The exocenter and type decomposition of a generalized pseudoeffect algebra

David J. Foulis; Silvia Pulmannová; Elena Vinceková

Discussiones Mathematicae - General Algebra and Applications (2013)

  • Volume: 33, Issue: 1, page 13-47
  • ISSN: 1509-9415

Abstract

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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.

How to cite

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David J. Foulis, Silvia Pulmannová, and Elena Vinceková. "The exocenter and type decomposition of a generalized pseudoeffect algebra." Discussiones Mathematicae - General Algebra and Applications 33.1 (2013): 13-47. <http://eudml.org/doc/270278>.

@article{DavidJ2013,
abstract = {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.},
author = {David J. Foulis, Silvia Pulmannová, Elena Vinceková},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {pseudoeffect algebra; generalized pseudoeffect algebra; center; exocenter; central orthocompleteness; type determining set; type decomposition; effect algebra; generalized effect algebra; orthosummable family; boolean algebra; hull system; exocentral cover; von Neumann algebra; types I/II/III.},
language = {eng},
number = {1},
pages = {13-47},
title = {The exocenter and type decomposition of a generalized pseudoeffect algebra},
url = {http://eudml.org/doc/270278},
volume = {33},
year = {2013},
}

TY - JOUR
AU - David J. Foulis
AU - Silvia Pulmannová
AU - Elena Vinceková
TI - The exocenter and type decomposition of a generalized pseudoeffect algebra
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2013
VL - 33
IS - 1
SP - 13
EP - 47
AB - 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.
LA - eng
KW - pseudoeffect algebra; generalized pseudoeffect algebra; center; exocenter; central orthocompleteness; type determining set; type decomposition; effect algebra; generalized effect algebra; orthosummable family; boolean algebra; hull system; exocentral cover; von Neumann algebra; types I/II/III.
UR - http://eudml.org/doc/270278
ER -

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