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Automorphism groups of orthomodular lattices. (English) Zbl 0538.06009

It is proved that every group is the automorphism group of some orthomodular lattice. This generalizes the result of G. Schrag who solved this problem in the case of finite groups [Proc. Am. Math. Soc. 55, 243-249 (1976; Zbl 0352.06007)]. The proof uses the following results: Birkhoff’s construction of a lattice with a given group as its automorphism group; the embedding of a lattice L in an orthomodular lattice generated by the chains in L; the existence of Boolean algebras with trivial automorphism groups; and the pasting of orthomodular lattices at atoms.
Reviewer: G.A.Fraser

MSC:

06C15 Complemented lattices, orthocomplemented lattices and posets
08A35 Automorphisms and endomorphisms of algebraic structures
20F29 Representations of groups as automorphism groups of algebraic systems

Citations:

Zbl 0352.06007
Full Text: DOI

References:

[1] Kalmbach, Orthomodular lattices (1983)
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[11] DOI: 10.2307/2041882 · Zbl 0352.06007 · doi:10.2307/2041882
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