Günter Bruns

Algebraic Aspects of Orthomodular Lattices

In this survey article we try to give an up-to-date account of certain aspects of the theory of ortholattices (abbreviated OLs), orthomodular lattices (abbreviated OMLs) and modular ortholattices (abbreviated MOLs), not hiding our own research interests. Since most of the questions we deal with have their origin in Universal Algebra, we start with a section discussing the basic concepts and results of Universal Algebra without proofs. In the next three sections we discuss, mostly with proofs, the basic results and standard techniques of the theory of OMLs. In the remaining five sections we work our way to the border of present day research, with no or only sketchy proofs. Section 5 deals with products and subdirect products, section 6 with free structures and section 7 with classes of OLs defined by equations. In section 8 we discuss embeddings of OLs into complete ones. The last section deals with questions originating in Category Theory, mainly amalgamation, epimorphisms and monomorphisms. The later sections of this paper contain an abundance of open problems. We hope that this will initiate further research.

Completions of orthomodular lattices II

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.

The fundamental duality of partially ordered sets

The representation of partially ordered sets by subsets of some set such that specified joins (meets) are taken to unions (intersections) suggests two categories, that of partially ordered sets with specified joins and meets, and that of sets equipped with suitable collections of subsets, and adjoint contravariant functors between them. This, in turn, induces a duality including, among several others, the two Stone Dualities and that between spatial locales and sober spaces.

Orthomodular lattices which can be covered by finitely many blocks

In our paper [3] we considered four finiteness conditions for an orthomodular lattice (abbreviated: OML) L and conjectured their equivalence. The only question left open in that paper was whether an OML L which can be covered by finitely many blocks (maximal Boolean subalgebras) has only finitely many blocks. In this paper we give an affirmative answer to this question , in fact , we prove the slightly stronger result:

Amalgamation of Ortholattices

We show that the variety of ortholattices has the strong amalgamation property and that the variety of orthomodular lattices has the strong Boolean amalgamation property, i.e. that two orthomodular lattices can be strongly amalgamated over a common Boolean subalgebra. We give examples to show that the variety orthomodular lattices does not have the amalgamation property and that the variety of modular ortholattices does not even have the Boolean amalgamation property. We further show that no non-Boolean variety of orthomodular lattices which is generated by orthomodular lattices of bounded height can have the Boolean amalgamation property.

Epimorphisms in Certain Varieties of Algebras

We prove a lemma which, under restrictive conditions, shows that epimorphisms in V are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by orthomodular lattices which are the horizontal sum of their blocks. The lemma can also be applied to obtain known results for discriminator varieties.