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.     

Self-dual bases for varieties of orthomodular lattices

For any finitely based variety of orthomodular lattices, we determine the sizes of all equational bases that are both irredundant and self-dual.     

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.     

Irreducible orthomodular lattices which are simple

It is well known that for a chain finite orthomodular lattice, all congruences are factor congruences, so any directly irreducible chain finite orthomodular lattice is simple. In this paper it is shown that the notions of directly irreducible and simple coincide in any variety generated by a set of orthomodular lattices that has a uniform finite upper bound on the lengths of their chains. The prototypical example of such a variety is any variety generated by a set ofn dimensional orthocomplemented projective geometries.Presented by B. Jónsson.Supported by a grant from NSERC.     

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.     

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.     

A characterization of state spaces of orthomodular lattices

It is shown that every rational polytope is affinely equivalent to the set of all states of a finite orthomodular lattice, and that every compact convex subset of a locally convex topological vector space is affinely homeomorphic to the set of all states of an orthomodular lattice.     

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.     

Lattice uniformities and modular functions on orthomodular lattices

It is shown that the lattice of all exhaustive lattice uniformities on an orthomodular latticeL is isomorphic to the centre of a natural completion (of a quotient) ofL, and is thus a complete Boolean algebra. This is applied to prove a decomposition theorem for exhaustive modular functions on orthomodular lattices, which generalizes Traynor's decomposition theorem [14].