The real unit interval and effect algebras

A Categorical Construction of the Real Unit Interval

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.The characterisation of the real unit interval then follows easily using a recent representation theorem for omega-complete effect monoids.