Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDistributive envelopes and topological duality for lattices via canonical extensions
137
0
0.0
(
0
)
Authors
Mai Gehrke
Ask ChatGPT about the research
No Arabic abstract
We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of admissibility to the finitary case. Finally, we show that the dual spaces of the distributive envelopes of a lattice coincide with completions of quasi-uniform spaces naturally associated with the lattice, thus giving a precise spatial meaning to the distributive envelopes.
rate research
Read More
In cite{CGH15} we introduced TiRS graphs and TiRS frames to create a new natural setting for duals of canonical extensions of lattices. In this continuation of cite{CGH15} we answer Problem 2 from there by characterising the perfect lattices that are dual to TiRS frames (and hence TiRS graphs). We introduce a new subclass of perfect lattices called PTi lattices and show that the canonical extensions of lattices are PTi lattices, and so are `more than just perfect lattices. We introduce morphisms of TiRS structures and put our correspondence between TiRS graphs and TiRS frames from cite{CGH15} into a full categorical framework. We illustrate our correspondences between classes of perfects lattices and classes of TiRS graphs by examples.
We construct a canonical extension for strong proximity lattices in order to give an algebraic, point-free description of a finitary duality for stably compact spaces. In this setting not only morphisms, but also objects may have distinct pi- and sigma-extensions.
The Vietoris monad on the category of compact Hausdorff spaces is a topological analogue of the power-set monad on the category of sets. Exploiting Manes characterisation of the compact Hausdorff spaces as algebras for the ultrafilter monad on sets, we give precise form to the above analogy by exhibiting the Vietoris monad as induced by a weak distributive law, in the sense of Bohm, of the power-set monad over the ultrafilter monad.
We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorv{c}evi{c} theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover various results of Kechris-Pestov-Todorv{c}evi{c}, Moore, Ngyuen Van Th{e}, in the context of automorphism groups of not necessarily countable structures, as well as Zucker.
We characterize the left-handed noncommutative frames that arise from sheaves on topological spaces. Further, we show that a general left-handed noncommutative frame $A$ arises from a sheaf on the dissolution locale associated to the commutative shadow of $A$. Both constructions are made precise in terms of dual equivalences of categories, similar to the duality result for strongly distributive skew lattices in arXiv:1206.5848.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
