There are orthomodular lattices without non-trivial group-valued states: A computer-based construction
type: Item
instance of: scholarly article
title: There are orthomodular lattices without non-trivial group-valued states: A computer-based construction
zbMATH Open document ID: 0797.06010
author: Hans Weber
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-08-22T00:00:00Z
review text: Greechie has shown that there are orthomodular lattices without non- trivial positive real-valued states. P. Pták posed the problem of whether there are also orthomodular lattices without non-trivial group- valued states. In this note we describe the construction of such an example. Recently, P. Pták informed me that Navara independently found such an example with a quite different construction. The example given is a finite orthomodular lattice having 73 atoms and is constructed by a computer calculation. The proof that the constructed orthomodular lattice does not admit any non-trivial group-valued state is reduced to the proof that a certain homogeneous system of linear equations in 74 unknowns has no non-trivial solution; because of the size of this system calculation by hand would be too lengthy (even though theoretically not impossible). The two programs (for constructing the orthomodular lattice and for solving the system of equations) are not explicitly included in this note, but can immediately be reproduced according to the description given here.
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 06-04
zbMATH DE Number: 599937
zbMATH Keywords: orthomodular lattices
zbMATH Keywords: group-valued states
MaRDI profile type: Publication
full work available at URL: jmaa.1994.1133
OpenAlex ID: W2073763940
DOI: 10.1006/JMAA.1994.1133
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description: scientific article; zbMATH DE number 599937