Past Analytic Topology in Mathematics and Computer Science

1 February 2012
16:00
Ramon Jansana
Abstract

 I will first present Priestley style topological dualities for 
several categories of distributive meet-semilattices
and implicative semilattices developed by G. Bezhanishvili and myself. 
Using these dualities I will introduce a topological duality for Hilbert 
algebras, 
the algebras that correspond to the implicative reduct of intuitionistic logic.

  • Analytic Topology in Mathematics and Computer Science
27 January 2012
09:00
Leonardo Cabrer
Abstract

Dualities of various types have been used by different authors to 
describe free and projective objects in a large
  number of classes of algebras. Particularly, natural dualities provide a 
general tool to describe free objects. In
  this talk we present two interesting applications of this fact. 
  We first provide a combinatorial classification of unification problems 
by their unification type for the
varieties of Bounded Distributive Lattices, Kleene algebras, De Morgan 
algebras. Finally we provide axiomatizations forsingle
and multiple conclusion admissible rules for the varieties of Kleene 
algebras, De Morgan algebras, Stone algebras.

  • Analytic Topology in Mathematics and Computer Science
30 November 2011
16:00
to
17:30
Umberto Rivieccio
Abstract
I will give an overview of some of the most interesting algebraic-lattice theoretical results on bilattices. I will focus in particular on the product construction that is used to represent a subclass of bilattices, the so-called 'interlaced bilattices', mentioning some alternative strategies to prove such a result. If time allows, I will discuss other algebras of logic related to bilattices (e.g., Nelson lattices) and their product representation.
  • Analytic Topology in Mathematics and Computer Science
12 October 2011
16:00
to
17:30
Steve Vickers
Abstract
<p>In topos-valid point-free topology there is a good analogue of regular measures and associated measure theoretic concepts including integration. It is expressed in terms of valuations, essentially measures restricted to the opens. A valuation $m$ is $0$ on the empty set and Scott continuous, as well as satisfying the modular law $$ m(U \cup V) + m(U \cap V) = m(U) + m(V). $$</p>\\ <p>Of course, that begs the question of why one would want to work with topos-valid point-free topology, but I'll give some general justification regarding fibrewise topology of bundles and a more specific example from recent topos work on quantum foundations.</p>\\ <p>The focus of the talk is the valuation locale, an analogue of hyperspaces: if $X$ is a point-free space (locale) then its valuation locale $VX$ is a point-free space whose points are the valuations on $X$. It was developed by Heckmann, by Coquand and Spitters, and by myself out of the probabilistic powerdomain of Jones and Plotkin.</p>\\ <p>I shall discuss the following results, proved in a draft paper "A monad of valuation locales" available at <a href="http://www.cs.bham.ac.uk/~sjv/Riesz.pdf">http://www.cs.bham.ac.uk/~sjv/Riesz.pdf</a>:<ul> <li> V is a strong monad, analogous to the Giry monad of measure theory.</li> <li> There is a Riesz theorem that valuations are equivalent to linear functionals on real-valued maps.</li> <li> The monad is commutative: this is a categorical way of saying that product valuations exist and there is a Fubini theorem.</li> </ul></p>\\ <p>The technical core is an analysis of simple maps to the reals. They can be used to approximate more general maps, and provide a means to reducing the calculations to finitary algebra. In particular the free commutative monoid $M(L)$ over a distributive lattice $L$, subject to certain relations including ones deriving from the modular law, can be got as a tensor product in a semilattice sense of $L$ with the natural numbers. It also satisfies the Principle of Inclusion and Exclusion (in a form presented without subtraction).</p>
  • Analytic Topology in Mathematics and Computer Science

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