An Analogue of Distributivity for Ungraded Lattices

n-distributivity, dimension and Carathéodory's theorem

A. Huhn proved that the dimension of Euclidean spaces can be characterized through algebraic properties of the lattices of convex sets. In fact, the lattice of convex sets of IEn is n+1-distributive but not n-distributive. In this paper his result is generalized for a class of algebraic lattices generated by their completely join-irreducible elements. The lattice theoretic form of Carathédory's...

Linear types and approximation

We study continuous lattices with maps that preserve all suprema rather than only directed ones. We introduce the (full) subcategory of FS-lattices, which turns out to be ∗-autonomous, and in fact maximal with this property. FS-lattices are studied in the presence of distributivity and algebraicity. The theory is extremely rich with numerous connections to classical Domain Theory, complete dist...

Many-Valued Complete Distributivity

Suppose (Ω, ∗, I) is a commutative, unital quantale. Categories enriched over Ω can be studied as generalized, or many-valued, ordered structures. Because many concepts, such as complete distributivity, in lattice theory can be characterized by existence of certain adjunctions, they can be reformulated in the many-valued setting in terms of categorical postulations. So, it is possible, by aid o...

Distributive Lattices from Graphs

Several instances of distributive lattices on graph structures are known. This includes c-orientations (Propp), α-orientations of planar graphs (Felsner/de Mendez) planar flows (Khuller, Naor and Klein) as well as some more special instances, e.g., spanning trees of a planar graph, matchings of planar bipartite graphs and Schnyder woods. We provide a characterization of upper locally distributi...

Measürability and Distributivity in the Theory of Lattices