MTL-代数的模糊理想
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摘要
将模糊集与理想概念相结合,在MTL-代数中引入模糊理想、模糊素理想、模糊Boolean理想、模糊关联理想、模糊超理想、模糊固执理想和模糊Gdel理想的概念,并考察它们的特征、性质和相互关系.证明了对MTL-代数的非常值模糊理想f,如下各条件等价:(i)f是模糊Boolean理想和模糊素理想;(ii)f是模糊关联理想和模糊素理想;(iii)f是模糊超理想;(iv)f是模糊固执理想.
Combining fuzzy sets with the ideals,the concepts of fuzzy ideals,fuzzy prime,Boolean,implicative,ultra,obstinate and Gdel ideals in MTL-algebras are introduced and their properties and relations are investigated.It is proved that the following conditions are equivalent to a non-constant fuzzy ideal fin a MTL-algebra:(i)fis fuzzy Boolean and fuzzy prime ideal;(ii)fis fuzzy implicative and fuzzy prime ideal;(iii)fis fuzzy ultra ideal;(iv)fis fuzzy obstinate ideal.
Combining fuzzy sets with the ideals,the concepts of fuzzy ideals,fuzzy prime,Boolean,implicative,ultra,obstinate and Gdel ideals in MTL-algebras are introduced and their properties and relations are investigated.It is proved that the following conditions are equivalent to a non-constant fuzzy ideal fin a MTL-algebra:(i)fis fuzzy Boolean and fuzzy prime ideal;(ii)fis fuzzy implicative and fuzzy prime ideal;(iii)fis fuzzy ultra ideal;(iv)fis fuzzy obstinate ideal.
引文
[1]WANG Guojun,ZHOU Hongjun.Introduction to Mathematical Logic and Resolution Principle[M].Beijing:Science Press,2009.
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[5]HJEK P.Observations on the monoidal t-norm based logic[J].Fuzzy Sets and Systems,2002,132(1):107-112.
[6]PEI Daowu.On equivalent forms of fuzzy logic systems NM and IMTL[J].Fuzzy Sets and Systems,2003,138(1):187-195.
[7]裴道武.MTL代数的特征定理[J].数学学报,2007,50(6):1201-1206.PEI Daowu.The characterizations of MTL-algebras[J].Acta Mathematica Sinica,2007,50(6):1201-1206.
[8]BORZOOEI R A,SHOAR S K,AMERI R.Some types of filters in MTL-algebras[J].Fuzzy Sets and Systems,2012,187:92-102.
[9]ZADEH L A.Fuzzy sets[J].Information&Control,1965(8):338-353.
[10]LIU Lianzhen,LI Kaitai.Fuzzy filters of BL-algebras[J].Information Sciences,2005,173:141-154.
[11]LIU Lianzhen,LI Kaitai.Fuzzy Boolean and positive implicative filters of BL-algebras[J].Fuzzy Sets and Systems,2005,152:333-348.
[12]刘春辉.BL代数的((∈,∈∨q)-,∈∨q)-模糊素滤子[J].浙江大学学报:理学版,2014,41(5):489-493.LIU Chunhui.(∈,∈∨q)-fuzzy prime filters in BLalgebras[J].Journal of Zhejiang University:Science Edition,2014,41(5):489-493.
[13]SAEID A B,ZAHIRI S.Radicals in MTL-algebras[J].Fuzzy Sets and Systems,2014,236:91-103.
[14]JUN Youngbat,XU Yang,ZHANG Xiaohong.Fuzzy filters of MTL-algebras[J].Information Sciences,2005,175:120-138.
[15]ZHANG Xiaohong,WANG Yongquan,LIU Yonglin.Fuzzy ultra filters and fuzzy G-filters of MTL-algebras[J].Fuzzy Systems and Knowledge Discovery,2005,3613:160-166.
[16]ZHANG Xiaohong,QIN Keyun,WIESLAW A.Ultra LI-ideals in lattice implication algebras and MTLalgebras[J].Czechoslovak Mathematical Journal,2007,57(132):591-605.
[2]HJEK P.Metamathematics of Fuzzy Logic[M].Dordrecht:Kluwer Academic Publishers,1998.
[3]HJEK P.Basic fuzzy logic and BL-algebras[J].Soft Computing,1998,2(3):124-128.
[4]ESTEVA F,GODO L.Monoidal t-norm based logic:Towards a logic for left-continuous t-MYMnorms[J].Fuzzy Sets and Systems,2001,124(3):271-288.
[5]HJEK P.Observations on the monoidal t-norm based logic[J].Fuzzy Sets and Systems,2002,132(1):107-112.
[6]PEI Daowu.On equivalent forms of fuzzy logic systems NM and IMTL[J].Fuzzy Sets and Systems,2003,138(1):187-195.
[7]裴道武.MTL代数的特征定理[J].数学学报,2007,50(6):1201-1206.PEI Daowu.The characterizations of MTL-algebras[J].Acta Mathematica Sinica,2007,50(6):1201-1206.
[8]BORZOOEI R A,SHOAR S K,AMERI R.Some types of filters in MTL-algebras[J].Fuzzy Sets and Systems,2012,187:92-102.
[9]ZADEH L A.Fuzzy sets[J].Information&Control,1965(8):338-353.
[10]LIU Lianzhen,LI Kaitai.Fuzzy filters of BL-algebras[J].Information Sciences,2005,173:141-154.
[11]LIU Lianzhen,LI Kaitai.Fuzzy Boolean and positive implicative filters of BL-algebras[J].Fuzzy Sets and Systems,2005,152:333-348.
[12]刘春辉.BL代数的((∈,∈∨q)-,∈∨q)-模糊素滤子[J].浙江大学学报:理学版,2014,41(5):489-493.LIU Chunhui.(∈,∈∨q)-fuzzy prime filters in BLalgebras[J].Journal of Zhejiang University:Science Edition,2014,41(5):489-493.
[13]SAEID A B,ZAHIRI S.Radicals in MTL-algebras[J].Fuzzy Sets and Systems,2014,236:91-103.
[14]JUN Youngbat,XU Yang,ZHANG Xiaohong.Fuzzy filters of MTL-algebras[J].Information Sciences,2005,175:120-138.
[15]ZHANG Xiaohong,WANG Yongquan,LIU Yonglin.Fuzzy ultra filters and fuzzy G-filters of MTL-algebras[J].Fuzzy Systems and Knowledge Discovery,2005,3613:160-166.
[16]ZHANG Xiaohong,QIN Keyun,WIESLAW A.Ultra LI-ideals in lattice implication algebras and MTLalgebras[J].Czechoslovak Mathematical Journal,2007,57(132):591-605.