Commutative BCK-Algebras and Quantum Structures

We study commutative BCK-algebras with the relative cancellation property, i.e.,if a ≤ x, y and x * a = y * a, then x = y. Such algebras generalize Booleanrings as well as Boolean D-posets (= MV-algebras). We show that any suchBCK-algebra X can be embedded into the positive cone of an Abelianlattice-ordered group. Moreover, this group can be chosen to be a universal group forX. We compare BCK-algebras with the relative cancellation property with knownquantum structures as posets with difference, D-posets, orthoalgebras, andquantum MV-algebras, and we show that in many cases we obtain MV-algebras.