Symmetries and retracts of quantum logics
The authors call a pair \(Q=(L,M)\) a quantum logic if L is an orthomodular \(\sigma\)-lattice and M is a \(\sigma\)-convex full set of states on L. A symmetry of Q is an automorphism \(\tau\) of L with \(\{\) \(m\circ \tau |\) \(m\in M\}=M\). Let \(\{G_ i|\) \(i\in I\}\) be a family of groups, \(\leq\) a partial order on I and Q a fixed quantum logic. It is shown that there exists a family \(\{Q_ i|\) \(i\in I\}\) of quantum logics such that Q is a sublogic of every \(Q_ i\), \(Q_ i\) has \(G_ i\) as group of symmetries and \(Q_ i\) is a retract of \(Q_ j\) for \(i\leq j\). For \(i\nleq j\), \(Q_ i\) is not a sublogic of \(Q_ j\). This result strengthens the result of \textit{G. Kalmbach} [Bull. Aust. Math. Soc. 29, 309-313 (1984; Zbl 0538.06009)].
scientific article; zbMATH DE number 4016973