新型软亚BCI-代数的进一步研究
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摘要
提出了两个亚BCI-代数的并代数、两个软集在并代数上的扩展交、软集的软平移以及两个软集的合成运算等新概念。举例说明了并代数以及在并代数上扩展交的存在性。证明了两个新型软亚BCI-代数在一定的条件下在并代数上的扩展交仍然是新型软亚BCI-代数。研究了新型软亚BCI-代数的软平移及投影等的相关性质。另外,利用两个软集的合成运算、软集的水平集以及广义特征函数分别给出了新型软亚BCI-代数的等价刻画。
The notions of union algebra of two weak-BCI-algebras, extended intersection of two soft sets on union algebra,soft translation of soft set and composite operation of two soft sets are proposed. Examples are given to illustrate the existence of union algebra and extended intersection on union algebra. It is proved that the extended intersection of two new type soft weak-BCI-algebras on union algebra is also the new type of soft weak-BCI-algebra under certain condition.The properties of soft translation and projection of the new type of soft weak-BCI-algebra are studied. Furthermore, by applying compositional operation, the level set of soft set and generalized characteristic function, the equivalent characterizations of the new type of soft weak-BCI-algebra are obtained respectively.
The notions of union algebra of two weak-BCI-algebras, extended intersection of two soft sets on union algebra,soft translation of soft set and composite operation of two soft sets are proposed. Examples are given to illustrate the existence of union algebra and extended intersection on union algebra. It is proved that the extended intersection of two new type soft weak-BCI-algebras on union algebra is also the new type of soft weak-BCI-algebra under certain condition.The properties of soft translation and projection of the new type of soft weak-BCI-algebra are studied. Furthermore, by applying compositional operation, the level set of soft set and generalized characteristic function, the equivalent characterizations of the new type of soft weak-BCI-algebra are obtained respectively.
引文
[1] Imai Y,Iseki K.On axiom systems of propositional calculi XIV[J].Proceedings of the Japan Academy,1966,42:19-22.
[2] Iseki K.An algebra related with a propositional calculus[J].Proceedings of the Japan Academy,1966,42:26-29.
[3]胡庆平,Iseki K.可结合的BCI-代数[J].科学通报,1982(12):714-716.
[4] Lin Y L.Some results on P-semisimple BCI-algebras[J].Math Japon,1992,37:79-81.
[5] Dvurecenskij A,Pulmanova S.New trends in quantum algebtas[M].Dordrecht:Kluwer Academic Publishers,2000.
[6] Cignoli R,Torrens A.Glivenko like theorems in natural expansions of BCK-logic[J].Math Logic Quarter,2004,50(2):111-125.
[7] Dymek G.On pseudo-BCI-algebras[J].Annales UMCS Mathematica,2015,69(1):59-71.
[8]陈露,蒲义书.亚BCI-代数及其理想[J].纯粹数学与应用数学,2005,21(3):250-254.
[9]彭家寅.亚BCI-代数的模糊理想[J].模糊系统与数学,2010,24(5):38-41.
[10] Molodtsov D.Soft set theory-first results[J].Computers and Mathematics with Applications,1999,37(4/5):19-31.
[11] Maji P K,Roy A R,Biswas R.An application of soft sets in a decision making problem[J].Comput Math Appl,2002:1077-1083.
[12] Feng F,?a?man N.Generalized uni-int decision making schemes based on choice value soft sets[J].European Journal of Operational Research,2012,220(1):162-170.
[13] Khameneh A Z,K?l??man A.Multi-attribute decisionmaking based on soft set theory:a systematic review[J].Soft Computing,2018:1-22.
[14] Feng F,Li Y M,Li C X,et al.Soft set based approximate reasoning:a quantitative logic approach[C]//Quantitative Logic and Soft Computing,2010,2:245-255.
[15] Pant J,Juyal A,Bahuguna S.Soft set,a soft computing approach for dimensionality reduction[J].International Journal of Innovative Science,Engineering and Technology,2015,2(4):728-735.
[16] Zou Yan,Xiao Zhi.Data analysis approaches of soft sets under incomplete information[J].Knowledge-Based Syst,2008,21(8):941-945.
[17] Aktas H,Cagman N.Soft sets and soft groups[J].Information Science,2007,177:2726-2735.
[18] Jun Y B.Soft BCK/BCI algebras[J].Computers and Mathematics with Applications,2008,56(5):1408-1413.
[19]伏文清,李生刚.软BCK代数[J].计算机工程与应用,2010,46(10):5-6.
[20]伏文清.软BCK代数再研究[J].计算机工程与应用,2012,48(11):22-25.
[21]刘春辉.滤子化软FI代数[J].山东大学学报(理学版),2012,47(11):94-98.
[22]彭家寅.软BCC-理想与理想的软BCC-代数[J].模糊系统与数学,2016(3):1-8.
[23]温永川.关于软集的研究[D].辽宁大连:辽宁师范大学,2008.
[24] Zheng Gaoping,Liao Zuhua,Wang Nini,et al.Soft lattice implication subalgebras[J].Appl Math Inform Sci,2013,7(3):1181-1186.
[25]廖祖华,王妮妮,郑高平,等.格蕴涵代数的软滤子[J].计算机工程与应用,2015,51(9):72-76.
[26]李巧燕,廖祖华,刘维龙.(α,β)-软BCK代数[J].计算机科学与探索,2016,10(10):1493-1500.
[27]廖翠萃,廖祖华,张龙祥,等.新型软亚BCI-代数[J].计算机科学,2016,43(1):22-24.
[28] Jun Y B,Ahn S S,Muhiuddin G.Hesitant fuzzy soft subalgebras and ideals in BCK/BCI-algebras[J].The Scientific World Journal,2014.
[29] Jun Y B,Song S Z.Hesitant fuzzy set theory applied to filters in MTL-algebras[J].Honam Mathematical Journal,2014,36(4):813-830.
[30] Jun Y B,Song S Z.Hesitant fuzzy prefilters and filters of EQ-algebras[J].Applied Mathematical Science,2015,9(11):515-532.
[31] Rezaei A,Saeid A B.Hesitant fuzzy filters in BEalgebras[J].International Journal of Computational Intelligence Systems,2016,9(1):110-119.
[32] Muhiuddin G,Kim H S,Song S Z,et al.Hesitant fuzzy translations and extensions of subalgebras and ideals in BCK/BCI-algebras[J].Journal of Intelligent and Fuzzy Systems,2017,32:43-48.
[33]彭家寅.BL-代数的区间值犹豫模糊理想[J].模糊系统与数学,2017,31(6):1-13.
[34]彭家寅.剩余格的犹豫模糊滤子理论[J].计算机科学与探索,2017,11(11):1860-1870.
[35] Alshehri H A,Abujabal H A,Alshehri N O.New types of hesitant fuzzy soft ideals in BCK-algebras[J].Soft Computing,2018,22:3675-3683.
[36] Maji P K,Biswas R,Roy A R.Soft set theory[J].Computers Mathematics with Applications,2003,45(4):555-562.
[37]奚悦,廖祖华,陈鑫梦,等.KU代数的新型软理想[J].模糊系统与数学,2017,31(2):13-21.
[2] Iseki K.An algebra related with a propositional calculus[J].Proceedings of the Japan Academy,1966,42:26-29.
[3]胡庆平,Iseki K.可结合的BCI-代数[J].科学通报,1982(12):714-716.
[4] Lin Y L.Some results on P-semisimple BCI-algebras[J].Math Japon,1992,37:79-81.
[5] Dvurecenskij A,Pulmanova S.New trends in quantum algebtas[M].Dordrecht:Kluwer Academic Publishers,2000.
[6] Cignoli R,Torrens A.Glivenko like theorems in natural expansions of BCK-logic[J].Math Logic Quarter,2004,50(2):111-125.
[7] Dymek G.On pseudo-BCI-algebras[J].Annales UMCS Mathematica,2015,69(1):59-71.
[8]陈露,蒲义书.亚BCI-代数及其理想[J].纯粹数学与应用数学,2005,21(3):250-254.
[9]彭家寅.亚BCI-代数的模糊理想[J].模糊系统与数学,2010,24(5):38-41.
[10] Molodtsov D.Soft set theory-first results[J].Computers and Mathematics with Applications,1999,37(4/5):19-31.
[11] Maji P K,Roy A R,Biswas R.An application of soft sets in a decision making problem[J].Comput Math Appl,2002:1077-1083.
[12] Feng F,?a?man N.Generalized uni-int decision making schemes based on choice value soft sets[J].European Journal of Operational Research,2012,220(1):162-170.
[13] Khameneh A Z,K?l??man A.Multi-attribute decisionmaking based on soft set theory:a systematic review[J].Soft Computing,2018:1-22.
[14] Feng F,Li Y M,Li C X,et al.Soft set based approximate reasoning:a quantitative logic approach[C]//Quantitative Logic and Soft Computing,2010,2:245-255.
[15] Pant J,Juyal A,Bahuguna S.Soft set,a soft computing approach for dimensionality reduction[J].International Journal of Innovative Science,Engineering and Technology,2015,2(4):728-735.
[16] Zou Yan,Xiao Zhi.Data analysis approaches of soft sets under incomplete information[J].Knowledge-Based Syst,2008,21(8):941-945.
[17] Aktas H,Cagman N.Soft sets and soft groups[J].Information Science,2007,177:2726-2735.
[18] Jun Y B.Soft BCK/BCI algebras[J].Computers and Mathematics with Applications,2008,56(5):1408-1413.
[19]伏文清,李生刚.软BCK代数[J].计算机工程与应用,2010,46(10):5-6.
[20]伏文清.软BCK代数再研究[J].计算机工程与应用,2012,48(11):22-25.
[21]刘春辉.滤子化软FI代数[J].山东大学学报(理学版),2012,47(11):94-98.
[22]彭家寅.软BCC-理想与理想的软BCC-代数[J].模糊系统与数学,2016(3):1-8.
[23]温永川.关于软集的研究[D].辽宁大连:辽宁师范大学,2008.
[24] Zheng Gaoping,Liao Zuhua,Wang Nini,et al.Soft lattice implication subalgebras[J].Appl Math Inform Sci,2013,7(3):1181-1186.
[25]廖祖华,王妮妮,郑高平,等.格蕴涵代数的软滤子[J].计算机工程与应用,2015,51(9):72-76.
[26]李巧燕,廖祖华,刘维龙.(α,β)-软BCK代数[J].计算机科学与探索,2016,10(10):1493-1500.
[27]廖翠萃,廖祖华,张龙祥,等.新型软亚BCI-代数[J].计算机科学,2016,43(1):22-24.
[28] Jun Y B,Ahn S S,Muhiuddin G.Hesitant fuzzy soft subalgebras and ideals in BCK/BCI-algebras[J].The Scientific World Journal,2014.
[29] Jun Y B,Song S Z.Hesitant fuzzy set theory applied to filters in MTL-algebras[J].Honam Mathematical Journal,2014,36(4):813-830.
[30] Jun Y B,Song S Z.Hesitant fuzzy prefilters and filters of EQ-algebras[J].Applied Mathematical Science,2015,9(11):515-532.
[31] Rezaei A,Saeid A B.Hesitant fuzzy filters in BEalgebras[J].International Journal of Computational Intelligence Systems,2016,9(1):110-119.
[32] Muhiuddin G,Kim H S,Song S Z,et al.Hesitant fuzzy translations and extensions of subalgebras and ideals in BCK/BCI-algebras[J].Journal of Intelligent and Fuzzy Systems,2017,32:43-48.
[33]彭家寅.BL-代数的区间值犹豫模糊理想[J].模糊系统与数学,2017,31(6):1-13.
[34]彭家寅.剩余格的犹豫模糊滤子理论[J].计算机科学与探索,2017,11(11):1860-1870.
[35] Alshehri H A,Abujabal H A,Alshehri N O.New types of hesitant fuzzy soft ideals in BCK-algebras[J].Soft Computing,2018,22:3675-3683.
[36] Maji P K,Biswas R,Roy A R.Soft set theory[J].Computers Mathematics with Applications,2003,45(4):555-562.
[37]奚悦,廖祖华,陈鑫梦,等.KU代数的新型软理想[J].模糊系统与数学,2017,31(2):13-21.