Geodesic
The exocenter and type decomposition of a generalized pseudoeffect algebra
Discussiones Mathematicae. General Algebra and Applications, Tome 33 (2013) no. 1, pp. 13-47

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We extend the notion of the exocenter of a generalized effect algebra (GEA) to a generalized pseudoeffect algebra (GPEA) and show that elements of the exocenter are in one-to-one correspondence with direct decompositions of the GPEA; thus the exocenter is a generalization of the center of a pseudoeffect algebra (PEA). The exocenter forms a boolean algebra and the central elements of the GPEA correspond to elements of a sublattice of the exocenter which forms a generalized boolean algebra. We extend the notion of central orthocompleteness to GPEA, prove that the exocenter of a centrally orthocomplete GPEA (COGPEA) is a complete boolean algebra and show that the sublattice corresponding to the center is a complete boolean subalgebra. We also show that in a COGPEA, every element admits an exocentral cover and that the family of all exocentral covers, the so-called exocentral cover system, has the properties of a hull system on a generalized effect algebra. We extend the notion of type determining (TD) sets, originally introduced for effect algebras and then extended to GEAs and PEAs, to GPEAs, and prove a type-decomposition theorem, analogous to the type decomposition of von Neumann algebras.
Keywords: pseudoeffect algebra, generalized pseudoeffect algebra, center, exocenter, central orthocompleteness, type determining set, type decomposition 
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Foulis, David; Pulmannová, Silvia; Vinceková, Elena. The exocenter and type decomposition of a generalized pseudoeffect algebra. Discussiones Mathematicae. General Algebra and Applications, Tome 33 (2013) no. 1, pp. 13-47. http://geodesic.mathdoc.fr/item/DMGAA_2013_33_1_a1/