J].Journal of Applied Non-Classical Logics,1983,7:15-24.
[2]Imai K,Iseki K.On axiom system of propositional calculus[J].Proceedings of the Japan Academy,1966,42:19-22.
[3]Iseki K,Tanaka S.An introduction to the theory of BCK-algebras[J].Mathematica Japonica,1978,23:1-26.
[4]Iseki K,Tanaka S.Ideal theory of BCK-algebras[J].Mathematica Japonica,1976,21:351-366.
[5]Peng J Y.Normed BCI-algebras[J].Chinese Quarterly Journal of Mathematics,2011,26(4):543-548.
[6]Meng J.On ideals in BCK-algebras[J].Mathematica Japonica,1994,1:143-154.
[7]Iseki K.On BCI-algebras[J].Mathematics Seminar Notes,1980,8:125-130.
[8]Lei T D,Xi C C.p-radical in BCI-algebras[J].Mathematica Japonica,1985,30:511-517.
[9]Xi C C.On a class of BCI-algebra[J].Mathematica Japonica,1990,35:13-17.
[10]Zhang X H,Hao J,Bhatti S A.On p-ideals of a BCI-algebra[J].Punjab University Journal of Mathematics,1994,27:121-128.
[11]Liu Y L,Xu Y,Meng J.BCI-implicative ideals of BCI-algebras[J].Information Sciences,2007,177(22):4987-4996.
[12]Liu Y L,Meng J,Zhang X H,et al.q-ideals and a-ideals in BCI-algebras[J].Southeast Asian Bulletin of Mathematics,2000,24:243-253.
[13]Meng J,Xin X L.Commutative BCI-algebras[J].Mathematica Japonica,1992,37:569-572.
[14]Zadeh L A.Fuzzy sets[J].Information and Control,1965,8(3):338-353.
[15]Li D F,Cheng C T.New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions[J].Pattern Recognition Letters,2002,23(1):221-225.
[16]Xu Z S.Theory and application of intuitionistic fuzzy information integration[M].Beijing:Science Press,2008.
[17]Liu H W,Wang G J.Multi-criteria decision-making methods based on intuitionistic fuzzy sets[J].European Journal of Operational Research,2007,179(1):220-233.
[18]Xi O G.Fuzzy BCK-algebras[J].Mathematica Japonica,1991,36(5):935-942.
[19]Meng J,Guo X E.On fuzzy ideals in BCK/BCI-algebras[J].Fuzzy Sets and Systems,2005,149(3):509-525.
[20]Jun Y B,Hong S M,Meng J N,et al.Characterizations of fuzzy positive implicative ideals in BCK-algebras[J].Mathematica Japonica,1994,40:503-507.
[21]Jun Y B,Roh E H.Fuzzy commutative ideals of BCK-algebras[J].Fuzzy Sets and Systems,1994,64(1):401-405.
[22]Peng J Y.Fuzzifying implicative ideals of BCK-algebras based on continuous-valued logic[J].Fuzzy Systems and Mathematics,2015,29(5):1-10.
[23]Peng J Y.Generalized(∈,∈∨q)-fuzzy implicative ideals of BCK-algebras[J].Journal of Frontiers of Computer Science and Technology,2018,12(4):662-670.
[24]Jun Y B,Meng J N.Fuzzy p-ideals in BCI-algebras[J].Mathematica Japonica,1994,40(2):271-282.
[25]Meng J.Fuzzy ideals of BCI-algebras[J].Southeast Asian Bulletin of Mathematics,1994,18(1):65-68.
[26]Atannassov K T.Intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1986,20(1):87-96.
[27]Zadeh L A.The concept of a linguistic variable and its application to approximate reasoning-I[J].Information Science,1975,8(3):199-249.
[28]Torra V.Hesitant fuzzy sets[J].International Journal of Intelligent Systems,2010,25(6):529-539.
[29]Gau W L,Buehrer D J.Vague sets[J].IEEE Transactions on Systens,Man,and Cybernetics,1993,23(2):610-614.
[30]Liu X,Chen T Y.Disturbing fuzzy logic and its“non operator”[C]//Proceeding of the 11th Annual Conference of Systems Engineering Society of China,Xiamen,2002:189-192.
[31]Chen T Y,Han Y.The algebra and generalized tautology in limited disturbing fuzzy propositional logic[J].Journal of Liaoning Normal University,2002,25(4):343-345.
[32]Chen T Y,Han Y,Liao S Z.The largest subalgebra of disturbed fuzzy logic I2and its generalized tautology[J].Journal of Engineering Mathematics,2003,20(2):118-121.
[33]Han Y,Chen S F.Algebraic structure of the disturbing fuzzy propositional logic and the properties of its generalized tautology[J].Applied Mathematics-A Journal of Chinese Universities,2005,20(4):483-488.
[34]Han Y,Chen S F,Chen S.Triple I algorithm of disturbingvalued fuzzy reasoning[J].Fuzzy Systems and Mathematics,2007,21(5):14-19.
[35]Peng J Y.Disturbing-valued fuzzy finite-state automata and their languages[J].Pattern Recognition and Artificial Intelligence,2016,29(4):298-312.
[36]Peng J Y.Disturbing fuzzy finite switchboard state machines[J].Computer Engineering and Applications,2016,52(22):1-8.
[37]Peng J Y.Disturbing fuzzy grammars and their languages[J].Journal of Neijiang Normal University,2017,32(10):26-33.
[38]Peng J Y.Transition structure of disturbing fuzzy finite switchboard statemachines[J].Computer Engineering and Applications,2017,53(16):1-9.
[39]Han Y,Chen S F,Chen S.The TD-disturbing-valued fuzzy normal subgroup[J].Fuzzy Systems and Mathematics,2006,20(5):26-29.
[40]Peng J Y.Disturbing fuzzy ideals of quasi BCK-algebra[J].Fuzzy Systems and Mathematics,2016,30(5):1-10.
[41]Han Y.Disturbing fuzzy theory and its application in reasoning decision[D].Nanjing:Southeast University,2009:6-14.
[42]Peng H Q.Disturbing fuzzy ideals of BCK-algebra[J].Journal of Neijiang Normal University,2016,31(6):7-15.
[16]徐泽水.直觉模糊信息集成理论及应用[M].北京:科学出版社, 2008.
[22]彭家寅.基于连续值逻辑之BCK-代数的不分明蕴涵理想[J].模糊系统与数学, 2015, 29(5):1-10.
[23]彭家寅. BCK-代数的广义(∈,∈∨q)-模糊蕴涵理想[J].计算机科学与探索, 2018, 12(4):662-670.
[30]刘心,陈图云.扰动模糊逻辑及其“非”算子[C]//中国系统工程学会模糊数学与模糊系统委员会第十一届年会,厦门, 2002:189-192.
[31]陈图云,韩莹.有限扰动模糊逻辑代数及其广义重言式[J].辽宁师范大学学报, 2002, 25(4):343-345.
[32]陈图云,韩莹,廖士中.扰动模糊逻辑I2的最大子代数及其广义重言式[J].工程数学学报, 2003, 20(2):118-121.
[33]韩莹,陈森发.扰动模糊命题逻辑的代数结构及其广义重言式性质[J].高校应用数学学报, 2005, 20(4):483-488.
[34]韩莹,陈森发,陈胜.扰动模糊推理的三I算法[J].模糊系统与数学, 2007, 21(5):14-19.
[35]彭家寅.扰动值模糊有限自动机及语言[J].模式识别与人工智能, 2016, 29(4):298-312.
[36]彭家寅.扰动模糊有限转换状态机[J].计算机工程与应用, 2016, 52(22):1-8.
[37]彭家寅.扰动模糊文法及其语言[J].内江师范学院学报,2017, 32(10):26-33.
[38]彭家寅.扰动模糊有限状态机的转移结构[J].计算机工程与应用, 2017, 53(16):1-9.
[39]韩莹,陈森发,陈胜. TD型扰动值模糊正规子群[J].模糊系统与数学, 2006, 20(5):26-29.
[40]彭家寅.拟BCK-代数的扰动模糊理想[J].模糊系统与数学, 2016, 30(5):1-10.
[41]韩莹.扰动模糊理论及其在推理决策中的应用[D].南京:东南大学, 2009:6-14.
[42]彭虹侨. BCK代数的扰动模糊理想[J].内江师范学院学报, 2016, 31(6):7-15.
[2]Imai K,Iseki K.On axiom system of propositional calculus[J].Proceedings of the Japan Academy,1966,42:19-22.
[3]Iseki K,Tanaka S.An introduction to the theory of BCK-algebras[J].Mathematica Japonica,1978,23:1-26.
[4]Iseki K,Tanaka S.Ideal theory of BCK-algebras[J].Mathematica Japonica,1976,21:351-366.
[5]Peng J Y.Normed BCI-algebras[J].Chinese Quarterly Journal of Mathematics,2011,26(4):543-548.
[6]Meng J.On ideals in BCK-algebras[J].Mathematica Japonica,1994,1:143-154.
[7]Iseki K.On BCI-algebras[J].Mathematics Seminar Notes,1980,8:125-130.
[8]Lei T D,Xi C C.p-radical in BCI-algebras[J].Mathematica Japonica,1985,30:511-517.
[9]Xi C C.On a class of BCI-algebra[J].Mathematica Japonica,1990,35:13-17.
[10]Zhang X H,Hao J,Bhatti S A.On p-ideals of a BCI-algebra[J].Punjab University Journal of Mathematics,1994,27:121-128.
[11]Liu Y L,Xu Y,Meng J.BCI-implicative ideals of BCI-algebras[J].Information Sciences,2007,177(22):4987-4996.
[12]Liu Y L,Meng J,Zhang X H,et al.q-ideals and a-ideals in BCI-algebras[J].Southeast Asian Bulletin of Mathematics,2000,24:243-253.
[13]Meng J,Xin X L.Commutative BCI-algebras[J].Mathematica Japonica,1992,37:569-572.
[14]Zadeh L A.Fuzzy sets[J].Information and Control,1965,8(3):338-353.
[15]Li D F,Cheng C T.New similarity measures of intuitionistic fuzzy sets and application to pattern recognitions[J].Pattern Recognition Letters,2002,23(1):221-225.
[16]Xu Z S.Theory and application of intuitionistic fuzzy information integration[M].Beijing:Science Press,2008.
[17]Liu H W,Wang G J.Multi-criteria decision-making methods based on intuitionistic fuzzy sets[J].European Journal of Operational Research,2007,179(1):220-233.
[18]Xi O G.Fuzzy BCK-algebras[J].Mathematica Japonica,1991,36(5):935-942.
[19]Meng J,Guo X E.On fuzzy ideals in BCK/BCI-algebras[J].Fuzzy Sets and Systems,2005,149(3):509-525.
[20]Jun Y B,Hong S M,Meng J N,et al.Characterizations of fuzzy positive implicative ideals in BCK-algebras[J].Mathematica Japonica,1994,40:503-507.
[21]Jun Y B,Roh E H.Fuzzy commutative ideals of BCK-algebras[J].Fuzzy Sets and Systems,1994,64(1):401-405.
[22]Peng J Y.Fuzzifying implicative ideals of BCK-algebras based on continuous-valued logic[J].Fuzzy Systems and Mathematics,2015,29(5):1-10.
[23]Peng J Y.Generalized(∈,∈∨q)-fuzzy implicative ideals of BCK-algebras[J].Journal of Frontiers of Computer Science and Technology,2018,12(4):662-670.
[24]Jun Y B,Meng J N.Fuzzy p-ideals in BCI-algebras[J].Mathematica Japonica,1994,40(2):271-282.
[25]Meng J.Fuzzy ideals of BCI-algebras[J].Southeast Asian Bulletin of Mathematics,1994,18(1):65-68.
[26]Atannassov K T.Intuitionistic fuzzy sets[J].Fuzzy Sets and Systems,1986,20(1):87-96.
[27]Zadeh L A.The concept of a linguistic variable and its application to approximate reasoning-I[J].Information Science,1975,8(3):199-249.
[28]Torra V.Hesitant fuzzy sets[J].International Journal of Intelligent Systems,2010,25(6):529-539.
[29]Gau W L,Buehrer D J.Vague sets[J].IEEE Transactions on Systens,Man,and Cybernetics,1993,23(2):610-614.
[30]Liu X,Chen T Y.Disturbing fuzzy logic and its“non operator”[C]//Proceeding of the 11th Annual Conference of Systems Engineering Society of China,Xiamen,2002:189-192.
[31]Chen T Y,Han Y.The algebra and generalized tautology in limited disturbing fuzzy propositional logic[J].Journal of Liaoning Normal University,2002,25(4):343-345.
[32]Chen T Y,Han Y,Liao S Z.The largest subalgebra of disturbed fuzzy logic I2and its generalized tautology[J].Journal of Engineering Mathematics,2003,20(2):118-121.
[33]Han Y,Chen S F.Algebraic structure of the disturbing fuzzy propositional logic and the properties of its generalized tautology[J].Applied Mathematics-A Journal of Chinese Universities,2005,20(4):483-488.
[34]Han Y,Chen S F,Chen S.Triple I algorithm of disturbingvalued fuzzy reasoning[J].Fuzzy Systems and Mathematics,2007,21(5):14-19.
[35]Peng J Y.Disturbing-valued fuzzy finite-state automata and their languages[J].Pattern Recognition and Artificial Intelligence,2016,29(4):298-312.
[36]Peng J Y.Disturbing fuzzy finite switchboard state machines[J].Computer Engineering and Applications,2016,52(22):1-8.
[37]Peng J Y.Disturbing fuzzy grammars and their languages[J].Journal of Neijiang Normal University,2017,32(10):26-33.
[38]Peng J Y.Transition structure of disturbing fuzzy finite switchboard statemachines[J].Computer Engineering and Applications,2017,53(16):1-9.
[39]Han Y,Chen S F,Chen S.The TD-disturbing-valued fuzzy normal subgroup[J].Fuzzy Systems and Mathematics,2006,20(5):26-29.
[40]Peng J Y.Disturbing fuzzy ideals of quasi BCK-algebra[J].Fuzzy Systems and Mathematics,2016,30(5):1-10.
[41]Han Y.Disturbing fuzzy theory and its application in reasoning decision[D].Nanjing:Southeast University,2009:6-14.
[42]Peng H Q.Disturbing fuzzy ideals of BCK-algebra[J].Journal of Neijiang Normal University,2016,31(6):7-15.
[16]徐泽水.直觉模糊信息集成理论及应用[M].北京:科学出版社, 2008.
[22]彭家寅.基于连续值逻辑之BCK-代数的不分明蕴涵理想[J].模糊系统与数学, 2015, 29(5):1-10.
[23]彭家寅. BCK-代数的广义(∈,∈∨q)-模糊蕴涵理想[J].计算机科学与探索, 2018, 12(4):662-670.
[30]刘心,陈图云.扰动模糊逻辑及其“非”算子[C]//中国系统工程学会模糊数学与模糊系统委员会第十一届年会,厦门, 2002:189-192.
[31]陈图云,韩莹.有限扰动模糊逻辑代数及其广义重言式[J].辽宁师范大学学报, 2002, 25(4):343-345.
[32]陈图云,韩莹,廖士中.扰动模糊逻辑I2的最大子代数及其广义重言式[J].工程数学学报, 2003, 20(2):118-121.
[33]韩莹,陈森发.扰动模糊命题逻辑的代数结构及其广义重言式性质[J].高校应用数学学报, 2005, 20(4):483-488.
[34]韩莹,陈森发,陈胜.扰动模糊推理的三I算法[J].模糊系统与数学, 2007, 21(5):14-19.
[35]彭家寅.扰动值模糊有限自动机及语言[J].模式识别与人工智能, 2016, 29(4):298-312.
[36]彭家寅.扰动模糊有限转换状态机[J].计算机工程与应用, 2016, 52(22):1-8.
[37]彭家寅.扰动模糊文法及其语言[J].内江师范学院学报,2017, 32(10):26-33.
[38]彭家寅.扰动模糊有限状态机的转移结构[J].计算机工程与应用, 2017, 53(16):1-9.
[39]韩莹,陈森发,陈胜. TD型扰动值模糊正规子群[J].模糊系统与数学, 2006, 20(5):26-29.
[40]彭家寅.拟BCK-代数的扰动模糊理想[J].模糊系统与数学, 2016, 30(5):1-10.
[41]韩莹.扰动模糊理论及其在推理决策中的应用[D].南京:东南大学, 2009:6-14.
[42]彭虹侨. BCK代数的扰动模糊理想[J].内江师范学院学报, 2016, 31(6):7-15.
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