# Orthmodular lattices whose MacNeille completions are not orthomodular

## Authors

- Received:
- Accepted:

DOI: 10.1007/BF00385817

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

- 42 Views

## 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.