There are orthomodular lattices without non-trivial group-valued states: A computer-based construction
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.
scientific article; zbMATH DE number 599937