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《山东大学学报(理学版)》 ›› 2022, Vol. 57 ›› Issue (11): 10-20.doi: 10.6040/j.issn.1671-9352.0.2022.125

• • 上一篇    

有界Heyting代数上基于理想的一致拓扑空间

刘春辉   

  1. 赤峰学院教育科学学院, 内蒙古 赤峰 024001
  • 发布日期:2022-11-10
  • 作者简介:刘春辉(1982— ),男,硕士,教授,研究方向为非经典数理逻辑、Domain理论及拓扑学. E-mail:chunhuiliu1982@163.com
  • 基金资助:
    内蒙古自治区高等学校科学研究项目(NJZY21138);赤峰学院一流扶持学科(数学及计算机科学与技术)建设专项经费资助项目

Uniform topological space base on ideals in bounded Heyting algebras

LIU Chun-hui   

  1. School of Educational Science, Chifeng University, Chifeng 024001, Inner Mongolia, China
  • Published:2022-11-10

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摘要： 为了利用拓扑学工具研究有界Heyting代数的性质和结构问题,基于由理想概念诱导的一类同余关系在有界Heyting代数(H,≤,→,0,1)上构造一致拓扑空间(H,τ)并考察其基本性质和拓扑性质,证明了(H,τ)是非连通的局部连通、局部紧、零维、第一可数的完全正则空间,(H,τ)是T1空间当且仅当(H,τ)是Hausdorff空间,获得了(H,τ)成为离散空间和紧致空间的充要条件,指出了(H,≤,→,0,1)中格运算和蕴涵运算关于一致拓扑τ都是连续的,从而构成拓扑有界Heyting代数。同时,讨论了(H,τ)的商空间性质。

关键词: 有界Heyting代数, 理想, 一致拓扑空间, 商空间

Abstract: In order to study the properties and structure of bounded Heyting algebras by using topological tools, based on a type of congruences induced by the notion of ideal, uniform topological space (H,τ) is established and some of its basic and topological properties are investigated in bounded Heyting algebra (H,≤,→,0,1). It is proved that (H,τ) is disconnected, locally connected, locally compact, zero-dimensional, first-countable and completely regular space. Moreover (H,τ) is a T1 space if it is a Hausdorff space. Some necessary and sufficient conditions for (H,τ)to be discrete and compact space are obtained. It is showed that the lattice and implication operations in (H,≤,→,0,1) are continuous under the uniform topology τ, make (H,≤,→,0,1) to be a topological bounded Heyting algebra. Meanwhile, some properties of the quotient space of (H,τ) are discussed.

Key words: bounded Heyting algebra, ideal, uniform topological space, quotient space

中图分类号: 

  • O141.1
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