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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1194,
author = {David J. Foulis and Silvia Pulmannov\'a and Elena Vincekov\'a},
title = {The exocenter and type decomposition of a generalized pseudoeffect algebra},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {33},
year = {2013},
pages = {13-47},
zbl = {1295.81016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1194}
}
David J. Foulis; Silvia Pulmannová; Elena Vinceková. The exocenter and type decomposition of a generalized pseudoeffect algebra. Discussiones Mathematicae - General Algebra and Applications, Tome 33 (2013) pp. 13-47. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1194/
[000] [1] P. Bush, P. Lahti and P. Mittelstaedt, The Quantum Theory of Measurement (Lecture Notes in Physics, Springer, Berlin-Heidelberg-New York, 1991).
[001] [2] J.C Carrega, G. Chevalier and R. Mayet, Direct decomposition of orthomodular lattices, Alg. Univers. 27 (1990) 480-496. doi: 10.1007/DF01188994. | Zbl 0715.06006
[002] [3] A. Dvurečenskij, Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras, J. Aust. Math. Soc. 74 (2003) 121-143. | Zbl 1033.03036
[003] [4] A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures (Kluwer, Dordrecht, 2000). | Zbl 0987.81005
[004] [5] A. Dvurečenskij and T. Vetterlein, Pseudoeffect algebras I. Basic properties, Int. J. Theor. Phys. 40 (2001) 685-701. doi: 10.1023/A:1004192715509. | Zbl 0994.81008
[005] [6] A. Dvurečenskij and T. Vetterlein, Pseudoeffect algebras II. Group representations, Int. J. Theor. Phys. 40 (2001) 703-726. doi: 10.1023/A:1004144832348. | Zbl 0994.81009
[006] [7] A. Dvurečenskij and T. Vetterlein, Algebras in the positive cone of po-groups, Order 19 (2002) 127-146. doi: 10.1023A:1016551707-476. | Zbl 1012.03064
[007] [8] A. Dvurečenskij and T. Vetterlein, Generalized pseudo-effect algebras, in: Lectures on Soft Computing and Fuzzy Logic; Adv. Soft Comput, (Ed(s)), (Physica, Heidelberg, 2001) 89-111. | Zbl 1012.03063
[008] [9] D.J. Foulis and M.K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994) 1331-1352. doi: 10.1007/BF02283036. | Zbl 1213.06004
[009] [10] D.J. Foulis and S. Pulmannová, Type-decomposition of an effect algebra, Found. Phys. 40 (2010) 1543-1565. doi: 10.1007/s10701-009-9344-3.
[010] [11] D.J Foulis and S. Pulmannová, Centrally orthocomplete effect algebras, Algebra Univ. 64 (2010) 283-307. doi: 10.1007/s00012-010-0100-5. | Zbl 1218.06005
[011] [12] D.J. Foulis and S. Pulmannová, Hull mappings and dimension effect algebras, Math. Slovaca 61 (2011) 485-522. doi: 10.2478/s12175-011-0025-2. | Zbl 1265.81003
[012] [13] D.J. Foulis and S. Pulmannová, The exocenter of a generalized effect algebra, Rep. Math. Phys. 61 (2011) 347-371. | Zbl 1257.81003
[013] [14] D.J. Foulis and S. Pulmannová, The center of a generalized effect algebra, to appear in Demonstratio Math. | Zbl 1318.06007
[014] [15] D.J. Foulis and S. Pulmannová, Hull determination and type decomposition for a generalized effect algebra, Algebra Univers. 69 (2013) 45-81. doi: 10.1007/s00012-012-0214-z. | Zbl 1275.03165
[015] [16] D.J. Foulis, S. Pulmannová and E. Vinceková, Type decomposition of a pseudoeffect algebra, J. Aust. Math. Soc. 89 (2010) 335-358. doi: 10.1017/S1446788711001042. | Zbl 1235.06011
[016] [17] K.R. Goodearl and F. Wehrung, The Complete Dimension Theory of Partially Ordered Systems with Equivalence and Orthogonality, Mem. Amer. Math. Soc. 831 (2005). | Zbl 1075.06001
[017] [18] R.J. Greechie, D.J. Foulis and S. Pulmannová, The center of an effect algebra, Order 12 (1995) 91-106. doi: 10.1007/BF01108592. | Zbl 0846.03031
[018] [19] G. Grätzer, General Lattice Theory (Academic Press, New York, 1978). | Zbl 0436.06001
[019] [20] J. Hedlíková and S. Pulmannová, Generalized difference posets and orthoalgebras, Acta Math. Univ. Comenianae 45 (1996) 247-279. | Zbl 0922.06002
[020] [21] M.F. Janowitz, A note on generalized orthomodular lattices, J. Natur. Sci and Math. 8 (1968) 89-94. | Zbl 0169.02104
[021] [22] G. Jenča, Subcentral ideals in generalized effect algebras, Int. J. Theor. Phys. 39 (2000) 745-755. doi: 10.1023?A:1003610426013. | Zbl 0957.03060
[022] [23] G. Kalmbach, Measures and Hilbert Lattices (World Scientific Publishing Co., Singapore, 1986).
[023] [24] G. Kalmbach and Z. Riečanová, An axiomatization for abelian relative inverses, Demonstratio Math. 27 (1994) 535-537. | Zbl 0826.08002
[024] [25] F. Kôpka and F. Chovanec, D-posets, Math. Slovaca 44 (1994) 21-34.
[025] [26] L.H. Loomis, The Lattice-Theoretic Background of the Dimension Theory of Operator Algebras (Mem. Amer. Math. Soc. No. 18, 1955). | Zbl 0067.08702
[026] [27] S. Maeda, Dimension functions on certain general lattices, J. Sci. Hiroshima Univ A 19 (1955) 211-237. | Zbl 0068.02502
[027] [28] A. Mayet-Ippolito, Generalized orthomodular posets, Demonstratio Math. 24 (1991) 263-274.
[028] [29] F.J. Murray and J. von Neumann, On Rings of Operators, J. von Neumann colected works, vol. III, Pergamon Press, Oxford, 1961, 6-321.
[029] [30] S. Pulmannová and E. Vinceková, Riesz ideals in generalized effect algebras and in their unitizations, Algebra Univ. 57 (2007) 393-417. doi: 10.1007/s00012-007-2043-z. | Zbl 1139.81007
[030] [31] S. Pulmannová and E. Vinceková, Abelian extensions of partially ordered partial monoids, Soft Comput. 16 (2012) 1339-1346. doi: 10.1007/s00500-012-0814-8. | Zbl 1284.06037
[031] [32] A. Ramsay, Dimension theory in complete weakly modular orthocomplemented lattices, Trans. Amer. Math. Soc. 116 (1965) 9-31. | Zbl 0163.26206
[032] [33] Z. Riečanová, Subalgebras, intervals, and central elements of generalized effect algebras, Int. J. Theor. Phys. 38 (1999) 3209-3220. doi: 10.1023/A:1026682215765. | Zbl 0963.03087
[033] [34] M.H. Stone, Postulates for Boolean algebras and generalized Boolean algebras, Amer. J. Math. 57 (1935) 703-732. | Zbl 61.0975.05
[034] [35] A. Wilce, Perspectivity and congruence in partial Abelian semigroups, Math. Slovaca 48 (1998) 117-135. | Zbl 0938.03094
[035] [36] Y. Xie and Y. Li, Riesz ideals in generalized pseudo effect algebras and in their unitizations, Soft Comput. 14 (2010) 387-398. doi: 10.1007/s00500-009-0412-6. | Zbl 1194.03053