On Halin-lattices in graphs

AbstractHalin [2] has shown that primitive sets with respect to a subset A of the vertex set of a connected graph G form a complete lattice (Halin-lattice). In this article special contractions are defined such that pairs (G, A) and these maps form a category HG and that a contravariant functor exists from HG to the category of complete lattices and lattice homomorphisms. Using this functor it is proved that the lattice homomorphism is a well-quasi ordering in the class of all Halin-lattices of rooted trees

A non-commutative Priestley duality

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras. From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and so-called override. Our duality shows that there is a canonical choice for this embedding. Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces.Comment: 20 page