  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.
