Mirko Navara, Josef Tkadlec
Automorphisms of concrete logics

Comment.Math.Univ.Carolinae 32,1 (1991) 15-26.

Abstract: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.

Keywords: orthomodular lattice, quantum logic, concrete logic, set representation, automorphism group of a logic, state space
AMS Subject Classification: Primary 06C15; Secondary 03G12, 81C10