MV-代数




在纯数学分支抽象代数中，MV-代数（多值代数）是带有二元运算 ⊕{displaystyle oplus }oplus 、一元运算 ¬{displaystyle neg }neg 和常量 0{displaystyle 0}{displaystyle 0} 的满足特定公理的代数结构。多值逻辑是 MV-代数的模型。




目录






  • 1 定义


  • 2 例子


  • 3 讨论


  • 4 引用


  • 5 外部链接


  • 6 参见





定义


设 A 是个集合。MV-代数是代数结构，带有型  ⟨2,1,0⟩{displaystyle langle 2,1,0rangle }{displaystyle langle 2,1,0rangle } 的标识(signature) ⟨A,⊕,¬,0⟩,{displaystyle leftlangle A,oplus ,lnot ,0rightrangle ,}{displaystyle leftlangle A,oplus ,lnot ,0rightrangle ,}，它满足如下恒等式：




  • (x⊕y)⊕z=x⊕(y⊕z){displaystyle (xoplus y)oplus z=xoplus (yoplus z)}{displaystyle (xoplus y)oplus z=xoplus (yoplus z)},


  • x⊕0=x{displaystyle xoplus 0=x}{displaystyle xoplus 0=x},


  • x⊕y=y⊕x{displaystyle xoplus y=yoplus x}{displaystyle xoplus y=yoplus x},


  • ¬¬x=x{displaystyle lnot lnot x=x}{displaystyle lnot lnot x=x},


  • x⊕¬0=¬0{displaystyle xoplus lnot 0=lnot 0}{displaystyle xoplus lnot 0=lnot 0},


  •  ¬(¬x⊕y)⊕y=¬(¬y⊕x)⊕x{displaystyle lnot (lnot xoplus y)oplus y=lnot (lnot yoplus x)oplus x}{displaystyle lnot (lnot xoplus y)oplus y=lnot (lnot yoplus x)oplus x}.


备注：通过前三个公理 ⟨A,⊕,0⟩{displaystyle leftlangle A,oplus ,0rightrangle }{displaystyle leftlangle A,oplus ,0rightrangle } 是交换幺半群。


或者作为替代，MV-代数是一个剩余格 A=⟨L,∧,∨,⊗,→,0,1⟩{displaystyle A=leftlangle L,wedge ,vee ,otimes ,rightarrow ,0,1rightrangle }{displaystyle A=leftlangle L,wedge ,vee ,otimes ,rightarrow ,0,1rightrangle } 满足额外恒等式



∀ x,y∈A:x∨y=(x→y)→y{displaystyle forall x,yin A:xvee y=(xrightarrow y)rightarrow y}{displaystyle forall x,yin A:xvee y=(xrightarrow y)rightarrow y}。

Hájek (1998)描述了这两个公式的等同。



例子


一个简单的例子是 A=[0,1]{displaystyle A=[0,1],}{displaystyle A=[0,1],}，带有定义为 x⊕y=min(x+y,1){displaystyle xoplus y=min(x+y,1)}{displaystyle xoplus y=min(x+y,1)} 和 ¬x=1−x{displaystyle lnot x=1-x}{displaystyle lnot x=1-x} 的运算。



讨论


在多值逻辑中，给定一个 MV-代数 A，一个 A-賦值就是从命题演算中公式的集合到 MV-代数的函数。如果对于所有 A-賦值这个函数把一个公式映射到 1（或 ¬{displaystyle lnot }lnot 0），则这个公式是一个 A-重言式。因此对于无穷值逻辑（比如模糊逻辑、武卡谢维奇逻辑），我们设 [0,1] 是 A 的下层集合来获得 [0,1]-賦值和 [0,1]-重言式（经常就叫做賦值和重言式）。


Chang 发明 MV-代数来研究波蘭數學家扬·武卡谢维奇（Jan Łukasiewicz）在 1920 年介入的多值逻辑。Chang 的完备定理(1958, 1959) 声称任何在 [0,1] 区间成立的 MV-代数等式也在所有 MV-代数中成立。通过这个定理，证明了无穷值的武卡谢维奇逻辑可以被 MV-代数所刻画。后来同样适用于模糊逻辑。这类似于在 {0,1} 成立的布尔代数等式在任何布尔代数中也成立，布尔代数因此刻画了标准二值逻辑。



引用



  • Chang, C. C. (1958) "Algebraic analysis of many-valued logics," Transactions of the American Mathematical Society 88: 476-90.

  • ------ (1959) "A new proof of the completeness of the Lukasiewicz axioms," Transactions of the American Mathematical Society 88: 74-80.

  • Cignoli, R. L. O., D'Ottaviano, I, M. L., Mundici, D., 2000. Algebraic Foundations of Many-valued Reasoning. Kluwer.

  • Di Nola A., Lettieri A. (1993) "Equational characterization of all varieties of MV-algebras," Journal of Algebra 221: 123-31.

  • Hájek, Petr (1998) Metamathematics of Fuzzy Logic. Kluwer.



外部链接



  • Stanford Encyclopedia of Philosophy："Many-valued logic" -- by Siegfried Gottwald.


参见



  • 布尔代数

  • 武卡谢维奇逻辑




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