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We prove a number of fundamental facts about the canonical order onprojections in C*-algebras of real rank zero. Specifically, we show that thisorder is separative and that arbitrary countable collections have equivalent(in terms of their lower bounds) decreasing sequences. Under the furtherassumption that the order is countably downwards closed, we show how tocharacterize greatest lower bounds of finite collections of projections, andtheir existence, using the norm and spectrum of simple product expressions ofthe projections in question. We also characterize the points at which thecanonical homomorphism to the Calkin algebra preserves least upper bounds ofcountable collections of projections, namely that this occurs precisely whenthe span of the corresponding subspaces is closed.

The Order on Projections in C*-Algebras of Real Rank Zero