One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups
type: Item
instance of: scholarly article
title: One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups
zbMATH Open document ID: 1416.51004
author: Carsten Dietzel
author: Wolfgang Rump
author: Xia Zhang
published in: Journal of Algebra
publication date: 2019-03-22T00:00:00Z
review text: We know from the second author [Forum Math. 30, No. 4, 973--995 (2018; Zbl 1443.06005)] that every orthomodular lattice $X$ admits a group $G(X)$ with a right invariant lattice order as a complete invariant. As a special case the projection lattice $X$ of a von Neumann algebra characterizes the algebra up to $*$-symmetry and trivial summands of type I2. In [Stud. Log. 106, No. 1, 85--100 (2018; Zbl 1412.06011)], the second author extended the connection to $G(X)$ to unbounded lattices. In this article, the authors extend the connection, firstly by dropping the symmetry of the orthogonality relation, and secondly they do not assume that $X$ is a lattice. They start with a $V$-semilattice called a $\perp$-lattice. They show that the structure of a $\perp$-lattice is expressed by a single binary operation $\rightarrow$ which can be conceived as quantum-logical implication. This implies that every $\perp$-lattice can be understood as a special type of $L$-algebra. The $L$-algebras occur in connection with lattice-ordered groups [the second author, J. Algebra 320, No. 6, 2328--2348 (2008; Zbl 1158.06009)], Garside groups [the second author, J. Algebra 439, 470--510 (2015; Zbl 1360.20030)], non-commutative prime factorization [the second author, J. Number Theory 190, 394--413 (2018; Zbl 1420.11140)] and von Neumann algebras [the second author, J. Algebra 439, 470--510 (2015; Zbl 1360.20030)]. A special class of $L$-algebras arises as intervals $[u-1,1]$ in right $l$-groups, that is, groups with a right invariant lattice order, where $u$ is a strong order unit. These $L$-algebras have been called by the second author, right bricks. Therefore, the authors prove that every bounded $\perp$-lattice and its corresponding $L$-algebra is a right brick. This enables them to embed every (not necessarily bounded) $\perp$-lattice $X$ into its structure group $G(X)$. They also show that every $\perp$-lattice $X$ coincides with the set of right singular elements in the positive cone of…
Mathematics Subject Classification ID: 51F20
Mathematics Subject Classification ID: 51D25
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 46L51
zbMATH DE Number: 7040784
zbMATH Keywords: orthogonality
zbMATH Keywords: orthomodular space
zbMATH Keywords: L-algebra
zbMATH Keywords: Garside group
zbMATH Keywords: right l-group
zbMATH Keywords: group valued measure
zbMATH Keywords: structure group
reviewed by: Boumediene Et-Taoui
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2019.02.012
OpenAlex ID: W2913785887
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Wikidata QID: Q128368478
DOI: 10.1016/J.JALGEBRA.2019.02.012
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