Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCanonical extensions of lattices are more than perfect
79
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
Let $2<n<mleq omega$. Let $CA_n$ denote the class of cylindric algebras of dimension $n$ and $RCA_n$ denote the class of representable $CA_n$s. We say that $Ain RCA_n$ is representable up to $m$ if $CmAtA$ has an $m$-square representation. An $m$ square represenation is locally relativized represenation that is classical locally only on so called $m$-squares. Roughly if we zoom in by a movable window to an $m$ square representation, there will become a point determinded and depending on $m$ where we mistake the $m$ square-representation for a genuine classical one. When we zoom out the non-representable part gets more exposed. For $2<n<m<lleq omega$, an $l$ square represenation is $m$-square; the converse however is not true. The variety $RCA_n$ is a limiting case coinciding with $CA_n$s having $omega$-square representations. Let $RCA_n^m$ be the class of algebras representable up to $m$. We show that $RCA_n^{m+1}subsetneq bold RCA_n^m$ for $mgeq n+2$.
All known structural extensions of the substructural logic $mathsf{FL_e}$, Full Lambek calculus with exchange/commutativity, (corresponding to subvarieties of commutative residuated lattices axiomatized by ${vee, cdot, 1}$-equations) have decidable theoremhood; in particular all the ones defined by knotted axioms enjoy strong decidability properties (such as the finite embeddability property). We provide infinitely many such extensions that have undecidable theoremhood, by encoding machines with undecidable halting problem. An even bigger class of extensions is shown to have undecidable deducibility problem (the corresponding varieties of residuated lattices have undecidable word problem); actually with very few exceptions, such as the knotted axioms and the other prespinal axioms, we prove that undecidability is ubiquitous. Known undecidability results for non-commutative extensions use an encoding that fails in the presence of commutativity, so and-branching counter machines are employed. Even these machines provide encodings that fail to capture proper extensions of commutativity, therefore we introduce a new variant that works on an exponential scale. The correctness of the encoding is established by employing the theory of residuated frames.
A clock is, from an information-theoretic perspective, a system that emits information about time. One may therefore ask whether the theory of information imposes any constraints on the maximum precision of clocks. Here we show a quantum-over-classical advantage for clocks or, more precisely, the task of generating information about what time it is. The argument is based on information-theoretic considerations: we analyse how the accuracy of a clock scales with its size, measured in terms of the number of bits that could be stored in it. We find that a quantum clock can achieve a quadratically improved accuracy compared to a purely classical one of the same size.
Surveys to find high-redshift radio galaxies deliberately exclude optically-bright objects, which may be distant radio-loud quasars. In order to properly determine the space density of supermassive black holes, the fraction of such objects missed must be determined within a quantitative framework for AGN unification. I briefly describe the receding torus model, which predicts that quasars should have more luminous ionizing continua than radio galaxies of similar radio luminosity, and present evidence to support it. I also suggest two further tests of the model which should constrain some of its parameters.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
