  We define and study an alternative partial order, called the spectral order,
on a synaptic algebra-a generalization of the self-adjoint part of a von
Neumann algebra. We prove that if the synaptic algebra A is norm complete (a
Banach synaptic algebra), then under the spectral order, A is Dedekind
sigma-complete lattice, and the corresponding effect algebra E is a
sigma-complete lattice. Moreover, E can be organized into a Brouwer-Zadeh
algebra in both the usual (synaptic) and spectral ordering; and if A is Banach,
then E is a Brouwer-Zadeh lattice in the spectral ordering. If A is of finite
type, then De Morgan laws hold on E in both the synaptic and spectral ordering.
