  Every reduced ring $R$ has a natural partial order defined by $a\le b$ if
$a^2=ab$; it generalizes the natural order on a boolean ring. The article
examines when $R$ is a lower semi-lattice in this order with examples drawn
from weakly Baer rings (pp-rings) and rings of continuous functions. Locally
connected spaces and basically disconnected spaces give rings $C(X)$ which are
such lower semi-lattices. Liftings of countable orthogonal (in this order) sets
over surjective ring homomorphisms are studied.
