  In this article, we only consider finite effect algebras. We define the
concepts of classical and quantum effect algebras and show that an effect
algebra $E$ is classical if and only if there exists an observable that
measures every effect of $E$. We next consider matrix representations of effect
algebras and prove an effect algebra is classical if and only if its matrix
representation has precisely one row. We then discuss sum table for effect
algebras. Although these are not as concise as matrix representations, they
give more immediate information about effect sums which are the basic
operations of an effect algebra. We subsequently study states on effect
algebras and prove that classical effect algebras are quantum effect algebras.
Finally, we consider composites of effect algebras. This allows us to study
interacting systems described by effect algebras. We show that two effect
algebras are classical if and only if their composite is classical. We point
out that scale effect algebras are not the only classical effect algebras and
stress the importance of atoms in this work.
