Tychonoff spaces and a ring theoretic order on $\text{C}(X)$

TL;DR

This paper explores the algebraic and topological properties of the reduced ring order on rings of continuous functions, characterizing spaces where this order admits infima and analyzing how these properties behave under products.

Contribution

It introduces the concept of rr-good spaces where the ring of continuous functions has infima in the reduced ring order and studies their topological and product stability properties.

Findings

Locally connected and basically disconnected spaces are rr-good.

The product of two rr-good spaces may not be rr-good.

The product of a P-space and an rr-good weakly Lindelöf space is rr-good.

Abstract

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., $\beta \mathbf{R}\times \beta \mathbf{R}$), however, the product of a $P$-space and an rr-good weakly Lindel\"of 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.