2025

Gudder, Stan; Heinosaari, Teiko

We investigate finite effect algebras and their classification. 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.