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Symmetries and retracts of quantum logics. (English) Zbl 0626.06013
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 G. Kalmbach [Bull. Aust. Math. Soc. 29, 309-313 (1984; Zbl 0538.06009)].
Reviewer: G.Kalmbach

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
46L30 States of selfadjoint operator algebras
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
20B27 Infinite automorphism groups
20F29 Representations of groups as automorphism groups of algebraic systems
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