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/

[1] P. Bush, P. Lahti and P. Mittelstaedt, The Quantum Theory of Measurement (Lecture Notes in Physics, Springer, Berlin-Heidelberg-New York, 1991).

[2] J.C Carrega, G. Chevalier and R. Mayet, Direct decomposition of orthomodular lattices, Alg. Univers. 27 (1990) 480-496. doi: 10.1007/DF01188994.

[3] A. Dvurečenskij, Central elements and Cantor-Bernstein's theorem for pseudo-effect algebras, J. Aust. Math. Soc. 74 (2003) 121-143.

[4] A. Dvurečenskij and S. Pulmannová, New Trends in Quantum Structures (Kluwer, Dordrecht, 2000).

[5] A. Dvurečenskij and T. Vetterlein, Pseudoeffect algebras I. Basic properties, Int. J. Theor. Phys. 40 (2001) 685-701. doi: 10.1023/A:1004192715509.

[6] A. Dvurečenskij and T. Vetterlein, Pseudoeffect algebras II. Group representations, Int. J. Theor. Phys. 40 (2001) 703-726. doi: 10.1023/A:1004144832348.

[7] A. Dvurečenskij and T. Vetterlein, Algebras in the positive cone of po-groups, Order 19 (2002) 127-146. doi: 10.1023A:1016551707-476.

[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.

[9] D.J. Foulis and M.K. Bennett, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994) 1331-1352. doi: 10.1007/BF02283036.

[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.

[11] D.J Foulis and S. Pulmannová, Centrally orthocomplete effect algebras, Algebra Univ. 64 (2010) 283-307. doi: 10.1007/s00012-010-0100-5.

[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.

[13] D.J. Foulis and S. Pulmannová, The exocenter of a generalized effect algebra, Rep. Math. Phys. 61 (2011) 347-371.

[14] D.J. Foulis and S. Pulmannová, The center of a generalized effect algebra, to appear in Demonstratio Math.

[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.

[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.

[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).

[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.

[19] G. Grätzer, General Lattice Theory (Academic Press, New York, 1978).

[20] J. Hedlíková and S. Pulmannová, Generalized difference posets and orthoalgebras, Acta Math. Univ. Comenianae 45 (1996) 247-279.

[21] M.F. Janowitz, A note on generalized orthomodular lattices, J. Natur. Sci and Math. 8 (1968) 89-94.

[22] G. Jenča, Subcentral ideals in generalized effect algebras, Int. J. Theor. Phys. 39 (2000) 745-755. doi: 10.1023?A:1003610426013.

[23] G. Kalmbach, Measures and Hilbert Lattices (World Scientific Publishing Co., Singapore, 1986).

[24] G. Kalmbach and Z. Riečanová, An axiomatization for abelian relative inverses, Demonstratio Math. 27 (1994) 535-537.

[25] F. Kôpka and F. Chovanec, D-posets, Math. Slovaca 44 (1994) 21-34.

[26] L.H. Loomis, The Lattice-Theoretic Background of the Dimension Theory of Operator Algebras (Mem. Amer. Math. Soc. No. 18, 1955).

[27] S. Maeda, Dimension functions on certain general lattices, J. Sci. Hiroshima Univ A 19 (1955) 211-237.

[28] A. Mayet-Ippolito, Generalized orthomodular posets, Demonstratio Math. 24 (1991) 263-274.

[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.

[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.

[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.

[32] A. Ramsay, Dimension theory in complete weakly modular orthocomplemented lattices, Trans. Amer. Math. Soc. 116 (1965) 9-31.

[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.

[34] M.H. Stone, Postulates for Boolean algebras and generalized Boolean algebras, Amer. J. Math. 57 (1935) 703-732.

[35] A. Wilce, Perspectivity and congruence in partial Abelian semigroups, Math. Slovaca 48 (1998) 117-135.

[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.