  The real unit interval is the fundamental building block for many branches of
mathematics like probability theory, measure theory, convex sets and homotopy
theory. However, a priori the unit interval could be considered an arbitrary
choice and one can wonder if there is some more canonical way in which the unit
interval can be constructed. In this paper we find such a construction by using
the theory of effect algebras. We show that the real unit interval is the
unique non-initial, non-final irreducible algebra of a particular monad on the
category of bounded posets. The algebras of this monad carry an order,
multiplication, addition and complement, and as such model much of the
operations we need to do on probabilities. On a technical level, we show that
both the categories of omega-complete effect algebras as well as that of
omega-complete effect monoids are monadic over the category of bounded posets
using Beck's monadicity theorem. The characterisation of the real unit interval
then follows easily using a recent representation theorem for omega-complete
effect monoids.
