软集合理论在模糊逻辑代数中的应用研究
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摘要
软集合理论是由Molodtsov在1999年提出的,它是一种新的处理模糊和不确定性模型的数学工具.由于软集合中的参数可以取任意形式,使得该理论在数学,经济学,工程学和物理学等领域中都得到了广泛应用.
本文将软集合理论运用到模糊逻辑代数中.研究并讨论了软BCI-代数的软关联理想、软正定关联理想和软可换理想,软BL-代数的软滤子、软蕴涵滤子、软正蕴涵滤子和软奇异滤子,并对它们的性质进行了研究.得到一系列有意义的结果,丰富了软集合理论的代数结构.具体内容如下:
(1)给出软BCI-代数的软关联理想、软正定关联理想和软可换理想的概念.讨论了软理想、软关联理想、软正定关联理想和软可换理想之间的关系,并对两个软关联理想(软正定关联理想,软可换理想)的扩展交、限制交、限制并、限制差分的运算性质和软相等关系进行了研究.得到两个软关联理想(软正定关联理想,软可换理想)的扩展交,限制交依然是一个软关联理想(软正定关联理想,软可换理想),同时,给出实例说明两个软关联理想(软正定关联理想,软可换理想)的限制并、限制差分不是软关联理想(软正定关联理想,软可换理想).
(2)在软BL-代数上提出了软滤子、软蕴涵滤子、软正蕴涵滤子和软奇异滤子的概念,讨论软滤子、软蕴涵滤子、软正蕴涵滤子和软奇异滤子之间的关系,并对两个软滤子(软蕴涵滤子、软正蕴涵滤子、软奇异滤子)的软相等关系和偏序关系进行了研究.
The soft set theory was introduced by D Molodtsov in 1999. It is a new mathematical tool for dealing with vagueness and uncertainty model. As the parameters of soft sets can take any form, the theory has been widely used in mathematics, economics, engineering, physics and other fields.
In this paper, the soft set theory is applied to fuzzy logical algebras. Soft implicative ideals, soft positive implicative ideals, soft commutative ideals of BCI-algebras, soft filters, soft implicative filters, soft positive implicative filters, soft fantastic filters of BL-algebras are studied. Meanwhile, some fundamental properties of them are discussed. The results obtained in this paper, enrich the algebraic structure of the soft set theory. We list the main results as follows:
(1) The definitions of soft implicative ideals, soft positive implicative ideals, soft commutative ideals of BCI-algebras are introduced. Then, the relations among the soft ideals, the soft implicative ideals, the soft positive implicative ideals and the soft commutative ideals are discussed. Furthermore, the extended intersection operation, the restricted intersection operation, the restricted union operation, the restricted difference operation and soft equality relation of two soft implicative ideals (soft positive implicative ideals, soft commutative ideals) are studied. And we have got that the extended intersection operation (the restricted intersection operation) of two soft implicative ideals (soft positive implicative ideals, soft commutative ideals) is still a soft implicative ideal (soft positive implicative ideal, soft commutative ideal). Moreover, we give some examples to illustrate that the restricted union operation, the restricted difference operation of two soft implicative ideals (soft positive implicative ideals, soft commutative ideals) may not be a soft implicative ideal (soft positive implicative ideal, soft commutative ideal).
(2) The notions of soft filters, soft implicative filters, soft positive implicative filters and soft fantastic filters in soft BL-algebras are proposed. Then, the relations among the soft filters, the soft implicative filters, the soft positive implicative filters and the soft fantastic filters are investigated. Both the soft equality relation and the partial ordering relation of two soft implicative filters (soft positive implicative filters, soft fantastic filters) are studied.
本文将软集合理论运用到模糊逻辑代数中.研究并讨论了软BCI-代数的软关联理想、软正定关联理想和软可换理想,软BL-代数的软滤子、软蕴涵滤子、软正蕴涵滤子和软奇异滤子,并对它们的性质进行了研究.得到一系列有意义的结果,丰富了软集合理论的代数结构.具体内容如下:
(1)给出软BCI-代数的软关联理想、软正定关联理想和软可换理想的概念.讨论了软理想、软关联理想、软正定关联理想和软可换理想之间的关系,并对两个软关联理想(软正定关联理想,软可换理想)的扩展交、限制交、限制并、限制差分的运算性质和软相等关系进行了研究.得到两个软关联理想(软正定关联理想,软可换理想)的扩展交,限制交依然是一个软关联理想(软正定关联理想,软可换理想),同时,给出实例说明两个软关联理想(软正定关联理想,软可换理想)的限制并、限制差分不是软关联理想(软正定关联理想,软可换理想).
(2)在软BL-代数上提出了软滤子、软蕴涵滤子、软正蕴涵滤子和软奇异滤子的概念,讨论软滤子、软蕴涵滤子、软正蕴涵滤子和软奇异滤子之间的关系,并对两个软滤子(软蕴涵滤子、软正蕴涵滤子、软奇异滤子)的软相等关系和偏序关系进行了研究.
The soft set theory was introduced by D Molodtsov in 1999. It is a new mathematical tool for dealing with vagueness and uncertainty model. As the parameters of soft sets can take any form, the theory has been widely used in mathematics, economics, engineering, physics and other fields.
In this paper, the soft set theory is applied to fuzzy logical algebras. Soft implicative ideals, soft positive implicative ideals, soft commutative ideals of BCI-algebras, soft filters, soft implicative filters, soft positive implicative filters, soft fantastic filters of BL-algebras are studied. Meanwhile, some fundamental properties of them are discussed. The results obtained in this paper, enrich the algebraic structure of the soft set theory. We list the main results as follows:
(1) The definitions of soft implicative ideals, soft positive implicative ideals, soft commutative ideals of BCI-algebras are introduced. Then, the relations among the soft ideals, the soft implicative ideals, the soft positive implicative ideals and the soft commutative ideals are discussed. Furthermore, the extended intersection operation, the restricted intersection operation, the restricted union operation, the restricted difference operation and soft equality relation of two soft implicative ideals (soft positive implicative ideals, soft commutative ideals) are studied. And we have got that the extended intersection operation (the restricted intersection operation) of two soft implicative ideals (soft positive implicative ideals, soft commutative ideals) is still a soft implicative ideal (soft positive implicative ideal, soft commutative ideal). Moreover, we give some examples to illustrate that the restricted union operation, the restricted difference operation of two soft implicative ideals (soft positive implicative ideals, soft commutative ideals) may not be a soft implicative ideal (soft positive implicative ideal, soft commutative ideal).
(2) The notions of soft filters, soft implicative filters, soft positive implicative filters and soft fantastic filters in soft BL-algebras are proposed. Then, the relations among the soft filters, the soft implicative filters, the soft positive implicative filters and the soft fantastic filters are investigated. Both the soft equality relation and the partial ordering relation of two soft implicative filters (soft positive implicative filters, soft fantastic filters) are studied.
引文
1. Molodtsov D.Soft set theory-First results[J]. Computers & Mathematics with Applications, 1999, 37: 19-31
2. Chen D, Tsang E C C , Yeung D S, Wang X. The parametrization reduction of soft sets and its applications[J]. Computers & Mathematics with Applications, 2005, 49: 757-763
3. Zou Y, Xiao Z. Data analysis approaches of soft sets under incomplete information[J]. Computers & Mathematics with Applications, 2008, 21(8):941-945
4. Kong Z, Gao L Q ,Wang L F, Steven Li.The normal parameter reduction of soft sets and its algorithm[J], Computers & Mathematics with Applications ,2008,56(12):3029-3037
5. Feng F, Li C X, Davvaz B, Ali M I. Soft sets combined with fuzzy sets and rough sets: a tentative approach [J].Soft Computing, 6:1433-7479
6. Maji P K, Roy A R, Biswas R. Fuzzy soft sets[J], Journal of Fuzzy Mathematics, 2001,9 (3):589-602
7. Maji P K, Roy A R, Biswas R. Intuitionistic fuzzy soft sets[J], Journal of Fuzzy Mathematics, 2001, 9 (3): 677-692
8. Maji P K, Roy A R, Biswas R.An application of soft sets in a decision making problem[J]. Computers & Mathematics with Applications, 2002, 44: 1077-1083
9. Maji P K, Roy A R, Biswas R. A fuzzy soft set theoretic approach to decision making problems[J]. Journal of Computational and Applied Mathematics, 2007, 203: 412-418
10. Maji P K, Roy A R, Biswas R.Soft set theory[J]. Computers & Mathematics with Applications, 2003, 45:555-562
11. Yang C F.A note on“Soft Set Theory”[Comput. Math. Appl. 45 (4–5) (2003) 555–562] [J].Computers & Mathematics with Applications, 2008, 56:1899–1900
12. Ali M I, Feng F, Liu X, Min W K , Shabir M.On some new operations in soft set theory[J]. Computers & Mathematics with Applications, 2009, 57: 1547-1553
13. Xu W, Ma J,Wang S Y,Hao G.Vague soft sets and their properties[J].Computers & Mathematics with Applications, 2010, 59:787-794
14. Akta? H, ?a?man N.Soft sets and soft groups[J]. Information Science, 2007, 177: 2726-2735
15. Aygüno?lu A, Aygün H.Introduction to fuzzy soft groups[J]. Computers & Mathematics with Applications, 2009, 58:1279-1286
16. Feng F, Jun B Y, Zhao X.Soft semirings[J]. Computers & Mathematics with Applications, 2008, 56: 2621- 2628
17. Sun Q M, Zhong Z L, Liu J.Soft sets and soft modules, Rough Sets and KnowledgeTechnology[M]. Springer press, 2008, 403-409
18. Acar U, Koyuncu F, Tanay B.Soft sets and soft rings[J]. Computers & Mathematics with Applications, article in press
19. Qin K, Hong Z Y.On soft equality[J].Journal of Computational and Applied Mathematics 2010,234:1347-1355
20. Jun Y B.Soft BCK/BCI-algebras[J]. Computers & Mathematics with Applications, 2008,
56: 1408-1413
21. Park C H, Jun Y B, ?Ozt?urk M A. Soft WS-algebras[J]. Communications of the Korean Mathematical Society, 2008, 23 (3):313-324
22. Jun Y B, Park C H. Applications of soft sets in ideal theory of BCK/BCI-algebras[J]. Information Science, 2008, 178: 2466-2475
23. Jun Y B , Lee K J, Park C H. Soft set theory applied to ideals in d-algebras[J]. Computers & Mathematics with Applications, 2009, 57: 367-378
24. Jun Y B, Lee K J, Zhan J M. Soft p-ideals of soft BCI-algebras[J]. Computers & Mathematics with Applications, 2009, 58:2060-2068
25. Jun Y B, Song S Z.Soft subalgebras and soft ideals of BCK/BCI-algebras related to fuzzy set theory[J]. Mathematical communications, 2009, 14(2):271-282
26. Jun Y B, Lee K J, Park C H.Fuzzy soft set theory applied to BCK/BCI-algebras[J]. Computers and Mathematics with Applications, 2010, 59:3180-3192
27. Jun Y B.BL-algebras based on soft set theory[J]. Kyungpook Mathematical Journal,2010,
50:123-129
28. Zhan J M, Jun B Y. Soft BL-algebras based on fuzzy sets[J]. Computers & Mathematics with Applications, 2010, 59(6): 2037-2046
29. Zhan J M, Dudek W A. Soft MTL-algebras based on fuzzy sets[J]. http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.4595v1
30.孟杰,刘用麟. BCI-代数引论[M].西安:陕西科学技术出版社,2001
31. Iseki K, Tanaka S.Ideal theory of BCK-algebras[J], Math. Japan. 1976,21:351-366
32. Iseki K. On BCI-algebras[J]. Mathametical Semester Notes, 1980, 8:125-130
33.胡庆平. BCI-代数[M].西安:陕西科学技术出版社,1987
33.张小红.模糊逻辑及其代数分析[M].北京:科学出版社,2008
34.王国俊.非经典数理逻辑与近似推理[M].北京:科学出版社,2000
35.王国俊.数理逻辑引论与归结原理[M].北京:科学出版社,2003
36. Meng J. On ideals in BCK-algebras[J].Mathematica Japonica, 1994, 40:143-154
37. Meng J.An ideal characterization of commutative BCI-algebras[J]. Pusan KyongnamMathematics, 1993, 9:1-6
38. Liu Y L, Zhang X H. Characterization of weakly positive implicative BCI-algebras[J]. Journal of Hanzhong Teachers College (Natural), 1994, 1: 4-8
39. Liu Y L, Meng J, Xu Y. BCI-implicative ideals of BCI-algebras[J]. Information Science, 2007, 177: 4987-4996
40. Huang Y S. BCI-algebra[M], Science Press, Beijing, 2006
41. Meng J, Guo X. On fuzzy ideals in BCK/BCI-algebras[J], Fuzzy Sets and Systems, 2005, 149: 509–525
42. Jun Y B, Roh E H. Fuzzy commutative ideals of BCK-algebras[J], Fuzzy Sets and Systems, 1994, 64(3): 401-405
43. Hajek P. Metamathematics of Fuzzy Logic[M]. Kluwer Academic Publishers, Dordrecht, 1988
44. Hájek P. Basic fuzzy logic and BL-algebras[J].Soft Computing,1998,2:124-128
45. Hájek P Basic fuzzy logic and BL-algebras[J].Soft Computing,2003,7:179-183
46. Turunen E. BL-algebras and Basic Fuzzy Logic[J].Mathware and Soft Computing, 1999,
6:49-61
47. Turunen E. Boolean deductive systems of BL-algebras[J].Archive for Mathematical Logic, 2001,40:467-473
48. Turunen E, Sessa S. Local BL-algebras[J]. Mult-Valued Logic, 2001, 6:229-249
49. Haveshki M, Saeid A B, Eslami E. Some types of filters in BL-algebras[J].Soft Computing, 2006, 10:657-664
50. Haveshki M, Saeid A B, Eslami E. Some types of filters in BL-algebras[J]. Soft Computing, 2007, 11:209
51. Kondo M, Dudek W A. Filter theory of BL algebras[J]. Soft Computing, 2008, 12: 419-423
52. Liu L Z, Li K T. Fuzzy boolean and positive implicative filters of BL-algebras[J]. Fuzzy Sets System, 2005, 152:333-348
53. Liu L Z, Li K T, Fuzzy Filters of BL-algebras[J]. Information Sciences, 2005, 173:141-154
54. Ma X L, Zhang J M, Xu Y. Generalized fuzzy filters of BL-algebras[J]. Appl. Math. J. Chinese Univ. Ser. B, 2007, 22(4):490-496
55. Jun Y B, Ko J M. Folding theory applied to BL-algebras[J]. Central European Journal of Mathematics, 2004, 2(4):584 -592
56. Haveshki M, Eslami E. n-fold filters in BL-algebras[J]. Mathematical Logic Quarterly, 2008, 54(2): 176 -186
57. Saeid A B, Motamed S.Normal Filters in BL-Algebras[J], World Applied Sciences Journal (Special Issue for Applied Math), 2009,7: 70-76,
58. Lele C. Folding theory of positive implicative/fuzzy positive implicative filters in BL-algebras[J], (To appear)
59. Lele C. Foldness of fantastic/fuzzy fantastic filters in BL-algebras [J], (To appear)
2. Chen D, Tsang E C C , Yeung D S, Wang X. The parametrization reduction of soft sets and its applications[J]. Computers & Mathematics with Applications, 2005, 49: 757-763
3. Zou Y, Xiao Z. Data analysis approaches of soft sets under incomplete information[J]. Computers & Mathematics with Applications, 2008, 21(8):941-945
4. Kong Z, Gao L Q ,Wang L F, Steven Li.The normal parameter reduction of soft sets and its algorithm[J], Computers & Mathematics with Applications ,2008,56(12):3029-3037
5. Feng F, Li C X, Davvaz B, Ali M I. Soft sets combined with fuzzy sets and rough sets: a tentative approach [J].Soft Computing, 6:1433-7479
6. Maji P K, Roy A R, Biswas R. Fuzzy soft sets[J], Journal of Fuzzy Mathematics, 2001,9 (3):589-602
7. Maji P K, Roy A R, Biswas R. Intuitionistic fuzzy soft sets[J], Journal of Fuzzy Mathematics, 2001, 9 (3): 677-692
8. Maji P K, Roy A R, Biswas R.An application of soft sets in a decision making problem[J]. Computers & Mathematics with Applications, 2002, 44: 1077-1083
9. Maji P K, Roy A R, Biswas R. A fuzzy soft set theoretic approach to decision making problems[J]. Journal of Computational and Applied Mathematics, 2007, 203: 412-418
10. Maji P K, Roy A R, Biswas R.Soft set theory[J]. Computers & Mathematics with Applications, 2003, 45:555-562
11. Yang C F.A note on“Soft Set Theory”[Comput. Math. Appl. 45 (4–5) (2003) 555–562] [J].Computers & Mathematics with Applications, 2008, 56:1899–1900
12. Ali M I, Feng F, Liu X, Min W K , Shabir M.On some new operations in soft set theory[J]. Computers & Mathematics with Applications, 2009, 57: 1547-1553
13. Xu W, Ma J,Wang S Y,Hao G.Vague soft sets and their properties[J].Computers & Mathematics with Applications, 2010, 59:787-794
14. Akta? H, ?a?man N.Soft sets and soft groups[J]. Information Science, 2007, 177: 2726-2735
15. Aygüno?lu A, Aygün H.Introduction to fuzzy soft groups[J]. Computers & Mathematics with Applications, 2009, 58:1279-1286
16. Feng F, Jun B Y, Zhao X.Soft semirings[J]. Computers & Mathematics with Applications, 2008, 56: 2621- 2628
17. Sun Q M, Zhong Z L, Liu J.Soft sets and soft modules, Rough Sets and KnowledgeTechnology[M]. Springer press, 2008, 403-409
18. Acar U, Koyuncu F, Tanay B.Soft sets and soft rings[J]. Computers & Mathematics with Applications, article in press
19. Qin K, Hong Z Y.On soft equality[J].Journal of Computational and Applied Mathematics 2010,234:1347-1355
20. Jun Y B.Soft BCK/BCI-algebras[J]. Computers & Mathematics with Applications, 2008,
56: 1408-1413
21. Park C H, Jun Y B, ?Ozt?urk M A. Soft WS-algebras[J]. Communications of the Korean Mathematical Society, 2008, 23 (3):313-324
22. Jun Y B, Park C H. Applications of soft sets in ideal theory of BCK/BCI-algebras[J]. Information Science, 2008, 178: 2466-2475
23. Jun Y B , Lee K J, Park C H. Soft set theory applied to ideals in d-algebras[J]. Computers & Mathematics with Applications, 2009, 57: 367-378
24. Jun Y B, Lee K J, Zhan J M. Soft p-ideals of soft BCI-algebras[J]. Computers & Mathematics with Applications, 2009, 58:2060-2068
25. Jun Y B, Song S Z.Soft subalgebras and soft ideals of BCK/BCI-algebras related to fuzzy set theory[J]. Mathematical communications, 2009, 14(2):271-282
26. Jun Y B, Lee K J, Park C H.Fuzzy soft set theory applied to BCK/BCI-algebras[J]. Computers and Mathematics with Applications, 2010, 59:3180-3192
27. Jun Y B.BL-algebras based on soft set theory[J]. Kyungpook Mathematical Journal,2010,
50:123-129
28. Zhan J M, Jun B Y. Soft BL-algebras based on fuzzy sets[J]. Computers & Mathematics with Applications, 2010, 59(6): 2037-2046
29. Zhan J M, Dudek W A. Soft MTL-algebras based on fuzzy sets[J]. http://arxiv.org/PS_cache/arxiv/pdf/1003/1003.4595v1
30.孟杰,刘用麟. BCI-代数引论[M].西安:陕西科学技术出版社,2001
31. Iseki K, Tanaka S.Ideal theory of BCK-algebras[J], Math. Japan. 1976,21:351-366
32. Iseki K. On BCI-algebras[J]. Mathametical Semester Notes, 1980, 8:125-130
33.胡庆平. BCI-代数[M].西安:陕西科学技术出版社,1987
33.张小红.模糊逻辑及其代数分析[M].北京:科学出版社,2008
34.王国俊.非经典数理逻辑与近似推理[M].北京:科学出版社,2000
35.王国俊.数理逻辑引论与归结原理[M].北京:科学出版社,2003
36. Meng J. On ideals in BCK-algebras[J].Mathematica Japonica, 1994, 40:143-154
37. Meng J.An ideal characterization of commutative BCI-algebras[J]. Pusan KyongnamMathematics, 1993, 9:1-6
38. Liu Y L, Zhang X H. Characterization of weakly positive implicative BCI-algebras[J]. Journal of Hanzhong Teachers College (Natural), 1994, 1: 4-8
39. Liu Y L, Meng J, Xu Y. BCI-implicative ideals of BCI-algebras[J]. Information Science, 2007, 177: 4987-4996
40. Huang Y S. BCI-algebra[M], Science Press, Beijing, 2006
41. Meng J, Guo X. On fuzzy ideals in BCK/BCI-algebras[J], Fuzzy Sets and Systems, 2005, 149: 509–525
42. Jun Y B, Roh E H. Fuzzy commutative ideals of BCK-algebras[J], Fuzzy Sets and Systems, 1994, 64(3): 401-405
43. Hajek P. Metamathematics of Fuzzy Logic[M]. Kluwer Academic Publishers, Dordrecht, 1988
44. Hájek P. Basic fuzzy logic and BL-algebras[J].Soft Computing,1998,2:124-128
45. Hájek P Basic fuzzy logic and BL-algebras[J].Soft Computing,2003,7:179-183
46. Turunen E. BL-algebras and Basic Fuzzy Logic[J].Mathware and Soft Computing, 1999,
6:49-61
47. Turunen E. Boolean deductive systems of BL-algebras[J].Archive for Mathematical Logic, 2001,40:467-473
48. Turunen E, Sessa S. Local BL-algebras[J]. Mult-Valued Logic, 2001, 6:229-249
49. Haveshki M, Saeid A B, Eslami E. Some types of filters in BL-algebras[J].Soft Computing, 2006, 10:657-664
50. Haveshki M, Saeid A B, Eslami E. Some types of filters in BL-algebras[J]. Soft Computing, 2007, 11:209
51. Kondo M, Dudek W A. Filter theory of BL algebras[J]. Soft Computing, 2008, 12: 419-423
52. Liu L Z, Li K T. Fuzzy boolean and positive implicative filters of BL-algebras[J]. Fuzzy Sets System, 2005, 152:333-348
53. Liu L Z, Li K T, Fuzzy Filters of BL-algebras[J]. Information Sciences, 2005, 173:141-154
54. Ma X L, Zhang J M, Xu Y. Generalized fuzzy filters of BL-algebras[J]. Appl. Math. J. Chinese Univ. Ser. B, 2007, 22(4):490-496
55. Jun Y B, Ko J M. Folding theory applied to BL-algebras[J]. Central European Journal of Mathematics, 2004, 2(4):584 -592
56. Haveshki M, Eslami E. n-fold filters in BL-algebras[J]. Mathematical Logic Quarterly, 2008, 54(2): 176 -186
57. Saeid A B, Motamed S.Normal Filters in BL-Algebras[J], World Applied Sciences Journal (Special Issue for Applied Math), 2009,7: 70-76,
58. Lele C. Folding theory of positive implicative/fuzzy positive implicative filters in BL-algebras[J], (To appear)
59. Lele C. Foldness of fantastic/fuzzy fantastic filters in BL-algebras [J], (To appear)