We prove a Representation Theorem for -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 which is only defined when and an involution 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 -complete, i.e. where every increasing sequence has a supremum. We show that any -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 -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.