Tychonoff空间与$\text{C}(X)$上的环论序
Tychonoff spaces and a ring theoretic order on $\text{C}(X)$
摘要：
约化环序（rr-序）是约化环$R$上的自然偏序，由$r\le_{\text{rr}} s$给出，如果$r^2=rs$。它可以在$\text{C}(X)$形式的环中进行代数或拓扑研究。这里的重点是那些减少环中的每对元素有一个下确界在r阶，这意味着什么$X$。如果$\text{C}(X)$具有此属性，则空间$X$称为rr-good。令人惊讶的是，局部连通空间和基本不连通空间都具有这种性质。在各种拓扑条件下，包括其行为下的笛卡尔产品的RR-良好的性能进行了研究。两个r-good空间的乘积可能不是r-good（例如，$β\mathbf{R}\times β\mathbf{R}$），但是，$P$ -空间和r-好弱Lindelöf空间的乘积总是r-好的。$P$ -空间、$F$ -空间和$U$ -空间起作用，Glicksberg定理和Comfort、Hindman和Negrepontis的工作也起作用。
英文摘要：
The reduced ring order (rr-order) is a natural partial order on a reduced ring $R$ given by $r\le_{\text{rr}} s$ if $r^2=rs$. It can be studied algebraically or topologically in rings of the form $\text{C}(X)$. The focus here is on those reduced rings in which each pair of elements has an infimum in the rr-order, and what this implies for $X$. A space $X$ is called rr-good if $\text{C}(X)$ has this property. Surprisingly both locally connected and basically disconnected spaces share this property. The rr-good property is studied under various topological conditions including its behaviour under Cartesian products. The product of two rr-good spaces can fail to be rr-good (e.g., $β\mathbf{R}\times β\mathbf{R}$), however, the product of a $P$-space and an rr-good weakly Lindelöf space is always rr-good. $P$-spaces, $F$-spaces and $U$-spaces play a role, as do Glicksberg's theorem and work by Comfort, Hindman and Negrepontis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
