 An orthoalgebra satisfying condition (b) is said to be orthocoherent. In other words: an orthoalgebra is orthocoherent if and only if finite pairwise summable subsets of \(\mathbf{L}\) are jointly summable. The lemma tells us that every orthocoherent orthoalgebra is, inter alia, an orthomodular poset. Conversely, an orthocomplemented poset is orthomodular iff \(a\oplus b = a\vee b\) is defined for all pairs with \(a\le b'\) and the resulting partial binary operation is associative—in which case the resulting structure \((\mathbf{L},\oplus ,0,1)\) is an orthocoherent orthoalgebra, the canonical ordering on which agrees with the given ordering on \(\mathbf{L}\). Thus, orthomodular posets (the framework for Mackey’s version of quantum logic) are equivalent to orthocoherent orthoalgebras.