Orthmodular lattices whose MacNeille completions are not orthomodular
 John Harding
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The only known example of an orthomodular lattice (abbreviated: OML) whose MacNeille completion is not an OML has been noted independently by several authors, see Adams [1], and is based on a theorem of Ameniya and Araki [2]. This theorem states that for an inner product space V, if we consider the ortholattice ℒ(V,⊥) = {A \( \subseteq \) V: A = A ^{⊥⊥}} where A ^{⊥} is the set of elements orthogonal to A, then ℒ(V,⊥) is an OML if and only if V is complete. Taking the orthomodular lattice L of finite or confinite dimensional subspaces of an incomplete inner product space V, the ortholattice ℒ(V,⊥) is a MacNeille completion of L which is not orthomodular. This does not answer the longstanding question Can every OML be embedded into a complete OML? as L can be embedded into the complete OML ℒ(V,⊥), where V is the completion of the inner product space V.
Although the power of the AmeniyaAraki theorem makes the preceding example elegant to present, the ability to picture the situation is lost. In this paper, I present a simpler method to construct OMLs whose Macneille completions are not orthomodular. No use is made of the AmeniyaAraki theorem. Instead, this method is based on a construction introduced by Kalmbach [7] in which the Boolean algebras generated by the chains of a lattice are glued together to form an OML. A simple method to complete these OMLs is also given.
The final section of this paper briefly covers some elementary properties of the Kalmbach construction. I have included this section because I feel that this construction may be quite useful for many purposes and virtually no literature has been written on it.
 D. H.Adams (1969) The completion by cuts of an orthocomplemented modular lattice. Bull. Austral. Math. Soc. 1, 279–280.
 I.Ameniya and H.Araki (1966) A remark on Piron's paper. Pub. Res. Inst. Math. Ser., Kyoto Univ., Ser. A2, 423–427.
 D. R.Balbes and P.Dwinger (1974) Distributive Lattices, University of Missouri Press, Columbia, Missouri.
 G.Bruns, R. J.Greechie, J.Harding, and M.Roddy (1990) Completions of orthomodular lattices. Order 7, 67–76.
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 R.Godowski and R. J.Greechie (1984) Some equations related to the states on orthomodular lattices, Dem. Math. 17, 241–250.
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 G.Kalmbach (1983) Orthomodular Lattices, Academic Press, London.
 M. D.MacLaren (1964) Atomic orthocomplemented lattices, Pac. J. Math. 14, 597–612.
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 Title
 Orthmodular lattices whose MacNeille completions are not orthomodular
 Journal

Order
Volume 8, Issue 1 , pp 93103
 Cover Date
 19910301
 DOI
 10.1007/BF00385817
 Print ISSN
 01678094
 Online ISSN
 15729273
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 06A23
 06C15
 Boolean algebra generated by a chain
 dispersion free states
 MacNeille completion
 ortholattice
 orthomodular lattice
 Industry Sectors
 Authors

 John Harding ^{(1)}
 Author Affiliations

 1. Department of Mathematics, McMaster University, L8S 4K1, Hamilton, Ontario, Canada