On two types of ultrafilter extensions of binary relations

Levels of Ultrafilters with Extension Divisibilities

To work more accurately with elements of the semigroup of the Stone Cech compactification of the discrete semigroup of natural numbers N under multiplication. We divided these elements into ultrafilters which are on finite levels and ultrafilters which are not on finite levels. For the ultrafilters that are on finite levels we prove that any element is irreducible or product of irreducible elements and all elements on higher levels are extension divided by some elements on lower levels. We characterize ultrafilters that are not on finite levels and the effect of extension divisibility on the u…

Construction of nearly pseudocompactification

A space is nearly pseudocompact if and only if $\upsilon X\backslash X$ is dense in $βX\backslash X$. If we denote $K=cl_{βX}(\upsilon X\backslash X)$, then $δX=X\cup(βX\backslash K)$ is referred by Henriksen and Rayburn \cite{hr80} as nearly pseudocompact extension of $X$. Henriksen and Rayburn studied the nearly pseudocompact extension using different properties of $βX$. In this paper our main motivation is to construct nearly pseudocompact extension of $X$ independently and not using any kind of extension property of $βX$. An alternative construction of $βX$ is made by taking the family of …

Zorn引理向任意二元关系的推广

Extension of Zorn's lemma to arbitrary binary relations

本文将Zorn引理推广至任意二元关系，使其在传递性缺失时仍能用于优化，并借此证明无极大元的二元关系的广义解概念存在性定理。

映射的双等变扩张

Bi-Equivariant Extensions Of Maps

本文研究二元$G$-空间上连续映射的双等变扩张问题，引入结构映射概念，并证明两个相关扩张定理。

Hindman 有限和定理及其在代数拓扑化中的应用

Hindman's finite sums theorem and its application to topologizations of algebras

本文概述了Hindman有限和定理及其基于幂等超滤子的证明技术，并利用该定理的多维版本证明了无限polyring的Zariski拓扑总是非离散的，且相关映射构成闭无处稠密子集，最后提出了代数拓扑化的若干开放问题。

二元运算群与二元 $G$-空间

Groups of Binary Operations and Binary $G$-Spaces

研究拓扑空间上连续二元运算群及其与同胚群的关系，构造二元 $G$-空间与双等变映射范畴，为该理论奠定基础。