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Blocks of homogeneous effect algebras

Published online by Cambridge University Press:  17 April 2009

Gejza Jenča
Gejza Jenča
Affiliation:
Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Ilkovicova 3, 812 19 Bratislava, Slovakia, e-mail: jenca@kmat.elf.stuba.sk
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Abstract

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Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalise some well known classes of algebraic structures (for example orthomodular lattices, MV algebras, orthoalgebras et cetera). In the present paper, we introduce a new class of effect algebras, called homogeneous effect algebras. This class includes orthoalgebras, lattice ordered effect algebras and effect algebras satisfying the Riesz decomposition property. We prove that every homogeneous effect algebra is a union of its blocks, which we define as maximal sub-effect algebras satisfying the Riesz decomposition property. This generalizes a recent result by Riec˘anová, in which lattice ordered effect algebras were considered. Moreover, the notion of a block of a homogeneous effect algebra is a generalisation of the notion of a block of an orthoalgebra. We prove that the set of all sharp elements in a homogeneous effect algebra E forms an orthoalgebra Es. Every block of Es is the centre of a block of E. The set of all sharp elements in the compatibility centre of E coincides with the centre of E. Finally, we present some examples of homogeneous effect algebras and we prove that for a Hilbert space ℍ with dim (ℍ) > 1, the standard effect algebra ℰ(ℍ) of all effects in ℰ is not homogeneous.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 1 , August 2001 , pp. 81 - 98
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bennett, M.K. and Foulis, D.J., ‘Phi-symmetric effect algebras’, Foundations of Physics 25 (1995), 1699–1722.CrossRefGoogle Scholar
[2]Beran, L., Orthomodular lattices. Algebraic approach. (D. Reidel Publishing Co., Dordrecht, 1985).CrossRefGoogle Scholar
[3]Bush, P., Grabowski, M., and Lahti, P., Operational quantum physics (Springer-Verlag, Berlin, Heidelberg, New York, 1995).CrossRefGoogle Scholar
[4]Chang, C.C., ‘Algebraic analysis of many-valued logics’, Trans. Amer. Math. Soc. 89 (1959), 74–80.Google Scholar
[5]Chevalier, G. and Pulmannová, S., ‘Some ideal lattices in partial abelian monoids’, (preprint, Mathematical Institute of Slovak Academy of Sciences, Bratislava, 1998).Google Scholar
[6]Chovanec, F. and Kôpka, F., ‘Boolean D-posets. Quantum structures’, Tatra Mt. Math. Publ. 10 (1997), 1–15.Google Scholar
[7]Cohen, D.W., An introduction to Hilbert space and quantum logic (Springer-Verlag, Berlin, Heidelberg, New York, 1989).CrossRefGoogle Scholar
[8]Foulis, D.J. and Bennett, M.K., ‘Effect algebras and unsharp quantum logics’, Found. Phys. 24 (1994), 1331–1352.CrossRefGoogle Scholar
[9]Foulis, D.J., Greechie, R. and Rütimann, G., ‘Filters and supports in othoalgebras’, Internat. J. Theoret. Phys. 35 (1995), 789–802.Google Scholar
[10]Giuntini, R. and Greuling, H., ‘Toward a formal language for unsharp properties’, Found. Phys. 19 (1994), 769–780.Google Scholar
[11]Greechie, R., Foulis, D. and Pulmannová, S., ‘The center of an effect algebra’, Order 12 (1995), 91–106.CrossRefGoogle Scholar
[12]Gudder, S. and Pulmannová, S., ‘Quotients of partial abelian monoids’, Algebra Universalis 38 (1998), 395–421.CrossRefGoogle Scholar
[13]Jenča, G. and Riečanová, Z., ‘On sharp elements in lattice ordered effect algebras’, BUSEFAL 80 (1999), 24–29.Google Scholar
[14]Kalmbach, G., Orthomodular lattices (Academic Press, New York, 1983).Google Scholar
[15]Kôpka, F. and Chovanec, F., ‘D-posets’, Math. Slovaca 44 (1994), 21–34.Google Scholar
[16]Ludwig, G., Foundations of quantum mechanics (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[17]Pták, P. and Pulmannov´, S., Orthomodular structures as quantum logics (Kluwer, Dordrecht, 1991).Google Scholar
[18]Ravindran, K., On a structure theory of effect algebras, (Ph.D. thesis) (Kansas State Univ., Manhattan, KS, 1996).Google Scholar
[19]Riečanová, Z., ‘Compatibility and central elements in effect algebras’, Tatra Mt. Math. Publ. 16 (1998), 1–8.Google Scholar
[20]Riečanová, Z., ‘A generalization of blocks for lattice effect algebras’, Int. J. Theor. Phys. 39 (2000), 855–865.CrossRefGoogle Scholar