JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (02): 75-82.doi: 10.6040/j.issn.1671-9352.0.2014.342

Previous Articles     Next Articles

Wajsberg's form of MTL algebras with applications

KOU Hai-yan, WU Hong-bo   

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, Shaanxi, China
  • Received:2014-07-25 Revised:2014-11-20 Online:2015-02-20 Published:2015-01-27

PDF (PC)

1020

Abstract: MTL algebra is an important basic logic algebra. Firstly, the classical algebras forms of NMTL algebra is given by taking Wajsberg's method and some parts of axioms of MTL logic system, and it is proved that NMTL algebra and MTL algebra have identical structure. Secondly, it is proved that an NMTL algebras L satisfying the condition: ∀x,y∈L, x→y=(y→0)→(x→0) is BR0 algebra. Finally, it is proved that IMTL algebra and BR0 algebra have identical structure and the Wajsberg forms of BR0 algebra and BL algebra are given.

Key words: fuzzy logic, residuated lattice, NMTL algebra, BR0 algebra, BL algebra, MTL algebra

CLC Number: 

  • O141.1
[1] PAVELKA J. On fuzzy logic Ⅰ: Many-valued rules of inference, Ⅱ: Enriched residuated lattice and semantics of propositionalcalculi, III: Semantical completeness of some many-valued propositional calculi[J]. Zeitschrf Math Logik und Grundlagender Math, 1979, 25:45-52; 119-134; 447-464.
[2] 徐扬. 格蕴涵代数[J]. 西南交通大学学报, 1993, 89(1):20-27. XU Yang. Lattice implication algebras[J]. Journal of Southwest Jiaotong University, 1993, 89(1):20-27.
[3] XU Yang, RUAN Da, QIN Keyun, et al. Lattice-valued logic[M]. Berlin Heidelberg: Springer-Verlag, 2003.
[4] HAJEK P. Metamathematics of fuzzy logic[M]. Dordrecht: Kluwer Academic Publishers, 1998.
[5] 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,2006. WANG Guojun. Non-classical mathematical logic and approximate reasoning[M]. 2nd.Beijing: Science Press, 2006.
[6] 吴洪博. 基础R0代数与基础L*系统[J]. 数学进展, 2003, 32(5):565-576. WU Hongbo. Baisis R0-algebra and basis L* system[J]. Advances in Mathematics, 2003, 32(5):565-576.
[7] 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.
[8] JUN Y B, XU, MA Y J. Fuzzy filters of MTL-algerbra[J]. Information Sciences, 2005, 175:120-138.
[9] 裴道武. MTL-代数的特征定理[J]. 数学学报:中文版,2007,50(6):1201-1206. PEI Daowu. The characterizations of MTL-algebras[J]. Acta Mathematics Sinica: Chinese Series, 2007, 50(6):1201-1206.
[10] 张小红,魏萍.DR0代数:由De Morgan 代数导出的正则剩余格[J].数学进展,2008,37(4):499-511. ZHANG Xiaohong, WEI Ping. DR0 algebras: a kind of regular residuated lattice via De Morgan algebras[J]. Advances in Mathematics, 2008, 37(4):499-511.
[11] 张小红. 模糊逻辑及其代数分析[M]. 北京: 科学出版社,2008. ZHANG Xiaohong. Fuzzy logic and algebraic analysis[M]. Beijing: Science Press, 2008.
[12] 刘敏,吴洪博.预线性剩余格与逻辑代数[J].工程数学学报,2008,25(2):199-203. LIU Min, WU Hongbo. Prelinearity residuated-lattice and logic algebras[J]. Chinese Journal of Engineering Mathematics, 2008, 25(2):199-203.
[13] 王娜, 吴洪博. MTL-代数的演绎系统和余零化子及其相互关系[J]. 模糊系统与数学,2014, 28(1):9-14. WANG Na, WU Hongbo. The deductive system and co-annihilator of MTL-algebras and the relation between them[J]. Fuzzy Systems and Mathematics, 2014, 28(1):9-14.
[14] Chang C C. Algebriac analysis of many-valued logics[J]. Trans Amer Math Soc, 1958, 88:467-490.
[15] 王国俊.数理逻辑引论与归结原理[M].2版.北京:科学出版社,2006. WANG Guojun. An introduction to mathematical logic and resolution principle[M]. 2nd.Beijing: Science Press, 2006.
[16] 吴望名. Fuzzy 蕴涵代数[J]. 模糊系统与数学, 1990,4(1):56-64. WU Wangming. Fuzzy implication algebras[J]. Fuzzy Systems and Mathematics, 1990, 4(1):56-64.
[17] 刘练珍, 李开泰. FI代数同构于一族全序FI代数的直积的子代数的条件[J]. 纯粹数学与应用数学, 2004,20(1):63-67. LIU Lianzhen, LI Kaitai. The conditions of FI algebra to be a subalgebra of direct of a system of linearly ordered FI algebras[J]. Pure and Applied Mathematics, 2004, 20(1):63-67.
[18] 刘春辉, 徐罗山. 格蕴涵代数的蕴涵表示定理[J]. 模糊系统与数学, 2010,24(4):26-32. LIU Chunhui, XU Luoshan. The representative theorems of lattice implication algebras by implication operator[J]. Fuzzy Systems and Mathematics, 2010, 24(4):26-32.
[19] FONT J, RODRIGUEZ A J, TORRENS A. Wajsberg algebras[J]. Stochastica, 1984, 8:5-31.
[20] 吴洪博,王昭海. BR0-代数的无序表示形式及WBR0-代数的性质[J].工程数学学报,2009, 26(3):456-460. WU Hongbo, WANG Zhaohai. The non-ordered Form of BR0-algeebras and properties of WBR0-algebras[J]. Chinese Journal of Engineering Mathematics, 2009, 26(3):456-460.
[21] 高李红, 吴洪博. QBL-代数及其与BL-代数的等价性[J].吉林大学学报:理学版,2011,49(1):41-46. GAO Lihong, WU Hongbo. Quasi-BL-algebras and their equivalence with BL-algebras[J]. Journal of Jili University: Science Edition, 2011, 49(1):41-46.
[22] 朱翔, 徐罗山. BL代数的等价刻画及更多性质[J]. 模糊系统与数学, 2011,25(1):13-18. ZHU Xiang, XU Luoshan. On characterizations and further properties of BL-algebras[J]. Fuzzy Systems and Mathematics, 2011, 25(1):13-18.
[1] ZHANG Yaning, YAO Wei. Relatively maximal ideals of co-residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2025, 60(2): 9-13.
[2] Chunhui LIU. Prime hesitant fuzzy filters in fuzzy implication algebras and its spectral spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2023, 58(12): 108-117.
[3] LIU Chun-hui. Hesitant fuzzy ideals in non-involutive residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2021, 56(2): 7-16.
[4] LIU Chun-hui, ZHANG Hai-yan, LI Yu-mao. Lattice of bipolar fuzzy ideals in negative non-involutive residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(9): 29-35.
[5] LIU Chun-hui, LI Yu-mao, ZHANG Hai-yan. Bipolar fuzzy ideals in negative non-involutive residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(5): 88-98.
[6] PENG Jia-yin. Pushdown automata and content-free grammars based on complete residuated lattice-valued logic [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(5): 112-126.
[7] LIU Chun-hui. Expand fuzzy filters in Heyting algebras [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(2): 57-65.
[8] LIU Chun-hui. Bipolar fuzzy prime ideals in negative non-involutive residuated lattices [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(11): 71-80.
[9] . Representative forms of commutative BR0-algebras on a set by implication operator [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 86-94.
[10] DUAN Jing-yao. Topological properties analysis of logical metric spaces on the residuated lattice [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 9-16.
[11] LIU Li-jun. Characterizations of n-fold positive implicative filter in residuated lattice [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(8): 48-52.
[12] LIANG Ying, CUI Yan-li, WU Hong-bo. Properties of residuated lattice of deduction systems set algebra in BL system [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(11): 65-70.
[13] LIU Chun-hui. Lattice of(,∨(-overq))-fuzzy filters in a BL-algebra [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(10): 104-110.
[14] QIAO Xi-min, WU Hong-bo. 〈,(-overQ)〉-fuzzy filter and its characterization of the non-commutative residual lattices on the interval sets [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2016, 51(2): 102-107.
[15] ZHOU Jian-ren, WU Hong-bo. A schematic extension of IMTL logic system [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2015, 50(12): 28-34.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd