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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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##### References:
 [1] Greechie, R. J. (1970). On generating pathological orthomodular structures, Technical Report No. 13, Kansas State University. [2] Kalmbach, G. (1983).Orthomodular Lattices, Academic Press, London. · Zbl 0512.06011 [3] Kalmbach, G. (1984). Automorphism groups of orthomodular lattices,Bulletin of the Australian Mathematical Society,29, 309-313. · Zbl 0538.06009 [4] Mackey, G. W. (1963).The Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, New York. · Zbl 0114.44002 [5] Pulmannová, S. (1977). Symmetries in quantum logics,International Journal of Theoretical Physics,16, 681-688. · Zbl 0388.06007 [6] Pultr, A., and Trnková, V. (1980).Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories, North-Holland, Amsterdam. · Zbl 0418.18004 [7] Sabidussi, G. (1957). Graphs with given group and given graph theoretical properties,Canadian Journal of Mathematics,9, 515-525. · Zbl 0079.39202
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