[1]张小红.源于非经典逻辑的代数结构研究综述[J].四川师范大学学报(自然科学版),2019,(01):1.[doi:10.3969/j.issn.1001-8395.2019.01.001]
 ZHANG Xiaohong.A Survey of Algebraic Structures Derived from Non-classical Logics[J].Journal of SichuanNormal University,2019,(01):1.[doi:10.3969/j.issn.1001-8395.2019.01.001]
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源于非经典逻辑的代数结构研究综述()
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《四川师范大学学报(自然科学版)》[ISSN:1001-8395/CN:51-1295/N]

卷:
期数:
2019年01期
页码:
1
栏目:
特约专稿
出版日期:
2018-12-15

文章信息/Info

Title:
A Survey of Algebraic Structures Derived from Non-classical Logics
文章编号:
1001-8395(2019)01-0001-14
作者:
张小红
陕西科技大学 中加数据智能与三支决策研究中心, 陕西 西安 710072
Author(s):
ZHANG Xiaohong
Research Center for Data Intelligence and Three-way Decision, Shaanxi University of Science and Technology, Xi'an 710072, Shaanxi
关键词:
非经典逻辑 模糊逻辑 代数结构 蕴涵代数 滤子
Keywords:
non-classical logic fuzzy logic algebraic structure implication algebra filter
分类号:
O141; O159
DOI:
10.3969/j.issn.1001-8395.2019.01.001
文献标志码:
A
摘要:
作为智能科学的数学基础之一,非经典逻辑(主要指非经典数理逻辑)及相关代数结构扮演着十分重要的角色.对源于非经典逻辑的代数结构进行全面系统总结,从蕴涵片段的视角梳理其中的内在联系,这些代数结构包括各种可换剩余格、非可换剩余格、非结合剩余格、剩余有序广群、BCK/BCI-代数、BCC/BZ-代数、伪BCK/BCI-代数等.同时介绍近年来非经典逻辑代数方向的最新研究进展,包括量子B-代数(quantum B-agebra)、EO-代数(extended-order algebra)及新近提出的基本蕴涵代数(basic implication algebra)等.
Abstract:
As one of the mathematical foundations of intelligent science, non-classical logic(mainly referring to non-classical mathematical logic)and related algebraic structures play an important role. In this paper, the algebraic structures derived from non-classical logics are summarized comprehensively and systematically, and their internal relations are sorted out from the perspective of implication fragments. These algebraic structures include various commutative residual lattices, non-commutative residual lattices, non-associative residual lattices, residual ordered groupoids, BCK/BCI-algebras, BCC/BZ-algebras, pseudo BCK/BCI-algebras, etc. At the same time, this paper introduced the latest research progress of non-classical logic algebras in recent years, including quantum B-algebras, EO-algebras and basic implication algebras.

参考文献/References:

[1] 王国俊. 理逻辑引论与归结原理[M]. 2版. 北京:科学出版社,2006.
[2] 王国俊. 非经典数理逻辑与近似推理[M]. 2版. 北京:科学出版社,2008.
[3] WANG G J, ZHOU H J. Introduction to Mathematical Logic and Resolution Principle[M]. Oxford:Alpha International Science Ltd,2009.
[4] HÁJEK P. Metamathematics of Fuzzy Logic[M]. Dordrecht:Kluwer,1998.
[5] KLEMENT E P, MESIAR R, PAP E. Triangular Norms[M]. Berlin:Springer-Verlag,2000.
[6] ESTEVA F, GODO L. Monoidal t-norm based logic:towards a logic for left-continuous t-norms[J]. Fuzzy Sets and Systems,2001,124(3):271-288.
[7] 裴道武. 基于三角模的模糊逻辑理论及其应用[M]. 北京:科学出版社,2013.
[8] 张小红. 模糊逻辑及其代数分析[M]. 北京:科学出版社,2008.
[9] 张小红,折延宏. 模糊量词及其积分语义[M]. 北京:科学出版社,2017.
[10] WARD M, DILWORTH R P. Residuated lattices[J]. Transactions of the American Mathematical Society,1939,45(3):335-354.
[11] DILWORTH R P. Non-commutative residuated lattices[J]. Transactions of the American Mathematical Society,1939,46(3):426-444.
[12] PEI D W. On equivalent forms of fuzzy logic systems NM and IMTL[J]. Fuzzy Sets and Systems,2003,138(1):187-195.
[13] PEI D W. Simplification and independence of axioms of fuzzy logic systems IMTL and NM[J]. Fuzzy Sets and Systems,2005,152(1):303-320.
[14] ABRUSCI V M, RUET P. Non-commutative logic I:the multiplicative fragment[J]. Annals of Pure & Applied Logic,1999,101(1):29-64.
[15] RUET P. Non-commutative logic II:sequent calculus and phase semantics[J]. Mathematical Structures in Computer Science,2000,10(2):277-312.
[16] KAMIDE N, MOURI M. Natural deduction systems for some non-commutative logics[J]. Logic and Logical Philosophy,2007,16:105-146.
[17] FLONDOR P, GEORGESCU G, IORGULESCU A. Pseudo-t-norms and pseudo-BL algebras[J]. Soft Computing,2001,5:355-371.
[18] RACH(。overU)NEK J. A non-commutative generalization of MV -algebras[J]. Czechoslovak Mathematical Journal,2002,52(2):255-273.
[19] HÁJEK P. Observations on non-commutative fuzzy logic[J]. Soft Computing,2003,8(1):38-43.
[20] JENEI S, MONTAGNA F. A proof of standard completeness for non-commutative monoidal t-norm logic[J]. Neural Network World,2003,5:481-489.
[21] LEUTEAN I. Non-commutative ukasiewicz propositional logic[J]. Archive for Mathematical Logic,2006,45(2):191-213.
[22] 张小红. 非可换模糊逻辑系统PL*及其完备性[J]. 数学学报,2007,50(2):421-442.
[23] 张小红. 基于左连续伪t-模的非可换模糊逻辑系统PUL*[J]. 数学进展,2007,36(3):295-308.
[24] CIUNGU L C. Non-Commutative Multiple-Valued Logic Algebras[M]. Berlin:Springer,2013.
[25] KAWAGUCHI M F, MIYAKOSHI M. Approximate reasoning with non-commutative and non-associative conjunctions[C]//Barcelona, Spain:Proceedings of IEEE 6th International Fuzzy Systems Conference,1997:163-169. DOI:10.1109/FUZZY.1997.616363.
[26] MARTINEZ J, MACIAS L, ESPER A, et al. Towards more realistic(e.g., non-associative)“and”- and “or”-operations in fuzzy logic[C]//Tucson:Man & Cybernetics, IEEE International Conference on Systems,2001:2187-2192. DOI:10.1007/s00500-003-0272-4.
[27] LIU H W. Distributivity and conditional distributivity of semi-uninorms over continuous t-conorms and t-norms[J]. Fuzzy Sets and Systems,2015,268:27-43.
[28] QIN F. Distributivity between semi-uninorms and semi-t-operators[J]. Fuzzy Sets and Systems,2016,299:66-88.
[29] BOTUR M. A non-associative generalization of Hájek's BL-algebras[J]. Fuzzy Sets and Systems,2011,178(1):24-37.
[30] CINTULA P, HORÍK R, NOGUERA C. Nonassociative substructural logics and their semilinear extensions:axiomatization and completeness properties[J]. The Review of Symbolic Logic,2013,6(3):394-423.
[31] YANG E. Weakening-free, non-associative fuzzy logics: Micanorm-based logics[J]. Fuzzy Sets and Systems,2015,276:43-58.
[32] CHAJDA I, LANGER H. A non-associative generalization of effect algebras[J]. Mathematica Slovaca,2007,57(4):301-312.
[33] BOTUR M, HALA R. Commutative basic algebras and non-associative fuzzy logics[J]. Arch Math Logic,2009,48:243-255.
[34] ZHANG X H. Commutative weak t-norm and non-associative residuated lattices[C]//Second International Symposium on Knowledge Acquisition and Modeling. Wuhan:IEEE Computer Soc,2009,223-226.DOI: 10.1109/KAM.2009.89.
[35] ZHANG X H, MA H. On filters of non-associative residuated lattices(commutativeresiduated lattice-ordered groupoids)[C]//Proceedings of the Ninth International Conferenceon Machine Learning and Cybernetics. Qingdao:IEEE Computer Soc,2010,2167-2173.DOI: 10.1109/ICMLC.2010.5580485.
[36] 梁聪,张小红. 预线性与对合非结合剩余格[J]. 模糊系统与数学,2012,26(3):17-23.
[37] ZHANG X H. Strong NMV-algebras, commutative basic algebras and naBL-algebras[J]. Mathematica Slovaca,2013,4:661-678.
[38] CHAJDA I, HALA R, LÄNGER H. Operations and structures derived from non-associative MV-algebras[J]. Soft Computing,2018. DOI:10.1007/s00500-018-3309-4.
[39] GABBAY D M, METCALFE G. Fuzzy logics based on [0,1)-continuous uninorms[J]. Arch Math Log,2007,46(5/6):425-449.
[40] MAS M, MONSERRAT M, TORRENS J. Two types of implications derived from uninorms[J]. Fuzzy Sets and Systems,2007,158(23):2612-2626.
[41] BALDI P. A note on standard completeness for some extensions of uninorm logic[J]. Soft Computing,2014,18(8):1463-1470.
[42] WANG Z D, FANG J X. Residual operations of left and right uninorms on a complete lattice[J]. Fuzzy Sets and Systems,2009,160(1):22-31.
[43] WANG Z D, FANG J X. Residual coimplicators of left and right uninorms on a complete lattice[J]. Fuzzy Sets and Systems,2009,160(14):2086-2096.
[44] AGUILÓ I, SUÑER J, TORRENS J. A characterization of residual implications derived from left-continuous uninorms[J]. Information Sciences,2010,180:3992-4005.
[45] 苏勇,王住登. 伪统一模与剩余格之间的关系[J]. 数学的实践与认识,2012,42(19):259-262.
[46] WANG S M. Uninorm logic with the n-potency axiom[J]. Fuzzy Sets and Systems,2012,205(1):116-126.
[47] WANG S M. Logics for residuated pseudo-uninorms and their residua[J]. Fuzzy Sets and Systems,2013,218(1):24-31.
[48] UKASIK R. A note on the mutual independence of the properties in the characterization of residual fuzzy implications derived from left-continuous uninorms[J]. Information Sciences,2014,260:209-214.
[49] XIE A F. On the extension of nullnorms and uninorms to fuzzy truth values[J]. Fuzzy Setsand Systems,2018,352(1):92-118.
[50] YANG E. Basic substructural core fuzzy logics and their extensions:mianorm-based logics[J]. Fuzzy Sets and Systems,2016,301:1-18.
[51] YANG E. Involutive basic substructural core fuzzy logics:involutivemianorm-based logics[J]. Fuzzy Sets and Systems,2017,320:1-16.
[52] LIU H W. Semi-uninorms and implications on a complete lattice[J]. Fuzzy Sets and Systems,2012,191(1):72-82.
[53] SUÁREZGARCIA F, GILÁLVAREZ P. Two families of fuzzy integrals[J]. Fuzzy Sets and Systems,1986,18(1):67-81.
[54] O'CARROLL L. A basis for the theory of residuated groupoids[J]. J London Math Soc,1971,3(2):7-20.
[55] 吴望名. 剩余广群与广义模糊矩阵[J]. 模糊系统与数学,1987,1(1):51-58.
[56] HAN S C, LI H X, WANG J Y. Generalized one-sided residuated groupoids[J]. Fuzzy Systems Math,2008,22(2):1-9.
[57] BLOK W J, VANALTEN C J. On the finite embeddability property for residuated ordered groupoids[J]. Transactions of theAmerican Mathematical Society,2004,357(10):4141-4157.
[58] BIRKHOFF G. Lattice Theory[M]. Providence:Am Math Soc,1967.
[59] GALATOS N, JIPSEN P, KOWALSKI T, et al. Residuated Lattices:an Algebraic Glimpse at Substructural Logics[M]. Amsterdam:Elsevier,2007.
[60] ZHANG X H. BCC-algebras and residuated partially-ordered groupoid[J]. Mathematica Slovaca,2013,63(3):397-410.
[61] ISEKIK. An algebra related with a propositional calculus[J]. Proc Jpn Acad,1966,42:26-29.
[62] BUNDER M. BCK and related algebras and their corresponding logics[J]. J Non-classical Logic,1983,2:15-24.
[63] KOMORI Y. The class of BCC-algebras is not a variety[J]. Mathematicae Japonicae,1984,29:391-394.
[64] BUNDER M, MEYER R K. A result for combinators, BCK logic and BCK algebras[J]. Logique et Analyse,1985,28:33-40.
[65] MENG J, JUN Y B. BCK-algebras[M]. Seoul:Kyung Moon Sa Co,1994.
[66] DUDEK W A, ZHANG X H. On ideals and congruences in BCC-algebras[J]. Czechoslovak Mathematical J,1998,48(123):21-29.
[67] GEORGESCU G, IORGULESCU A. Pseudo-BCK algebras:an extension of BCK algebras, Combinatorics, computability and logic[C]//Proceedings of the Third International Conference on Combinatorics, Computability and Logic. London:Springer-Verlag,2001:97-114.
[68] ZHANG X H, LI W H. On filters of pseudo-BL algebras and BCC-algebras[J]. Soft Computing,2006,10(10):941-952.
[69] IORGULESCU A. Classes of pseudo-BCK algebras-part I[J]. J Multiple-Valued Logic and Soft Computing,2006,12(1/2):71-130.
[70] ZHANG X H, DUDEK W A. BIK+-logic and non-commutative fuzzy logics[J]. Fuzzy Systems Math,2009,23(1):8-20.
[71] IORGULESCU A. Implicative-groups vs. Groups and Generalizations[M]. Bucuresti:Matrix Room,2018.
[72] DUDEK W A, JUN Y B. Pseudo-BCI algebras[J]. East Asian Mathematical J,2008,24(2):187-190.
[73] ZHANG X H, YE R F. BZ-algebra and group[J]. J Mathematical and Physical Sciences,1995,29:223-233.
[74] DUDEK W A, ZHANG X H, WANG Y Q. Ideals and atoms of BZ-algebras[J]. Mathematica Slovaca,2009,59(4):387-404.
[75] DUDEK W A, THOMYS J. On some generalizations of BCC-algebras[J]. International J Computer Mathematics,2012,89(12):1596-1616.
[76] ZHANG X H, JUN Y B. Anti-grouped pseudo-BCIalgebras and anti-grouped filters[J]. Fuzzy Systems Math,2014,28(2):21-33.
[77] RUMP W, YANG Y. Non-commutative logical algebras and algebraic quantales[J]. Ann Pure Appl Log,2014,165(2):759-785.
[78] RUMP W. Quantum B-algebras[J]. Cent Eur J Math,2013,11(11):1881-1899.
[79] BOTUR M, PASEKA J. Filters on some classes of quantum B-algebras[J]. International J Theoretical Physics,2015,54(12):4397-4409.
[80] HAN S W, XU X T, QIN F. The unitality of quantum B-algebras[J]. International J Theoretical Physics,2018,57(5):1582-1590.
[81] RUMP W. Quantum B-algebras:their omnipresence in algebraic logic and beyond[J]. Soft Computing,2017,21:2521-2529.
[82] HAN S W, WANG R R, XU X T. On the injective hulls of quantum B-algebras[J/OL]. Fuzzy Sets and Systems,2018:[2018-05-11].https://doi.org/10.1016/j.fss.
[83] FODOR J. On fuzzy implication operators[J]. Fuzzy Sets and Systems,1991,42(3):293-300.
[84] BACZYNSKI M, JAYARAM B. Fuzzy Implications,Studies in Fuzziness and Soft Computing[M]. Berlin:Springer-Verlag,2008.
[85] YAO O Y. On fuzzy implications determined by aggregation operators[J]. Information Sciences,2012,193:153-162.
[86] LI D C, QIN S J. The quintuple implication principle of fuzzy reasoning based on interval-valued S-implication[J]. J Logic Algebraic Meth Program,2018,100:185-194.
[87] SU Y, LIU H W, PEDRYCZ W. A method to construct fuzzy implications-rotation construction[J]. Int J Appr Reason,2018,92:20-31.
[88] LUO M X, ZHOU K Y. Logical foundation of the quintuple implication inference methods[J]. Int J Appr Reason,2018,101:1-9.
[89] ZHANG X H, BORZOOEI R A, JUN Y B. Q-filters of quantum B-algebras and basic implication algebras[J/OL]. Symmetry,2018,10(11):573[2018-10-10]. https://doi.org/10.3390/sym10110573.
[90] GUIDO C, TOTO P. Extended-order algebras[J]. J Applied Logic,2008,6(4):609-626.
[91] DELLA STELLA M E, GUIDO C. Extended-order algebras and fuzzy implicators[J]. Soft Computing,2012,16:1883-1892.
[92] MCCUNE W, SANDS A D. Computer and human reasoning:single implicative axioms for groups and for Abelian groups[J]. The American Mathematical Monthly,1996,103(10):888-892.
[93] MENG J. Implicative commutative semigroups are equivalent to a class of BCK algebras[J]. Semigroup Forum,1995,50(1):89-96.

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备注/Memo

备注/Memo:
收稿日期:2018-11-01 接受日期:2018-11-10
基金项目:国家自然科学基金(61573240、61473239)和陕西省教育厅2018年度专项科学研究计划(18JK0099)
作者简介:张小红(1965—),男,教授,博士生导师,主要从事代数学、模糊逻辑与粗糙集理论、数据智能、不确定性数学及其在管理决策中的应用研究,E-mail:zhangxiaohong@sust.edu.cn
更新日期/Last Update: 2018-12-15