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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

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