The Macneille Completion of a Uniquely Complemented Lattice

Problem 36 of the third edition of Birkhoo's Lattice theory 2] asks whether the MacNeille completion of uniquely complemented lattice is necessarily uniquely complemented. We show that the MacNeille completion of a uniquely complemented lattice need not be complemented. Questions regarding the axiomatics of Boolean algebras led Huntington to conjecture , in 1904, that every uniquely complemented lattice was distributive. By 1940, Huntington's conjecture had been veriied for the classes of modular lattices, atomic lattices, and complemented lattices which satisfy DeMorgan's laws. Then, a 1945 paper of Dilworth 3] proved the quite unexpected result that any lattice could be embedded into a uniquely complemented lattice. It is presently unknown whether a complete uniquely complemented lattice must be distributive. This question has been answered in the aarmative for the classes of continuous lattices (and therefore algebraic lattices), complete lattices with compact unit, as well as the classes mentioned above. The construction of Dilworth seems to have shed little light on this subject, as the uniquely complemented lattices constructed by his method need not be complete. For a thorough description of the results mentioned above and of the history of Huntington's conjecture, see 6] and 1]. Glivenko's theorem states that the MacNeille completion (also known as the completion by cuts) of a Boolean algebra is a Boolean algebra. One might hope for a generalization of this result to uniquely complemented lattices. Indeed, Birkhoo raised this question in the third edition of Lattice theory 2] as did Sali i in Lattices with unique complements 6]. We show that the MacNeille completion of a uniquely complemented lattice is not necessarily complemented. The example given here is based on Dilworth's original construction of uniquely complemented lattices given in 3], and we will assume a knowledge of this paper.