The structure group of a non-degenerate effect algebra

被引：0
作者：
Carsten Dietzel
Wolfgang Rump
机构：
[1] University of Stuttgart,Institute for Algebra and Number Theory
来源：
Algebra universalis | 2020年 / 81卷
关键词：
Effect algebra; L-algebra; Structure group; Self-similarity; Group-valued measure; 08A55; 81P10; 06F15; 46L51;
D O I：
暂无
中图分类号：
学科分类号：
摘要：
A (non-commutative) structure group G(E) is associated to an arbitrary effect algebra E, and a concept of non-degeneracy is introduced. If E is non-degenerate, G(E) has a right invariant partial order, E embeds as an interval into G(E), and the negative cone of G(E) is a self-similar partial L-algebra. In the degenerate case, the possible anomalies are explained. Lattice effect algebras, 2-divisible effect algebras, and the effect algebra of a Hilbert space, are shown to be non-degenerate. As an application, using a strengthened concept of sharpness, the theory of block decomposition is extended to arbitrary effect algebras, and it is shown that the “very sharp” elements always form a sub-effect algebra and an orthomodular poset.
引用
收藏
相关论文
共 86 条
[1]  
Baer R(1949)Free sums of groups and their generalizations. An analysis of the associative law Am. J. Math. 71 706-742
[2]  
Bennett MK(1997)Interval and scale effect algebras Adv. Appl. Math. 19 200-215
[3]  
Foulis DJ(1936)The logic of quantum mechanics Ann. Math. 37 823-843
[4]  
Birkhoff G(1972)Artin–Gruppen und Coxeter–Gruppen Invent. Math. 17 245-271
[5]  
von Neumann J(2000)Effect algebras and para-Boolean manifolds Int. J. Theoret. Phys. 39 551-564
[6]  
Brieskorn E(1958)Algebraic analysis of many valued logics Trans. Am. Math. Soc. 88 467-490
[7]  
Saito K(1959)A new proof of the completeness of the Łukasiewicz axioms Trans. Am. Math. Soc. 93 74-80
[8]  
Cattaneo G(1975)A factor not anti-isomorphic to itself Bull. Lond. Math. Soc. 7 171-174
[9]  
Dalla Chiara ML(2002)Groupes de Garside Ann. Sci. École Norm. Sup. (4) 35 267-306
[10]  
Giuntini R(1999)Gaussian groups and Garside groups, two generalisations of Artin groups Proc. Lond. Math. Soc. (3) 79 569-604
123456789