I-滤波器的统一特性与新子类探讨

66 浏览量更新于2024-07-14 收藏 314KB PDF 举报

本文主要探讨了剩余格中的I-滤波器及其在逻辑系统和相关代数结构中的作用。I-滤波器作为一类特殊的过滤器，因其在研究非经典逻辑、余积格（Residuated Lattices）中的重要性而备受关注。作者首先回顾了已有的过滤器理论，并强调了I-滤波器在理解这些逻辑代数中的核心地位。I-滤波器的特征包括但不限于暗示滤波器、奇异滤波器和布尔滤波器等，它们各自具有独特的性质，有助于深入分析这些滤波器的结构和行为。在论文的核心部分，作者对I-滤波器进行了详尽的特征刻画，不仅揭示了它们的基本特性，还探讨了非平凡的子类，如具有乘法性质（如implicative滤波器）、非凡性质（fantastic滤波器）以及布尔性质的I-滤波器。此外，文中还讨论了与I-滤波器相关的同源逻辑代数，这些代数结构与滤波器的性质密切相关。接下来，作者引入了三个新的I-滤波器概念：可分滤波器、强滤波器和n-压缩滤波器。通过深入研究，作者证明了n-倍隐含BL-代数和n-收缩BL-代数实际上是等价的概念。这些新的滤波器类别为研究非传统逻辑系统提供了新颖的视角和工具。最后，文章着重探讨了这些特殊I-滤波器之间的关系。一个关键发现是，一个滤波器同时满足可分性和常规性条件时，它才能被称为理想的滤波器。这一结果对于理解和分类剩余格中的I-滤波器有着重要的理论价值。这篇论文不仅扩展了对I-滤波器的理解，还通过引入新概念和提供特征刻画，推动了逻辑系统和余积格理论的发展，对于从事此领域研究的学者来说是一篇重要的参考资料。

Z.M. Ma, B.Q. Hu / Fuzzy Sets and Systems 247 (2014) 92–107 95

The following alternative deﬁnitions of ﬁlters are useful and give us an inspiration.

Deﬁnition 2.8. (See [3,10–12,20,22,30,35–38].)

Let F beaﬁlterofL. Then F is called

• an

implicative ﬁlter if x → x

∈ F for all x ∈ L ,

• a

regular ﬁlter if ¬¬x → x ∈ F for all x ∈ L,

• a

fantastic ﬁlter if (y → x) →{[(x → y) → y]→x}∈F for all x,y ∈ L,

• a

Boolean ﬁlter if x ∨¬x ∈ F for all x ∈ L,

• a

prime ﬁlter if F is either x → y ∈ F or y → x ∈ F for all x,y ∈ L,

• a

prime ﬁlter of the second kind if F is proper such that x ∨ y ∈ F implies x ∈ F or y ∈ F for all x,y ∈ L,

• a

prime ﬁlter of the third kind if (x → y) ∨ (y → x) ∈ F for all x,y ∈ L, equivalently, x → (y ∨ z) ∈ F implies

(x → y) ∨ (x → z) for all x,y,z ∈ L [25].

Note 2.9. An

implicative ﬁlter is also called a G-ﬁlter [34,38] or a positive implicative ﬁlter [10,22,23] and a Boolean

ﬁlter is also called a positive implicative ﬁlter [12,20] or an implicative ﬁlter [30].

Lemma 2.10. (See [12,35,37,38].)

Let F beaﬁlterofL. Then the following assertions are equivalent, respectively

for all x,y,z ∈ L.

• (I1) F is

an implicative ﬁlter.

(I2) z → (y → x) ∈ F implies (z → y) → (z → x) ∈ F .

(I3) y → (y → x) ∈ F implies y → x ∈ F .

• (R1) F is

a regular ﬁlter.

(R2) ¬x →¬y ∈ F implies y → x ∈ F .

(R3) ¬x → y ∈ F implies ¬y → x ∈ F .

• (F1) F is

a fantastic ﬁlter.

(F2) y → x ∈ F implies [(x → y) → y]→x ∈ F .

(F3) [(x → y) → y]→[(y → x) → x]∈F .

• (B1) F is

a Boolean ﬁlter.

(B2) (¬x → x) → x ∈ F .

(B3) (x → y) → x ∈ F implies x ∈ F .

Deﬁnition 2.11. (See [10].)

Let S be a set and C be a subset of its powerset P(S). We say that C satisﬁes the extension

property if and only if C is closed under supersets that are ﬁlters, i.e. S

∈ C whenever S

∈ C and S

is a ﬁlter with

⊆ S

⊆ S.

In [10],

it was pointed out that the ﬁlters deﬁned in Deﬁnition 2.8 satisfy the extension property except for prime

ﬁlters of the second kind.

3. I -ﬁlters and their characterizations

In this section, we will propose a formal deﬁnition and characterizations for several classes of ﬁlters.

Deﬁnition 3.1. Let F b

eaﬁlterofL. For all x,y ∈ L, if there exist terms t(x,y),t



(x, y) of L such that t



(x, y) 

t(x,y) and t(x,y) → t



(x, y) ∈ F , then F is called an I -ﬁlter w.r.t. t(x,y) → t



(x, y), and I-ﬁlter for short.

I-ﬁlter

can be considered as a formal deﬁnition for some types of ﬁlters such as implicative ﬁlters, regular ﬁlters,

fantastic ﬁlters, Boolean ﬁlters and prime ﬁlters of the third kind in Deﬁnition 2.8 when t(x,y) → t



(x, y) is x → x

¬¬x → x, (y → x) →{[(x → y) → y]→x},1→ (x ∨¬x) and 1 →[(x → y) ∨ (y → x)], respectively. Although

it is identical with that of ﬁlter deﬁned by identity form [20], it possesses a speciﬁc form in which the basic operation

“residuum” is included, and provides us with a meaningful method to characterize these ﬁlters and their corresponding

algebras.

剩余15页未读，继续阅读

weixin_38502292

粉丝: 5
资源: 965

I-滤波器的统一特性与新子类探讨

父类赋值给子类

MFC 数字滤波器程序设计

子类化 超类化 全局子类

wxPython教程：改变网格颜色与滤波器设计

wxPython入门指南：构建网格控件与滤波器设计

重构wxPython代码：lc滤波器与螺旋滤波器设计中的问题

树控件样式探讨：lc滤波器与螺旋滤波器在wxPython中的应用

VisualFoxPro数据库管理系统入门：子类特征与继承性

wxPython入门：尺寸调整与对齐行为标记在lc滤波器和螺旋滤波器设计中的应用

如何在Android studio中在类中添加子类

最新资源

子类化超类化全局子类