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Keywords:
orthomodular lattice; quantum logic; concrete logic; set representation; automorphism group of a logic; state space
orthomodular lattice; quantum logic; concrete logic; set representation; automorphism group of a logic; state space
Summary:
The main result of this paper is Theorem 3.3: Every concrete logic (i.e., every set-representable orthomodular poset) can be enlarged to a concrete logic with a given automorphism group and with a given center. Since every sublogic of a concrete logic is concrete, too, and since not every state space of a (general) quantum logic is affinely homeomorphic to the state space of a concrete logic [8], our result seems in a sense the best possible. Further, we show that every group is an automorphism group of a concrete lattice logic and, on the other hand, we prove that this is not true for Boolean logics with a dense center. As a technical tool for pursuing the latter type of problems, we investigate the correspondence between homomorphisms of concrete logics and pointwise mappings of their domain. We prove that in a suitable topological representation of concrete logics, every automorphism is carried by a homeomorphism.
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