Finite (quantum) effect algebras

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Abstract We investigate finite effect algebras and their classi-
fication. We show that an effect algebra with n elements has at
least n− 2 and at most (n− 1)(n− 2)/2 nontrivial defined sums. We
characterize finite effect algebras with these minimal and maximal
number of defined sums. The latter effect algebras are scale effect
algebras (i.e., subalgebras of [0,1]), and only those. We prove that
there is exactly one scale effect algebra with n elements for every
integer n ≥ 2. We show that a finite effect algebra is quantum
effect algebra (i.e. a subeffect algebra of the standard quantum
effect algebra) if and only if it has a finite set of order-determining
states. Among effect algebras with 2-6 elements, we identify all
quantum effect algebras.