Blocks of homogeneous effect algebras
Blocks of homogeneous effect algebras
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DOI：
10.1017/s0004972700019705
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发表时间：
2001-08
影响因子：
0.7
通讯作者：
Gejza Jenvca
中科院分区：
文献类型：
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作者：
Gejza Jenvca
效应代数由 Foulis 和 Bennett 于 1994 年提出，是一种偏代数，它概括了一些众所周知的代数结构类别（例如正交模格、MV 代数、正交代数等）。在本文中，我们介绍一类新的效应代数，称为齐次效应代数。此类包括正交代数、晶格有序效应代数和满足 Riesz 分解性质的效应代数。我们证明每个齐次效应代数都是其块的并集，我们将其定义为满足 Riesz 分解性质的最大子效应代数。这概括了 Riec˘anová 最近的结果，其中考虑了晶格有序效应代数。此外，齐次效应代数块的概念是正交代数块概念的推广。我们证明齐次效应代数 E 中所有尖锐元素的集合形成正交代数 Es。 Es 的每个块都是 E 块的中心。E 的相容中心中所有尖锐元素的集合与 E 的中心重合。最后，我们给出了齐次效应代数的一些例子，并证明对于昏暗 (ℍ) > 1 的希尔伯特空间 ℍ，ℰ 中所有效应的标准效应代数 ℰ(ℍ) 不是齐次的。
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.