剩余格上几类n-重模糊滤子的系统结构

剩余格上几类 n -重模糊滤子的系统结构 Structure of Several Kinds of n -Fold Fuzzy Filter in the Residuated Lattice first-author 刘莉君, 女, 讲师, 现从事模糊数学与逻辑代数方面的研究.E-mail：lljgsjys@163.com 刘莉君, 女, 讲师, 现从事模糊数学与逻辑代数方面的研究.E-mail： lljgsjys@163.com lljgsjys@163.com 陕西理工大学 数学与计算机科学学院, 陕西 汉中 723000 陕西理工大学 数学与计算机科学学院, 陕西 汉中 723000 School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong 723000, Shaanxi, China School of Mathematics and Computer Science, Shaanxi University of Technology, Hanzhong 723000, Shaanxi, China 在研究剩余格的代数结构中滤子理论起着非常重要的作用.通过研究剩余格上 n -重模糊蕴涵滤子, n -重模糊极滤子和 n -重模糊布尔滤子的概念的特征及性质, 获得了剩余格上这几类 n -重模糊滤子之间的关系, 以及它们的若干等价刻画, 研究结果进一步拓展了剩余格上的模糊滤子理论, 并为其揭示剩余格的代数结构奠定了理论基础. Filter theory plays a very important role in the study of the structure in the residuated lattice. We study the concept of n -fold fuzzy implicative filter, n -fold fuzzy fantastic filter and n -fold fuzzy Boolean filter. The relation and characterization theorem among these n-fold fuzzy filter are obtained systematically. The results of this study not only enrich the fuzzy filter theory of the residuated lattice, but also provide a theoretical basis for the structure of the residuated lattice. 剩余格 n -重模糊蕴涵滤子 n -重模糊极滤子 n -重模糊布尔滤子 residuated lattice n -fold fuzzy implicative filter n -fold fuzzy fantastic filter n -fold fuzzy boolean filter 引言 1 在非经典逻辑中, Pavelka引入的剩余格是一种非常基本的代数结构, 同时, 作为非经典逻辑系统的语义系统的各种逻辑代数也被广泛研究.目前, 大多数学者都认同剩余格为一种最广泛的逻辑代数结构, 其中格蕴涵代数, BL代数, MV代数, MTL代数等 [ 1 1 , 2 2 ] 均是剩余格的特殊情况, 而滤子是非经典逻辑代数研究领域的一个重要概念, 它们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.其中, 文献[ 3 3 ]研究了剩余格上的 n -重滤子, 文献[ 4 4 ]研究了剩余格上的广义 n -重模糊滤子, 文献[ 5 5 ]在BL代数上引入 n -重滤子并研究了它们的性质.因此, 系统地分析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, 基于此目的本文在上述工作的基础上, 受文献[ 6 6 ~ 10 10 6~10 ]的启发, 将模糊滤子的重理论进一步推广到剩余格上，通过研究剩余格上 n -重模糊蕴涵滤子, n -重模糊极滤子和 n -重模糊布尔滤子的特征及性质, 获得了剩余格上这几类 n -重模糊滤子之间的关系, 以及这几类 n -重模糊滤子的若干等价刻画.研究结果不但使剩余格上的模糊滤子理论得到进一步充实和丰富，概念间的层次关系更加的清晰和完善, 而且也能为研究基于剩余格的逻辑系统的结构特征提供理论基础上的支持和保障. 定义 1 下面先给出本文将用到的几个定义. 定义1 [ 2 2 ]   称(2, 2, 2, 2, 0, 0)-型代数 L =( M , ∧, ∨, ⊗, →, 0, 1)为剩余格，若以下条件成立: 1) ( M , ∧, ∨，0, 1)是有界格; 2) ( M , ⊗，1)是交换的幺半群; 3) 对于任意的 x , y , z ∈ M , x ⊗ y ≤ z ⇔ y ≤ x → z . 性质1 [ 2 2 ]   设 L =( M , ∧, ∨, ⊗, →, 0, 1)为剩余格，对于任意的 x , y , z ∈ L , 则以下条件成立: 1) x → y =1当且仅当 x ≤ y ; 2) 1→ x = x , x → x =1, x →1=1, 0→ x =1; 3) x →( y → z )= y →( x → z )=( x ⊗ y )→ z ; 4) ( x → y )⊗( y → z )≤ x → z ; 5) y → x ≤( z → y )→( z → x ), x → y ≤( y →z)→( x → z ); 6) 约定 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459566&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459566&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459566&type=middle , x n → x =1; 7) ¬ x = x →0, x ⊗ x =0, ¬( x ⊗ y )= x →¬ y = y →¬ x , x ∨ y ≤( y → x )→ x ; 8) x ≤¬ x → y , ¬ x ≤ x → y , x ∨ y =(( x → y )→ y )∧(( y → x )→ x ); 9) ( x ∨ y )→ z =( x → z )∧( y → z ), x →( y ∧ z )=( x → y )∧( x → z ). 定义2 [ 5 5 ]   设 L =( M , ∧, ∨, ⊗, →, 0, 1)为剩余格， μ : L →[0, 1]为 L 上的一个模糊集, 则模糊集 μ 被称为剩余格 L 上的模糊滤子, 如果对于任意的 x , y ∈ L , 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459333&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459333&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459333&type=middle 性质2 [ 5 5 ]   设 L =( M , ∧, ∨, ⊗, →, 0, 1)为剩余格，模糊集 μ 为剩余格 L 上的模糊滤子, 对于任意的 x , y , z ∈ L , 下列性质成立: 1) 如果 x ≤ y , 则 μ ( x )≤ μ ( y ), 即 μ 是保序的; 2) 若 μ ( x → y )= μ (1), 则 μ ( x )≤ μ ( y ); 3) μ ( x → y )≤ μ ( y → z )→ μ ( x → z ); 4) μ ( y → x )≤ μ ( z → y )→ μ ( z → x ); 5) μ ( y ⊗ x )= μ ( x ∧ y )= μ ( x )∧ μ ( y ); 6) μ (( x → y )⊗( y → z ))≤ μ ( x → z ). 定义3 [ 6 6 ]   设 L 是剩余格, μ 是 L 上的模糊滤子, 则 μ 是 L 上的一个 n -重模糊蕴涵滤子( n =1, 2, …), 如果对于任意的 x , y , z ∈ L , 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459338&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459338&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459338&type=middle 定义4 [ 7 7 ]   设 L 是剩余格, μ 是 L 上的模糊滤子, 则 μ 是 L 上的一个 n -重模糊极滤子( n =1, 2, …), 如果对于任意的 x , y , z ∈ L , 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459342&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459342&type=middle 定义5 [ 8 8 ]   设 L 是剩余格, μ 是 L 上的模糊滤子, 则 μ 是 L 上的一个 n -重模糊布尔滤子( n =1, 2, …), 如果对于任意的 x ∈ L , 有 μ (1)≥ μ ( x )且 μ ( x ∨¬ x n )= μ (1). 主要定理 1 引理1   设 L 是剩余格, 对于任意的 x , y ∈ L , 则下列等式成立: 1) ( x ∨ y ) n → y = x n → y ; 2) ( x ∨¬ x n )→( x n → x n +1 )=1. 证   1) ( x ∨ y ) n → y =( x ∨ y ) n －1→[( x ∨ y )→ y ]=( x ∨ y ) n －1→( x → y )= x →[( x ∨ y ) n －1→ y ] 连续重复 n -1次上述过程, 则( x ∨ y ) n → y = x n → y .证毕. 2) 设 L 是剩余格, 对于任意的 x ∈ L , 由性质1可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459347&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459347&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459347&type=middle 引理2   设 L 是剩余格, μ 为 L 上的 n -重模糊蕴涵滤子, 则对于任意的 x ∈ L , 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459352&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459352&type=middle 证   因为 μ 为 L 上的 n -重模糊蕴涵滤子, 则对于任意的 x ∈ L , 就有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459357&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459357&type=middle 即 μ ( x n → x 2 n )≥ μ (1), 结合(FF1)可知, μ ( x n → x 2 n )= μ (1), 证毕. 定理1   设 L 是剩余格, μ 为 L 上的模糊集, 则下列条件等价: 1) μ 是 L 上的 n -重模糊蕴涵滤子; 2) μ 是 L 上的模糊滤子, 且有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459362&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459362&type=middle 证   1)⇒2) 因为模糊集 μ 是 L 上的 n -重模糊蕴涵滤子, 由定义3可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459367&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459367&type=middle 在该式中令 x =1, 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459372&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459372&type=middle 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459377&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459377&type=middle 故 μ 是 L 上的模糊滤子.而另一方面, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459382&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459382&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459382&type=middle 对于任意的 x , y ∈ L , 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459387&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459387&type=middle 则 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459392&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459392&type=middle 故可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459397&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459397&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459397&type=middle 综上，由定义3可得 μ 是 L 上的 n -重模糊蕴涵滤子.证毕. 定理2   设 L 是剩余格, μ 为 L 上的模糊集, 则下列条件等价: 1) μ 是 L 上的 n -重模糊极滤子; 2) μ 是 L 上的模糊滤子, 且对于任意的 x , y ∈ L , 则有 μ (( x n → y )→ y )≤ μ ( x ∨ y ). 证   1)⇒2) 因为模糊集 μ 是 L 上的 n -重模糊极滤子, 由定义4可知， http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459402&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459402&type=middle 在该式中令 x = x ∨ y , 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459407&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459407&type=middle 又由引理1可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459409&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459409&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459409&type=middle 且 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459416&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459416&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459416&type=middle 从而可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459419&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459419&type=middle 再结合(FF1)式 μ (1)= μ ((( x n → y )→ y )→ x ∨ y ), 再由性质2可得对于任意的 x , y ∈ L , 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459424&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459424&type=middle 2) ⇒1) 对于任意的 x , y ∈ L , 因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459431&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459431&type=middle 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459437&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459437&type=middle 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459440&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459440&type=middle 再结合(FF1)式 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459444&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459444&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459444&type=middle 即可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459449&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459449&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459449&type=middle 综上可知 μ 是 L 上的 n -重模糊极滤子, 证毕. 定理3   设 L 是剩余格, μ 为 L 上的模糊滤子, 则下列条件等价: 1) μ 是 L 上的 n -重模糊布尔滤子; 2) μ ( x ∨( x n → y ))= μ (1); 3) μ ((( x n → y )→ x )→ x )= μ (1). 证   1)⇒2) 因为 μ 是 L 上的 n -重模糊布尔滤子, 则对于任意的 x , y ∈ L , 有¬ x n ≤ x n → y , 从而可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459452&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459452&type=middle 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459456&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459456&type=middle 再结合(FF1)式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459461&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459461&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459461&type=middle 2) ⇒3) 对于任意的 x , y ∈ L , 有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459465&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459465&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459465&type=middle 因此可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459469&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459469&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459469&type=middle 结合(FF1)式可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459472&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459472&type=middle 3) ⇒1) 因为 μ ((( x n → y )→ x )→ x )= μ (1), 在该式中令 y =0, 从而就有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459477&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459477&type=middle 结合(FF1)式可得: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459479&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459479&type=middle 综上可知 μ 是 L 上的 n -重模糊布尔滤子, 证毕. 定理4   设 L 是剩余格, 模糊集 μ 为 L 上的模糊滤子, 若 μ 为剩余格 L 上的 n -重模糊布尔滤子, 则对于任意的 x , y ∈ L , μ 也为剩余格 L 上的 n -重模糊极滤子. 证   设模糊集 μ 为剩余格 L 上的 n -重模糊布尔滤子, 则由定义5知, 对于任意的 x ∈ L , 都有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459485&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459485&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459485&type=middle 又根据性质1和2可得: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459487&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459487&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459487&type=middle 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459491&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459491&type=middle 再结合(FF1)式可知 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459495&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459495&type=middle 由性质2可得: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459498&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459498&type=middle 即 μ (1→( y → x ))∧ μ (1)≤ μ ((( x n → y )→ y )→ x ). 由定义4知 μ 也为剩余格 L 上的 n -重模糊极滤子.证毕. 定理5   设 L 是剩余格, 模糊集 μ 为 L 上的模糊滤子, 若 μ 为剩余格 L 上的 n -重模糊布尔滤子, 则对于任意的 x , y ∈ L , μ 为 L 上的 n -重模糊蕴涵滤子. 证   设 L 是剩余格, 对于任意的 x ∈ L , 由引理1可知, 在剩余格 L 上有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459503&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459503&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459503&type=middle 因此可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459508&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459508&type=middle 又因为 μ 为剩余格 L 上的 n -重模糊布尔滤子, 即可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459512&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459512&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459512&type=middle 从而 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459514&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459514&type=middle 则 μ ( x n →( x → x n +1 ))∧ μ ( x n → x )≤ μ ( x n → x n +1 ). 综上，由定义3可知模糊滤子 μ 为 L 上的 n -重模糊蕴涵滤子.证毕. 注   剩余格 L 上的 n -重模糊极滤子与剩余格 L 上的 n -重模糊蕴涵滤子之间没有必然等价关系. 例1   设 M 1 ={0, a , b , 1}，其中0≤ a ≤ b ≤1, 令x∧ y =min{ x , y }, x ∨ y =max{ x , y }, 且在 M 中定义二元运算“→”和“⊗”如 表 1 表 1 所示. → 0 a b 1 ⊗ 0 a b 1 0 1 1 1 1 0 0 0 0 0 a 1 1 a 1 a 0 a 0 a b a b a 1 b 0 0 b b 1 0 a b 1 1 0 a b 1 则 L =( M 1 , ∧, ∨, ⊗, →, 0, 1)是一个剩余格, 在 L 上定义模糊集 μ : L →[0, 1]使 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459518&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459518&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459518&type=middle 其中0≤ β ＜ α ≤1.可以验证模糊集 μ 为剩余格 L 上的 n -重模糊极滤子, 但非 n -重模糊蕴涵滤子, 这是因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459522&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459522&type=middle 显然, μ ( a n → b )＜ μ ( a n →(0→ b ))∧ μ ( a n →0). 例2   设 M 2 ={0, a , b , 1}，其中0≤ a ≤ b ≤1, 令 x ∧ y =min{ x , y }, x ∨ y =max{ x , y }, 且在 M 中定义二元运算“→”和“⊗”如 表 2 表 2 所示. → 0 a b 1 ⊗ 0 a b 1 0 1 1 1 1 0 0 0 0 0 a 0 1 a 1 a 0 a a a b 0 a 1 1 b 0 a b b 1 0 a b 1 1 0 a b 1 则 L =( M 2 , ∧, ∨, ⊗, →, 0, 1)是一个剩余格, 在 L 上定义模糊集 μ : L →[0, 1]使 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459527&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459527&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459527&type=middle 其中0≤ β ＜ α ≤1.可以验证模糊集 μ 为剩余格 L 上的 n -重模糊蕴涵滤子, 但非 n -重模糊极滤子, 这是因为 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459532&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459532&type=middle 显然, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459537&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459537&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459537&type=middle 定理6   设 L 是剩余格, 映射 μ 和 δ 是 L 上的模糊滤子, 且满足 μ ≤ δ , μ (1)= δ (1), 若映射 μ 是 L 上的 n -重模糊蕴涵(极, 布尔)滤子, 则映射 δ 是 L 上的 n -重模糊蕴涵(极, 布尔)滤子. 证   设 L 是剩余格, 对于任意的 x , y ∈ L , 令 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459541&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459541&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459541&type=middle 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459545&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459545&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459545&type=middle 因为映射 μ 是 L 上的 n -重模糊蕴涵滤子, 故由定义3可知: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459549&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459549&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459549&type=middle 又因为 μ ≤ δ , 则有 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459553&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459553&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459553&type=middle 即 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459557&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459557&type=middle 再结合(FF1)式可知: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459560&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459560&type=middle 因此可得 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459563&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459563&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4459563&type=middle 综上，由定义3可知映射 δ 也是 L 上的 n -重模糊蕴涵滤子. 同理可证得映射 δ 也是 L 上的 n -重模糊极滤子和 n -重模糊布尔滤子.证毕. 结论 1 在研究剩余格的代数结构中滤子理论起着非常重要的作用.本文在剩余格中引入了 n -重模糊蕴涵滤子, n -重模糊极滤子和 n -重模糊布尔滤子的概念, 通过研究它们的特征及性质, 系统分析了这几类 n -重模糊滤子概念之间的相互关系及其等价刻画.在下一步工作中我们将继续深入研究剩余格上其他的滤子, 为揭示剩余格的代数结构发挥作用. 参考文献 HAVESHKI M. Some types of filters in BL-algebras[J].Soft Computing, 2006, 10:657-664. 周红军.概率计量逻辑及其应用[M].北京:科学出版社, 2015. ZHOU H J. Probability Measurement Logic and Approximate Reasoning [M]. Beijing: Science Press, 2015(Ch). KADJI A, LELE C, TONGA M. N -fold filters in residuated lattice[J/OL].[2017-01-03] . http://arxiv.org/abs /13081878 v 1 http://arxiv.org/abs/13081878v1 . MA Z M.Some types of generalized fuzzy n -fold filters in residuated lattices[J]. Abstract and Applied Analysis, 2013, 20(3):1-8. DOI:10.1155/2013/736872. 孟彪龙, 辛小龙, 杨永伟.BL代数的 n -重模糊蕴涵滤子[J].计算机工程与应用, 2014, 50(11):62-65. MENG B L, XIN X L, YANG Y W. N -fold fuzzy filter on BL-algebra[J]. Computer Engineering and Applications, 2014, 50(11):62-65(Ch). KADJI A, LELE C, NGANOU J B, et al. Folding theory applied to residuated lattice [J].International Journal of Mathematics and Mathematical Sciences, 2014, 22(5):1-12. DOI: 10.1155/2014/42890. BORZOOEI R A. Fuzzy n -fold fantastic filters in BL-algebra [J].Neural Computing and Application, 2014, 18(3):378-385. DOI:10.1007/S13370-013-0169-z. DUMITRU B. Some types of filters in residuated lattices[J]. Soft Comput, 2014, 18(4):825-837.DOI: 10.1007/S00500-013-11846. MASOUD H. N -fold filters in BL-algebras[J]. Neural Computing and Application, 2011, 20(6):461-472. DOI: 10.1002/malq.200710029. TURUNEN E, TCHIKAPA N. N -fold implicative basic logic is Godel logic [J]. Soft Comput, 2012, 16(4):177-183. DOI:10.1007/S00500-011-0761-9. 10.14188/j.1671-8836.2017.06.012 O159 whdxxblxb-63-6-538 国家自然科学基金(11401357), 陕西省教育厅专项科学研究资助项目(17JK0145) 2017-01-02 2017-12-24 2020-12-09 版权所有© 《武汉大学学报(理学版)》编辑部2017 Copyright © 2017 Journal of Wuhan University(Natural Science Edition). All rights reserved. 2017 武汉大学学报(理学版) 6 63