  Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors
of type I. Extending Kadison's anti-lattice theorem, we show that each of these
factors is a disjointness free anti-lattice. We characterise disjointness,
bands, and disjointness preserving bijections with disjointness preserving
inverses in direct sums of disjointness free anti-lattices and, therefore, in
atomic JBW-algebras. We show that in unital JB-algebras the algebraic centre
and the order theoretical centre are isomorphic. Moreover, the order
theoretical centre is a Riesz space of multiplication operators. A survey of
JBW-algebra factors of type I is included.
