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The exocenter and type decomposition of a generalized pseudoeffect algebra

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Foulis D., Pulmannová S., Vinceková E.

Discussiones Mathematicae - General Algebra and Applications

2013

tom 33

nr 1

13-47

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.

Ograniczanie wyników

1 Discussiones Mathematicae - General Algebra and Applications

1 Foulis D.

1 Pulmannová S.

1 Vinceková E.

1 2013

Wyniki wyszukiwania

The exocenter and type decomposition of a generalized pseudoeffect algebra