n-Orthodistributivity in Orthomodular Lattices

A useful generalization of distributivity in lattices n-distributivity, \(n \in \mathbb{N}\) , was introduced in Huhn (Acta Sci. Math. 33:297–305, 1972). In Mayet and Roddy (Contrib. Gen. Algebra 5:285–294, 1987), ‘orthogonalized’ versions, n-orthodistributivity, \(n \in \mathbb{N}\) , of these equations were introduced and discussed. The discussion and results of Mayet and Roddy (Contrib. Gen. Algebra 5:285–294, 1987) centered on the class of modular ortholattices. In this paper we discuss and present some preliminary results for these conditions in orthomodular lattices. In particular, we completely classify the n-(ortho)distributive orthomodular lattices arising from Greechie’s classical 1971 construction, and we prove that a certain simple atomless orthomodular lattice, presented in Roddy (Algebra Univers. 29:564–597, 1992), is 4-orthodistributive. It is not 3-orthodistributive.     

Three Ultrafilters in a Modular Logic

We have conjectured that the equational theory of the modular logics is completely determined by its finite-dimensional members. Being able to embed an arbitrary modular logic into an atomistic one would almost certainly settle this Conjecture positively. A natural method of embedding a complemented modular lattice into an atomistic one is provided by the Frink embedding. In the case of a modular logic, much of the orthogonality structure can be carried through the embedding as well. Unfortunately, not enough of it to produce a modular logic as the codomain of the resulting ortho-embedding. The main technical result of this paper is an example which proves this.     

On geometric representations of modular ortholattices

A pre-orthogonality on a projective geometry is a symmetric binary relation, ⊥, such that for each point ${p, p^{\perp} = \{q | p \perp q \}}$ is a subspace. An orthogonality is a pre-orthogonality such that each p ^⊥ is a hyperplane. Such ⊥ is called anisotropic iff it is irreflexive. For projective geometries with an anisotropic pre-orthogonality, we show how to find a (large) projective subgeometry with a natural embedding for the lattices of subspaces and with an orthogonality induced by the given pre-orthogonality. We also discuss (faithful) representations of modular ortholattices within this context and derive a condition which allows us to transform a representation by means of an anisotropic pre-orthogonality into an anisotropic orthogeometry by means of an anisotropic orthogonality.