Generalized Effect Algebras Research Articles

Algebras in the Positive Cone of po-Groups

We systemize a number of algebras that are especially known in the field of quantum structures and that in particular arise from the positive cones of partially ordered groups. Generalized effect algebras, generalized difference posets, cone algebras, commutative BCK-algebras with the relative cancellation property, and positive minimal clans are included in the text. All these structures are conveniently characterizable as special cases of generalized pseudoeffect algebras, which we introduced in a previous paper. We establish the exact relations between all mentioned structures, thereby adding new structures whenever necessary to make the scheme of order complete. Generalized pseudoeffect algebras were under certain conditions proved to be representable by means of a po-group. From this fact, we will easily establish representation theorems for all of the structures included in discussion.

Subcentral Ideals in Generalized Effect Algebras

In this paper, we introduce subcentral ideals in the class of cancellative positivepartial abelian monoids (CPAMs). Every complementary pair of subcentral idealsin a CPAM P corresponds to a subdirect decomposition of P. If this decompositionis direct, the corresponding ideals are called central. Subcentral ideals arecharacterized as central elements in the lattice of the recently introducedso-called R1-ideals. Every subcentral ideal is a central element in the lattice of allideals. A subcentral ideal I is central iff I is Riesz ideal. In an upper-directedCPAM, every subcentral ideal is central.

Subalgebras, Intervals, and Central Elements of Generalized Effect Algebras

The relation between generalized effect algebrasand D-algebras and their subalgebras are discussed. Forgeneralized effect algebras the notion of centralelements is introduced and some of their properties are shown.