Automorphisms of concrete logics

Navara, Mirko ; Tkadlec, Josef

Commentationes Mathematicae Universitatis Carolinae, Tome 32 (1991) no. 1, pp. 15-25.

Résumé

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.

Détail

Classification : 03G12, 06C15, 81C10, 81P10
Keywords: orthomodular lattice; quantum logic; concrete logic; set representation; automorphism group of a logic; state space