Orthmodular lattices whose MacNeille completions are not orthomodular

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 ?(V,⊥) = {A \( \subseteq \) V: A = A ^⊥⊥} where A ^⊥ is the set of elements orthogonal to A, then ?(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 ?(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 ?(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.     

Commutator-finite orthomodular lattices

The class of orthomodular lattices which have only finitely many commutators is investigated. The following theorems are proved: contains the block-finite orthomodular lattices. Every irreducible element of is simple. Every element of is a direct product of a Boolean algebra and finitely many simple orthomodular lattices. The irreducible elements of which are modular, or are M-symmetric with at least one atom, have height two or less.     

Projective orthomodular lattices II

In this paper we continue to investigate projectivity in orthomodular lattices. We prove the somewhat surprising result that no uncountable Boolean algebra is projective in the variety of orthomodular lattices. Received January 7, 1994; accepted in final form July 16, 1996.     

Completions of orthomodular lattices

If K is a variety of orthomodular lattices generated by a finite orthomodular lattice the MacNeille completion of every algebra in K again belongs to K.     

Completions of orthomodular lattices II

If is a variety of orthomodular lattices generated by a set of orthomodular lattices having a finite uniform upper bound om the length of their chains, then the MacNeille completion of every algebra in again belongs to .The author gratefully acknowledges the support of NSERC.     

Boolean quotients of orthomodular lattices

Relations between ideals and commutative group valued finitely additive functions on orth-omodular lattices are studied. Nontrivial examples of orthomodular lattices with all nontrivial quotients being Boolean are found.Dedicated to the memory of Alan Day.Presented by J. Sichler.     

Decision problem for orthomodular lattices

This paper answers a question of H. P. Sankappanavar who asked whether the theory of orthomodular lattices is recursively (finitely) inseparable (question 9 in [10]). A very similar question was raised by Stanley Burris at the Oberwolfach meeting on Universal Algebra, July 15–21, 1979, and was later included in G. Kalmbach’s monograph [6] as the problem 42. Actually Burris asked which varieties of orthomodular lattices are finitely decidable. Although we are not able to give a full answer to Burris’ question we have a contribution to the problem.   Note here that each finitely generated variety of orthomodular lattices is semisimple arithmetical and therefore directly representable. Consequently each such a variety is finitely decidable. (For a generalization of this, i.e. a characterization of finitely generated congruence modular varieties that are finitely decidable see [5].) In section 3, we give an example of finitely decidable variety of orthomodular lattices that is not finitely generated. Received June 28, 1995; accepted in final form June 27, 1996.     

Irreducible polynomials which are locally reducible everywhere

For any positive integer , there exist polynomials of degree which are irreducible over and reducible over for all primes if and only if is composite. In fact, this result holds over arbitrary global fields.

     

Commutators and decompositions of orthomodular lattices

We show that in any complete OML (orthomodular lattice) there exists a commutatorc such that [0,c ] is a Boolean algebra. This fact allows us to prove that a complete OML satisfying the relative centre property is isomorphic to a direct product [0,a] × [0,a ] wherea is a join of two commutators, [0,a] is an OML without Boolean quotient and [0,a ] is a Boolean algebra. The proof uses a new characterization of the relative centre property in complete OMLs. In a final section, we specify the previous direct decomposition in the more particular case of locally modular OMLs.     

On finite lattices which are embeddable in subsemigroup lattices

Communicated by L. N. Shevrin     

Associativity of operations on orthomodular lattices

It is known that orthomodular lattices admit 96 binary operations, out of which 16 are commutative. We clarify which of them are associative.     

Blocks and commutators in orthomodular lattices

Presented by Alan Day.