Order

, Volume 8, Issue 1, pp 93–103

Orthmodular lattices whose MacNeille completions are not orthomodular

  • John Harding
Article

DOI: 10.1007/BF00385817

Cite this article as:
Harding, J. Order (1991) 8: 93. doi:10.1007/BF00385817

Abstract

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 Ameniya-Araki 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 Ameniya-Araki 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.

AMS subject classifications (1991)

06A2306C15

Key words

Boolean algebra generated by a chaindispersion free statesMacNeille completionortholatticeorthomodular lattice

Copyright information

© Kluwer Academic Publishers 1991

Authors and Affiliations

  • John Harding
    • 1
  1. 1.Department of MathematicsMcMaster UniversityHamiltonCanada