We prove a Representation Theorem for \(\omega\)-complete effect monoids.

In computer science, especially when dealing with quantum computing
or other non-standard models of computation, basic notions in
probability theory like “a predicate” vary wildly. There seems to
be one constant: the only useful example of an algebra of probabilities
is the real unit interval (or a subalgebra of some product of them).
In this paper we try to explain this phenomenon. We will show that
the structure of the real unit interval naturally arises from a few
reasonable assumptions. We do this by studying *effect monoids*,
an abstraction of the algebraic structure of the real unit interval:
it has an addition \(x+y\) which is only defined when \(x+y\leq 1\) and
an involution \(x\mapsto 1-x\) which make it an effect algebra, in
combination with an associative (possibly non-commutative)
multiplication. Examples include the unit intervals of ordered
rings and Boolean algebras.

We present a structure theory for effect monoids that are \(\omega\)-complete, i.e. where every increasing sequence has a supremum. We show that any \(\omega\)-complete effect monoid embeds into the direct sum of a Boolean algebra and the unit interval of a commutative unital C*-algebra. Intuitively then, each such effect monoid splits up into a ‘sharp’ part represented by the Boolean algebra, and a ‘probabilistic’ part represented by the commutative C*-algebra.

Some consequences of this characterization are that the multiplication must always be commutative, and that the unique \(\omega\)-complete effect monoid without zero divisors and more than 2 elements must be the real unit interval. Our results give an algebraic characterization and motivation for why any physical or logical theory would represent probabilities by real numbers.