西南大学学报 (自然科学版)  2019, Vol. 41 Issue (8): 58-63.  DOI: 10.13718/j.cnki.xdzk.2019.08.010
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  • 非交换剩余格上模糊极滤子的特征与性质    [PDF全文]
    刘莉君, 王树勋, 刘丽华     
    陕西理工大学 数学与计算机科学学院, 陕西 汉中 723000
    摘要:运用模糊集的运算方法和原理,在非交换剩余格上引入了模糊极滤子的概念,并研究了其表示定理和特征性质,获得了在一定条件下非交换剩余格上模糊极滤子与模糊子正蕴涵滤子相互等价的结论.研究结果进一步拓展了非交换剩余格上的模糊滤子理论,为其在逻辑代数及计算机信息处理等方面的应用奠定了理论基础.
    关键词模糊逻辑    非交换剩余格    模糊滤子    模糊极滤子    

    自从布尔代数作为经典二值逻辑所对应的代数系统被提出以来,各种不同逻辑系统所对应的代数系统受到研究人员的广泛关注,并取得了大量的研究成果.在信息科学、计算机科学、控制理论、人工智能等很多重要的领域中,逻辑代数是其推理机制的代数基础.为给不确定信息处理理论提供可靠且合理的逻辑基础,许多学者提出并研究了非经典逻辑系统.目前,大多数学者都认同非交换剩余格为一种最广泛的非可换逻辑代数结构,其中伪BL代数、伪MV代数、伪MTL代数等[1-3]均是非交换剩余格的特殊情况.而滤子是非经典逻辑代数研究领域的一个重要概念,它对各种逻辑系统及与之匹配的逻辑代数的完备性问题的研究发挥着极其重要的作用.近几年,学者们已经在各种逻辑代数框架下提出了多种滤子概念,并获得了许多有价值的研究结果.文献[4]通过讨论模糊正规滤子和模糊布尔滤子之间的关系,解决了伪BL代数上的公开问题.文献[5]研究了剩余格上几类模糊滤子的性质,使剩余格上滤子的结构研究更为清楚.文献[6]在非交换剩余格上引入了子模糊弱布尔滤子的概念,并研究了其特征刻画.因此,在众多滤子理论研究的基础上系统地分析出各种滤子概念之间的相互关系及层次结构就显得尤为重要[7-10].基于此目的,本文运用模糊集的运算方法和原理,在非交换剩余格上引入了模糊极滤子的概念,并研究了其表示定理和特征性质,获得了在一定条件下非交换剩余格上模糊极滤子与模糊子正蕴涵滤子相互等价的结论.研究结果不但使非交换剩余格上的模糊滤子理论得到进一步充实和丰富,还使得概念间的层次关系更加的清晰和完善,而且为研究基于非交换剩余格的逻辑系统的结构特征提供了理论基础.

    1 预备知识

    下面先给出本文将用到的几个定义.

    定义1[5]  称(2,2,2,2,2,0,0)-型代数L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)为非交换剩余格,若以下条件成立:

    (a) (M,∧,∨,0,1)是有界格;

    (b) (M,⊗,1)是以1为单位元的半群;

    (c) 对任意的xyzMxyzxyzyx$\circlearrowleft$z.

    性质1[5]  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是一个非交换剩余格,对于任意的xyzM,下列性质成立:

    (1°) xyxy=1⇔x$\circlearrowleft$y=1;

    (2°) xx=x$\circlearrowleft$x=1,1→x=1$\circlearrowleft$x=xx→1=x$\circlearrowleft$1=1;

    (3°) x≤(xy)$\circlearrowleft$yx≤(x$\circlearrowleft$y)→y

    (4°) x→(y$\circlearrowleft$z)=y$\circlearrowleft$(xz);

    (5°)若xy,则xzyzzxzy

    (6°)若xy,则zxzyz$\circlearrowleft$xz$\circlearrowleft$y

    (7°)若xy,则yzxzy$\circlearrowleft$zx$\circlearrowleft$z

    (8°) xy≤(zx)→(zy),x$\circlearrowleft$y≤(z$\circlearrowleft$x)$\circlearrowleft$(z$\circlearrowleft$y);

    (9°) xy≤(yz)$\circlearrowleft$(xz),x$\circlearrowleft$y≤(y$\circlearrowleft$z)→(x$\circlearrowleft$z);

    (10°) (xy)$\circlearrowleft$z=y$\circlearrowleft$(x$\circlearrowleft$z),(xy)→z=x→(yz);

    (11°) xy≤((xy)$\circlearrowleft$y)∧((yx)$\circlearrowleft$x),xy≤((x$\circlearrowleft$y)→y)∧((y$\circlearrowleft$x)→x).

    注1  在本文中,我们将用L代表一个非交换剩余格,并约定运算∨,∧,⊗优先于运算→,$\circlearrowleft$.

    定义2[7]  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)为非交换剩余格,μL[0, 1]为L上的模糊集,如果对于任意的xyL,有

    (a) μ(1)≥μ(x);

    (b) μ(x)∧μ(xy)≤μ(y)(或μ(x)∧μ(x$\circlearrowleft$y)≤μ(y)).

    μL上的模糊滤子.

    性质2[7]  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)为非交换剩余格,μL上的模糊滤子,对于任意的xyzL,下列性质成立:

    (1°)如果xy,则μ(x)≤μ(y),即μ是保序的;

    (2°)如果x→(yz)=1或x$\circlearrowleft$(y$\circlearrowleft$z)=1,则μ(z)≥μ(x)∧μ(y);

    (3°)如果μ(xy)=μ(1)或μ(x$\circlearrowleft$y)=μ(1),则μ(x)≤μ(y);

    (4°) μ(yx)=μ(xy)=μ(x)∧μ(y),μ(0)=μ(x)∧μ(x→0);

    (5°) μ(xz)≥μ(yz)∧μ(xy),μ(x$\circlearrowleft$z)≥μ(y$\circlearrowleft$z)∧μ(x$\circlearrowleft$y).

    定义3[8]  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,FL上的非空子集,如果对于任意的xyzL,满足下列条件(a),(b)或条件(a),(c):

    (a) 1∈F

    (b) z→(y$\circlearrowleft$x)∈FzF蕴涵((x$\circlearrowleft$y)→y)$\circlearrowleft$xF

    (c) z$\circlearrowleft$(yx)∈FzF蕴涵((xy)$\circlearrowleft$y)→xF.

    则称集合FL上的极滤子.

    定义4[6]  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,FL上的非空子集,对于任意的xyzL,如果满足下列条件(a),(b)或条件(a),(c):

    (a) 1∈F

    (b) ((xy)⊗z)$\circlearrowleft$((y$\circlearrowleft$x)→x),zF蕴涵((xy)$\circlearrowleft$y)∈F

    (c) (z⊗(x$\circlearrowleft$y))→((yx)$\circlearrowleft$x),zF蕴涵((x$\circlearrowleft$y)→y)∈F.

    则称集合FL上的子正蕴涵滤子.

    利用非交换剩余格上滤子与模糊滤子的关系,我们给出非模糊子正蕴涵滤子和模糊极滤子的概念.

    定义5  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,μL上的模糊集,对于任意的xyzL,如果满足下列条件(a),(b)或条件(a),(c):

    (a) μ(1)≥μ(x);

    (b) μ((xy)$\circlearrowleft$y)≥μ(((xy)⊗z)$\circlearrowleft$((y$\circlearrowleft$x)→x))∧μ(z);

    (c) μ((x$\circlearrowleft$y)→y)≥μ((z⊗(x$\circlearrowleft$y))→((yx)$\circlearrowleft$x))∧μ(z).

    则称μL上的模糊子正蕴涵滤子.

    定义6  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,μL上的模糊集,对于任意的xyzL,如果满足下列条件(a),(b)或条件(a),(c):

    (a) μ(1)≥μ(x);

    (b) μ(((x$\circlearrowleft$y)→y)$\circlearrowleft$x)≥μ(z→(y$\circlearrowleft$x))∧μ(z);

    (c) μ(((xy)$\circlearrowleft$y)→x)≥μ(z$\circlearrowleft$(yx))∧μ(z).

    则称μL上的模糊极滤子.

    2 主要结论

    引理1  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,对于任意的xyzL,有

    $ \left( {\left( {x \circlearrowleft z} \right) \to z} \right) \circlearrowleft ((y \circlearrowleft z) \to z) \geqslant x \circlearrowleft y $

      对于任意的xyzL,由性质1中(3°)和(4°)可知

    $ \begin{gathered} \left( {\left( {x \circlearrowleft z} \right) \to z} \right) \circlearrowleft ((y \circlearrowleft z) \to z) = (y \circlearrowleft z) \to \left( {\left( {\left( {x \circlearrowleft z} \right) \to z} \right) \circlearrowleft z} \right) \geqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(y \circlearrowleft z) \to \left( {x \circlearrowleft z} \right) \geqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \circlearrowleft y \hfill \\ \end{gathered} $

    引理2  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,模糊集μL上的模糊滤子,对于任意的xyL,下列条件等价:

    (ⅰ)模糊滤子μL上的模糊极滤子;

    (ⅱ) μ(y$\circlearrowleft$x)≤μ(((x$\circlearrowleft$y)→y)$\circlearrowleft$x);

    (ⅲ) μ(yx)≤μ(((xy)$\circlearrowleft$y)→x).

      (ⅰ)⇒(ⅱ)因为模糊滤子μ是非交换剩余格L上的模糊极滤子,则

    $ \mu \left( {\left( {\left( {x \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right) \geqslant \mu \left( {z \to \left( {y \circlearrowleft x} \right)} \right) \wedge \mu \left( z \right) $

    z=1,即可得

    $ \mu \left( {y \circlearrowleft x} \right) \leqslant \mu \left( {\left( {\left( {x \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right) $

    (ⅱ)⇒(ⅰ)因为模糊集μL上的模糊滤子,故由定义2可得μ(1)≥μ(x).又因为

    $ \begin{gathered} \mu \left( {\left( {\left( {x \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right) \geqslant \mu \left( {y \circlearrowleft x} \right) = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {1 \to \left( {y \circlearrowleft x} \right)} \right) \wedge \mu \left( 1 \right) \hfill \\ \end{gathered} $

    从而由定义6可得:模糊滤子μ是非交换剩余格L上的模糊极滤子.

    同理可证(ⅰ)与(ⅲ)相互等价,综上可知结论成立.

    引理3  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,模糊集μL上的模糊滤子,对于任意的xyzL,下列条件等价:

    (ⅰ)模糊滤子μL上的模糊子正蕴涵滤子;

    (ⅱ)μ(y)≥μ((yz)$\circlearrowleft$(xy))∧μ(x);

    (ⅲ)μ(y)≥μ((y$\circlearrowleft$z)→(x$\circlearrowleft$y))∧μ(x).

      (ⅰ)⇒(ⅱ)对于任意的yzL,因为z→(yz)=1,因此由性质1可知zyz,从而(yz)$\circlearrowleft$yz$\circlearrowleft$y.再由性质2可得

    $ \mu \left( {\left( {y \to z} \right) \circlearrowleft y} \right) \leqslant \mu \left( {z \circlearrowleft y} \right) $

    又因为

    $ \begin{gathered} \mu \left( {\left( {y \to z} \right) \circlearrowleft y} \right) \leqslant \mu \left( {\left( {y \to z} \right) \circlearrowleft \left( {\left( {z \circlearrowleft y} \right) \to y} \right)} \right) = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {\left( {y \to z} \right) \circlearrowleft z} \right) \leqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {\left( {z \circlearrowleft y} \right) \to \left( {\left( {y \to z} \right) \circlearrowleft z} \right)} \right) = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {\left( {z \circlearrowleft y} \right) \to y} \right) \hfill \\ \end{gathered} $

    因此

    $ \mu \left( {\left( {y \to z} \right) \circlearrowleft y} \right) \leqslant \mu \left( {z \circlearrowleft y} \right) \wedge \mu \left( {\left( {z \circlearrowleft y} \right) \to y} \right) \leqslant \mu \left( y \right) $

    而又因为

    $ \mu \left( {\left( {y \to z} \right) \circlearrowleft \left( {x \to y} \right)} \right) \wedge \mu \left( x \right) = \mu \left( {x \to \left( {\left( {y \to z} \right) \circlearrowleft y} \right)} \right) \wedge \mu \left( x \right) \leqslant \mu \left( {\left( {y \to z} \right) \circlearrowleft y} \right) $

    $ \mu \left( {\left( {y \to z} \right) \circlearrowleft \left( {x \to y} \right)} \right) \wedge \mu \left( x \right) \leqslant \mu \left( y \right) $

    (ⅱ)⇒(ⅰ)对于任意的xyL,因为

    $ x \leqslant \left( {x \to y} \right) \circlearrowleft y $

    故有

    $ \left( {\left( {x \to y} \right) \circlearrowleft y} \right) \circlearrowleft x \leqslant x \circlearrowleft x $

    $ \left( {y \circlearrowleft x} \right) \to \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \circlearrowleft x} \right) \leqslant \left( {y \circlearrowleft x} \right) \to \left( {x \circlearrowleft x} \right) = 1 $

    又因为

    $ \left( {y \circlearrowleft x} \right) \to \left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft x} \right) = \left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right) = 1 $

    $ \left( {y \circlearrowleft x} \right) \to \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \circlearrowleft x} \right) \leqslant \left( {y \circlearrowleft x} \right) \to \left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft x} \right) $

    从而可得

    $ ((x \to y) \circlearrowleft y)\circlearrowleft x \leqslant ((y\circlearrowleft x )\to x) \circlearrowleft x $

    $ \left( {y \circlearrowleft x} \right) \to x \leqslant \left( {x \to y} \right) \circlearrowleft y $

    又因为

    $ \left( {x \to y} \right) = \left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to y $

    因此

    $ \left( {x \to y} \right) \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right) \leqslant \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft \left( {\left( {x \to y} \right) \circlearrowleft y} \right) $

    $ \begin{gathered} \mu \left( {\left( {x \to y} \right) \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right)} \right) \leqslant \mu \left( {\left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft \left( {\left( {x \to y} \right) \circlearrowleft y} \right)} \right) = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {\left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft \left( {1 - \left( {\left( {x \to y \circlearrowleft y} \right)} \right)} \right)} \right) \wedge \mu \left( 1 \right) \leqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {\left( {x \to y} \right) \circlearrowleft y} \right) \hfill \\ \end{gathered} $

    $ \mu \left( {\left( {x \to y} \right) \circlearrowleft y} \right) \geqslant \mu \left( {\left( {\left( {x \to y} \right) \otimes 1} \right) \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right)} \right) \wedge \mu \left( 1 \right) $

    即满足定义5(b).由于μL上的模糊滤子,因此μ(x)≤μ(1).综上可得μL上的模糊子正蕴涵滤子.

    同理可证(ⅰ)与(ⅲ)相互等价.综上可知结论成立.

    引理4  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,模糊集μL上的模糊滤子,则对于任意的xyL,下列条件等价:

    (ⅰ) μL上的模糊子正蕴涵滤子;

    (ⅱ) μ((xy)$\circlearrowleft$x)=μ(x);

    (ⅲ) μ((x$\circlearrowleft$y)→x)=μ(x).

      (ⅰ)⇒(ⅱ)因为μL上的模糊子正蕴涵滤子,由引理3可知

    $ \mu \left( {\left( {x \to y} \right) \circlearrowleft x} \right) = \mu \left( {\left( {x \to y} \right) \circlearrowleft \left( {1 \to x} \right)} \right) \wedge \mu \left( 1 \right) \leqslant \mu \left( x \right) $

    另一方面,对于任意的xyL,有

    $ x \leqslant \left( {x \to y} \right) \circlearrowleft x $

    从而

    $ \mu \left( x \right) \leqslant \mu \left( {\left( {x \to y} \right) \circlearrowleft x} \right) $

    $ \mu \left( x \right) = \mu \left( {\left( {x \to y} \right) \circlearrowleft x} \right) $

    (ⅱ)⇒(ⅰ)因为

    $ \mu \left( {\left( {y \to z} \right) \circlearrowleft \left( {x \to y} \right)} \right) \wedge \mu \left( x \right) = \mu \left( {x \to \left( {\left( {y \to z} \right) \circlearrowleft y} \right)} \right) \wedge \mu \left( x \right) \leqslant \mu \left( {\left( {y \to z} \right) \circlearrowleft y} \right) = \mu \left( y \right) $

    $ \mu \left( {\left( {y \to z} \right) \circlearrowleft \left( {x \to y} \right)} \right) \wedge \mu \left( x \right) \leqslant \mu \left( y \right) $

    由引理3可得μL上的模糊子正蕴涵滤子.

    同理可证(ⅰ)与(ⅲ)相互等价,综上可知结论成立.

    定理1  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,模糊集φδ都是L上的模糊滤子,且满足φδφ(1)=δ(1),则当φL上的模糊极滤子时,δ也是L上的模糊极滤子.

      对于任意的xyL,因为

    $ y \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right) = \left( {y \circlearrowleft x} \right) \to x\left( {y \circlearrowleft x} \right) = 1 $

    $ \varphi \left( 1 \right) = \varphi \left( {y \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right)} \right) $

    又因为φL上的模糊极滤子,且满足φδφ(1)=δ(1),故由引理2可得

    $ \begin{gathered} \varphi \left( 1 \right) = \varphi \left( {y \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right)} \right) \leqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\varphi \left[ {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right)} \right] = \hfill \\ \;\;\;\;\;\;\;\;\;\;\varphi \left[ {\left( {y \circlearrowleft x} \right) \to \left( {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right)} \right] \leqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\delta \left[ {\left( {y \circlearrowleft x} \right) \to \left( {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right)} \right] \hfill \\ \end{gathered} $

    $ \delta \left( 1 \right) \leqslant \delta \left[ {\left( {y \circlearrowleft x} \right) \to \left( {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right)} \right] $

    结合定义2(a)可得

    $ \delta \left( 1 \right) = \delta \left[ {\left( {y \circlearrowleft x} \right) \to \left( {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right)} \right] $

    从而

    $ \delta \left( {y \circlearrowleft x} \right) \leqslant \delta \left[ {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right] $

    又因为

    $ \begin{array} [c]{l} \left[ {\left( {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right) \to \left( {\left( {\left( {x \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right)} \right] = \hfill \\ \delta \left[ {\left( {\left( {x \circlearrowleft y} \right) \to y} \right) \circlearrowleft \left( {\left( {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right) \to x} \right)} \right] \geqslant \hfill \\ \delta \left[ {\left( {\left( {x \circlearrowleft y} \right) \to y} \right) \circlearrowleft \left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right)} \right] \geqslant \hfill \\ \delta \left[ {x \circlearrowleft \left( {\left( {y \circlearrowleft x} \right) \to x} \right)} \right] = \hfill \\ \delta \left( {\left( {y \circlearrowleft x} \right) \to \left( {x \circlearrowleft x} \right)} \right) = \delta \left( 1 \right) \hfill \\ \end{array} $

    结合定义2(a)可得

    $ \delta \left( 1 \right) = \delta \left[ {\left( {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right) \to \left( {\left( {\left( {x \circlearrowleft y} \right) - y} \right) \circlearrowleft x} \right)} \right] $

    从而

    $ \delta \left[ {\left( {\left( {\left( {\left( {y \circlearrowleft x} \right) \to x} \right) \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right] \leqslant \delta \left( {\left( {\left( {x \circlearrowleft y} \right) - y} \right) \circlearrowleft x} \right) $

    $ \delta \left( {y \circlearrowleft x} \right) \leqslant d\left( {\left( {\left( {x \circlearrowleft y} \right) \to y} \right) \circlearrowleft x} \right) $

    由引理2可知δ也是L上的模糊极滤子.

    定理2  设L=(M,∧,∨,⊗,→,$\circlearrowleft$,0,1)是非交换剩余格,模糊集μL上的模糊滤子,若μL上的模糊子正蕴涵滤子,则μ也是L上的模糊极滤子.

      对于任意的xyL,因为

    $ x \leqslant \left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to x $

    $ \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to x} \right) \to y \leqslant x \to y $

    又因为模糊滤子μL上的模糊子正蕴涵滤子,故由引理4可得

    $ \begin{gathered} \mu \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to x} \right) = \mu \left( {\left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to x} \right) \to y} \right) \circlearrowleft \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to x} \right) \geqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {\left( {x \to y} \right) \circlearrowleft \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to x} \right)} \right) = \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to \left( {\left( {x \to y} \right) \circlearrowleft x} \right)} \right) \geqslant \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mu \left( {y \to x} \right) \hfill \\ \end{gathered} $

    综上可知

    $ \mu \left( {\left( {\left( {x \to y} \right) \circlearrowleft y} \right) \to x} \right) \geqslant \mu \left( {y \to x} \right) $

    即由引理2可得μL上的模糊极滤子.

    3 结语

    滤子是研究逻辑代数的有效工具,文章在非交换剩余格中引入了模糊极滤子的概念,通过研究其特征及性质,获得了模糊极滤子的等价刻画.在下一步的工作中,我们将继续研究非交换剩余格上的其它模糊滤子的特征及性质,使其为我们深入研究非交换剩余格的结构奠定基础.

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    The Characteristics of Fuzzy Fantastic Filters on the Non-Commutative Residuated Lattice
    LIU Li-jun, WANG Shu-xun, LIU Li-hua     
    School of Mathematics and Computer Sciences, Shaanxi University of Technology, Hanzhong Shaanxi 723000, China
    Abstract: By using the principles and methods of fuzzy sets, in this paper, the concept of "fuzzy fantastic filter" is introduced in the non-commutative residuated lattice. By studying its properties and characterizations, the equivalent representation theorems under certain conditions are given between the fuzzy fantastic filter and the fuzzy sub positive implicative filter. The results of the study further extend the fuzzy filter theory of the non-commutative residuated lattice, and lay a theoretical foundation for the application of algebraic logic and computer information processing.
    Key words: fuzzy logic    non-commutative residuated lattice    fuzzy filter    fuzzy fantastic filter    
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