  This is the first of a series of papers which define and study structures
called rootoids, which are groupoids equipped with a representation in the
category of Boolean rings and with an associated 1-cocycle. The axioms for
rootoids are abstracted from formal properties of Coxeter groups with their
root systems and weak orders. They imply that each of the weak orders of a
rootoid embeds as an order ideal in a complete ortholattice. This first paper
is concerned only with the most basic definitions, facts and examples; the main
results, which are new even for Coxeter groups, will be stated and proved in
subsequent papers. They involve certain categories of rootoids and especially a
notion of functor rootoid.
