ZHOU Jianren and WU Hongbo, “Schematic Extensions of MTL by Adding Weak Divisibility Axiom,” Chinese Journal of Electronics, vol. 25, no. 5, pp. 824-831, 2016, doi: 10.1049/cje.2016.06.032
 Citation: ZHOU Jianren and WU Hongbo, “Schematic Extensions of MTL by Adding Weak Divisibility Axiom,” Chinese Journal of Electronics, vol. 25, no. 5, pp. 824-831, 2016,

# Schematic Extensions of MTL by Adding Weak Divisibility Axiom

##### doi: 10.1049/cje.2016.06.032
Funds:  This work is supported by National Natural Science Foundation of China (No.61572016, No.11531009), and the Fundamental Research Funds for the Central Universities (No.GK201501001).
• Rev Recd Date: 2014-09-10
• Publish Date: 2016-09-10
• MTL is a Monoidal t-norm based logic introduced by Esteva and Godo by omitting divisibility axiom from Hájek's Basic logic (BL). Many logics can be obtained by adding axioms to MTL logic. Logic system WBL is obtained by adding weak divisibility axiom to logic system MTL. Logic system WMV is obtained by adding involution axiom to logic system WBL. WBL-algebra corresponding to logic system WBL and WMV-algebra to logic systemWMV are defined respectively. It is proved that the both of logic system Luk and logic system Nilpotent minimum (NM) are the schematic extensions of logic system WMV. Weak Wajsberg algebra and the simplified form of logic system WMV are obtained.
•  P. Hájek, Metamathematics of Fuzzy Logic, Kluwer Academic Publishers, Boston, 1998. F. Esteva and L. Godo, "Monoidal t-norm based logic:Towards a logic for left-continuous t-norms", Fuzzy Sets and Systems, Vol.124, No.3, pp.271-288, 2001. F. Esteva, L. Godo and C. Noguera, "On expansions of WNM t-norm based logics with truth-constants", Fuzzy Sets and Systems, Vol.161, No.3, pp.347-368, 2010. S. Jenei and F. Montagna, "A proof of standard completeness for F. Esteva and L. Godo's logic MTL", Studia Logica, Vol.70, No.2, pp.183-192, 2002. E. Turunen, Mathematics Behind Fuzzy Logic, Physica-Verlag, Heidelberg, 1999. Y. Xu, D. Ruan, K.Y. Qin and J. Liu, Lattice-Valued Logic, Springer-Verlag, Heidelberg, 2003. Guo-jun Wang, Non-Classical Mathematical Logics and Approximate Reasoning, Science press, Beijing, 2008. S.M. Wang, B.S. Wang and X.Y. Wang, "A characterization of truth functions in the Nilpotent minimum logic", Fuzzy Sets and Systems, Vol.145, No.2, pp.253-266, 2004. San-min Wang, Bao-shu Wang and Guo-jun Wang, "A triangular norm-based propositional fuzzy logic", Fuzzy Sets and Systems, Vol.136, No.1, pp.55-70, 2003. S.M. Wang, B.S. Wang and F. Ren, "NM L, a schematic extension of F. Esteva and L. Godo's logic MTL", Fuzzy Sets and Systems, Vol.149, No.2, pp.285-295, 2005. San-min Wang and Ming-yan Wang, "Undefinability of minconjunction in MTL", Fuzzy Sets and Systems, Vol.157, No.5, pp.670-676, 2006. San-min Wang and Ming-yan Wang, "Disjunctive elimination rule and its application in MTL", Fuzzy Sets and Systems, Vol.157, No.24, pp.3169-3176, 2006. Zhang Xiaohong, Fuzzy Logics With Its Algebra Analysis, Science press, Beijing, 2008. Zhang Xiaohong, "Non-commutative fuzzy logic system PL* and its completeness", Acta Mathematica Sinica (Chinese Series), Vol.50, No.2, pp.421-442, 2007. (in Chinese) Zhang Xiaohong, "DR0 algebras:A kind of regular residuated lattice via De Morgan algebras", Advanced Mathematics Sinica (Chinese Series), Vol.37, No.4, pp.499-511, 2008. (in Chinese) Pei Daowu, "The characterizations of MTL algebras", Acta Mathematica Sinica (Chinese Series), Vol.50, No.6, pp.1201-1206, 2007. (in Chinese) Pei Daowu and Wang Guojun, "Completeness and applications of the formal system L*", Science in China (series F), Vol.45, No.1, pp.40-50, 2002. Pei Daowu, "On equivalent forms of fuzzy systems NM and IMTL", Fuzzy Sets and Systems, Vol.138, No.1, pp.187-195, 2003. H.B. Wu, "The basic R0 algebra and the basic L* system", Advanced in Mathematics, Vol.32, No.5, pp.565-576, 2003. (in Chinese) J. Font, A.J. Rodriguez and A. Torrens, "Wajsberg algebras", Stochastica, Vol.8, No.1, pp.5-31, 1984.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142