Partial abelian monoids are structures \((P;\perp ,\oplus ,0)\) where \(\oplus \) is a partially defined binary operation with domain \(\perp \) which is commutative and associative in a restricted sense, and \(0\) is the neutral element. In the paper, partial abelian monoids with the Riesz decomposition property are studied. Relations with abelian groups, dimension equivalence and \(K_0\) for \(AF\;C^*\)-algebras are discussed.