Boolean quotients of orthomodular lattices
Let \(L\) be an orthomodular lattice and \(J\) a proper \(p\)-ideal of \(L\) (i.e. a lattice ideal such that \(a \in L\), \(b \in L \Rightarrow (a \vee b') \wedge b \in L)\). The present paper investigates the properties of those orthomodular lattices for which there exist nontrivial Boolean quotients \(L/J\). In particular, the following result is shown: If \(G\) is any commutative group not of order 2, then \(L\) has a nontrivial Boolean quotient if and only if there exists a function \(m : L \to \{0, \lambda\}\), where \(\lambda \in G\) is not of order 2, such that \(m (a \vee b) + m(a \wedge b) = m(a) + m(b)\) for any \(a\), \(b \in L\). This implies a generalization to commutative groups of a well-known no-go theorem concerning the nonexistence of dispersion-free states in Quantum Mechanics. The authors also construct an example of a non-Boolean orthomodular lattice with all nontrivial quotients Boolean.
scientific article; zbMATH DE number 836829
scientific article