Eﬀect algebras, or even just convex eﬀect algebras are very general structures. So, as in
GPTs where additional assumptions are often needed on the physical theory to make interesting
statements, we will impose some additional structure on eﬀect algebras.
The eﬀects of a quantum system, i.e. the positive subunital operators on some Hilbert space or,
more generally, positive subunital elements of a C
-algebra or Jordan algebra, have some interesting
algebraic structure (aside from the eﬀect algebra operations). The one we focus on in this paper
is the sequential product, also known as the L¨uders product
a. This product models
the act of ﬁrst measuring the eﬀect a and then measuring the eﬀect b.
In order to study the sequential product in the abstract, Gudder and Greechie introduced in
2002  the notion of a sequential eﬀect algebra (SEA), that models the sequential product on
an eﬀect algebra as a binary operation satisfying certain properties. SEAs have been studied by
several authors, see e.g. [16–18, 21, 22, 28, 38, 39, 49, 52, 53, 55, 58, 61].
Adopting the language of GPTs (and speciﬁcally that of reconstructions of quantum theory) we
can view the assumed existence of a sequential product on our eﬀect algebras as a physical postulate
(indeed, such a sequential product has been used for a reconstruction of quantum theory before ).
But as is the case for GPTs, we will also need some structural ‘background’ assumption. Indeed,
in most work dealing with GPTs it is assumed that the sets of states and eﬀects are closed in a
suitable topology. This models the operational assumption that when we can arbitrarily closely
approximate an eﬀect, that this eﬀect is indeed a physical eﬀect itself. Such assumptions are
routinely used to, for instance, remove the possibility of inﬁnitesimal eﬀects.
In a (sequential) eﬀect algebra we however have no natural notion of topology and hence no
direct way to require such a closure property. But as an eﬀect algebra is a partially ordered set, we
can require that this order is directed complete, meaning that any upwards directed subset has a
supremum. Indeed, this condition is routinely used in theoretical computer science to give meaning
to algorithms with loops or recursion in a branch called domain theory . In the ﬁeld of operator
algebras, one possible characterisation of von Neumann algebra’s is that they are C
are bounded directed-complete and have a separating set of normal states [40, 46, 60].
Motivated by the structure of von Neumann algebras, we call a sequential eﬀect algebra normal
when every directed set has a supremum, the sequential product preserves these suprema in the
second argument, and an eﬀect commutes with such a suprema provided it commutes with all
elements in the directed set. The set of eﬀects on a Hilbert space is a convex normal SEA. More
generally, the unit interval of any JBW-algebra (and so in particular that of any von Neumann
algebra) is also a convex normal SEA . As a rather diﬀerent example, any complete Boolean
algebra is a normal SEA, which is not convex.
A priori, a (normal) SEA is a rather abstract object and there could potentially be rather exotic
examples of them. The main result of this paper is to show that for normal SEAs this is, in a
sense, not the case.
We will show that any normal SEA is isomorphic to a direct sum E
, where E
a complete Boolean algebra, E
is a convex normal SEA and E
is a normal SEA that is purely
a-convex, a new type of eﬀect algebra we will deﬁne later on. We will show there is no overlap: for
instance, there is no normal SEA that is both purely a-convex and Boolean.
In much the same way as a C
-algebra is the union of its commutative subspaces, we show
that a purely a-convex SEA is a union of convex SEAs. Hence, normal SEAs come essentially in
two main types: Boolean algebras that model classical deterministic logic, and convex SEAs that
ﬁt into the standard GPT framework. Hence, rather than having convex structure as a starting
assumption, we show that it can be derived as a consequence of our other assumptions.
Along the way we will establish several smaller results that might be of independent interest.
We show any SEA where all elements are sharp (i.e. idempotent) is a Boolean algebra, and that
consequently any SEA with a ﬁnite number of elements is a Boolean algebra. We introduce a
new axiom for SEAs that exclude the seemingly pathological purely a-convex SEAs. Combining
this with our main result shows that normal SEAs satisfying this additional axiom neatly split up
into a Boolean algebra and a convex normal SEA. Finally, we study SEAs where the sequential
product is associative. We ﬁnd that associative normal SEAs satisfying our additional axiom must
be commutative, and we are able to completely classify the associative normal purely a-convex
factors, i.e. SEAs with trivial center.
This work relies on recent advances made in the representation theory of directed-complete
Accepted in Quantum 2020-12-22, click title to verify. Published under CC-BY 4.0. 2