Compressible effect algebras
type: Item
instance of: scholarly article
title: Compressible effect algebras
zbMATH Open document ID: 1075.81011
DOI: 10.1016/S0034-4877(04)80008-9
published in: Reports on Mathematical Physics
publication date: 2005-05-09T00:00:00Z
review text: The author sets out to vastly generalize results from \textit{D. J. Foulis}' papers on ``Compressions on partially ordered abelian groups'' [Proc. Am. Math. Soc. 132, 3581--3587 (2004; Zbl 1063.47003)], and on ``Compressible groups'' [Math. Slovaca 53, 433--455 (2003; Zbl 1068.06018)] to arbitrary effect algebras \((E,0,1,\oplus)\). By \(a'\) one denotes the supplement of \(a\in E\), i. e. \(a\oplus a'=0\). If \(a\wedge a' = 0\) then \(a\) is called sharp, \(E_s\) denotes the set of sharp elements of \(E\). A retraction \(J\) is an additive map \(E \rightarrow E\) such that \(a\leq J(1)\) implies \(J(a)=a\). \(J(1)\) is called the focus of \(J\). If \(J(a)=0\) implies \(a\leq J(1)'\) a retraction \(J\) is called a compression. A retraction \(J\) is called direct if \(J(a)\leq a\) for all \(a\in E\). Direct retractions \(J\) induce an isomorphism between \(E\) and the cartesian product algebra \(\text{Ker}(J)\oplus J[E]\). Thus, they are equivalent to projections of a cartesian product onto a factor (Thm. 3.1, 3.2). If in an effect algebra \(E\) every compression is uniquely determined by its focus \(p\) , and has a supplement, then \(E\) is called compressible. A finite cartesian product of effect algebras is compressible iff every component is compressible, whereas an analogous statement for horizontal sums does not hold. If \(p\in E\) is the focus of some compression, then it is called a projection; \(P\) denotes the set of all projections in \(E\). In compressible effect algebras compressions may be uniquely denoted by \(J_p\) where \(p\in P\) is chosen such that \(J_p(1)=p\). Using the binary operation \(\circ\) on \(P\times E\) defined by \(p\circ a = J_p(a)\), an element \(a\in E\) is called compatible with \(p\) if \(a= (p\circ a) \oplus (p'\circ a)\), and \(C(p)= \{a\in E: a= (p\circ a) \oplus (p'\circ a)\}\) is called the commutant of the projection \(p\). The operation \(\circ\) is examined in some detail. Among others the author proves for \(p,q\in P\): \…
reviewed by: Horst Szambien
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 05E25
Mathematics Subject Classification ID: 62L99
Mathematics Subject Classification ID: 06D35
zbMATH DE Number: 2166557
zbMATH Keywords: effect algebra
zbMATH Keywords: compressible
zbMATH Keywords: sequential product
zbMATH Keywords: projection
zbMATH Keywords: retraction
zbMATH Keywords: orthomodular poset
zbMATH Keywords: orthomodular lattice
zbMATH Keywords: Lüders map
author: Stan Gudder
MaRDI profile type: Publication
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full work available at URL: s0034-4877(04)80008-9
OpenAlex ID: W2016571829
description: scientific article; zbMATH DE number 2166557