Quotients of Partial Abelian Monoids

The basic algebraic structure that is studied in this chapter is a partial Abelian monoid (PAM in short) (cf. [Wil 1], [Wil 2], [Pul 4], [GuPu]). A PAM is a structure (P; 0, ⊕), where e is a commutative, associative partial binary operation on P and 0 is a neutral element. Beginning with a PAM at the lowest level, we shall consider a hierarchy of partial algebraic structures. The second level is a cancellative PAM (CPAM), the third level is a generalized effect algebra/generalized difference poset, which coincide with a cancellative, positive PAM. Commutative positive minimal clans and BCK-algebras are also included. An effect algebra is a unital generalized effect algebra/D-poset. On higher levels in the hierarchy we find orthoalgebras, orthomodular posets and lattices, MV-algebras, Boolean algebras.