  A sequential effect algebra (SEA) is an effect algebra equipped with a
sequential product operation modeled after the L\"uders product $(a,b)\mapsto
\sqrt{a}b\sqrt{a}$ on C*-algebras. A SEA is called normal when it has all
suprema of directed sets, and the sequential product interacts suitably with
these suprema. The effects on a Hilbert space and the unit interval of a von
Neumann or JBW algebra are examples of normal SEAs that are in addition convex,
i.e. possess a suitable action of the real unit interval on the algebra.
Complete Boolean algebras form normal SEAs too, which are convex only when
$0=1$.
  We show that any normal SEA $E$ splits as a direct sum $E\equiv E_b\oplus E_c
\oplus E_{ac}$ of a complete Boolean algebra $E_b$, a convex normal SEA $E_c$,
and a newly identified type of normal SEA $E_{ac}$ we dub purely almost-convex.
Along the way we show, among other things, that a SEA which contains only
idempotents must be a Boolean algebra; and we establish a spectral theorem
using which we settle for the class of normal SEAs a problem of Gudder
regarding the uniqueness of square roots. After establishing our main result,
we propose a simple extra axiom for normal SEAs that excludes the seemingly
pathological a-convex SEAs. We conclude the paper by a study of SEAs with an
associative sequential product. We find that associativity forces normal SEAs
satisfying our new axiom to be commutative, shedding light on the question of
why the sequential product in quantum theory should be non-associative.
