This paper analyses the notion of a positivity relation of Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finall...

There are several different standard conventions for specifying a locale, and the first decision is which convention to use. PhET specifies a locale using codes defined by the International Standards Organization (ISO). ISO 639-1 defines language codes [3], while ISO 3166-1 defines country codes [4]. While there are other standards, these are the most commonly-used definitions. An example of a ...

We present specification languages that naturally capture exactly the regular and ω-regular properties that are stutter invariant. Our specification languages are variants of the classical regular expressions and of the core of PSL, a temporal logic, which is widely used in industry and which extends the classical linear-time temporal logic LTL by semi-extended regular expressions.

We give two constructions for semi-regular relative difference sets (RDSs) in groups whose order is not a prime power, where the order u of the forbidden subgroup is greater than 2. No such RDSs were previously known. We use examples from the first construction to produce semi-regular RDSs in groups whose order can contain more than two distinct prime factors. For u greater than 2 these are the...

The exchange and storage of XML data is becoming increasingly important. In contrast to conventional semistructured data 4, 1], the labels in a document-oriented representation such as XML are ordered. Furthermore, regular expressions (DTDs) describe the horizontal (and vertical) structure of the data. Traditional query languages for semi-structured data ignore the horizontal order and are ther...

Searching for an object model is considered to be one of the most desirable and yet difficult searches. The problem is made difficult by the presence of clutter in a scene, as well as the fact that objects may be imaged under different lighting conditions. We have developed a feature localization scheme that finds a set of locales in an image. Our object search method matches image locales with...

We describe the isomorphism class of the torus centralizing a regular, semi-simple, stable conjugacy class in a simply-connected, semi-simple group. Let k be a field, and let G be a semi-simple, simply-connected algebraic group, which is quasi-split over k. The theory of semi-simple conjugacy classes in G is well understood, from work of Steinberg [S] and Kottwitz [K]. Any semi-simple conjugacy...

This paper investigates aspects of measure and randomness in the context of locale theory (point-free topology). We prove that every measure (σ-continuous valuation) μ, on the σ-frame of opens of a fitted σ-locale X, extends to a measure on the lattice of all σ-sublocales of X (Theorem 1). Furthermore, when μ is a finite measure with μ(X) = M , the σ-locale X has a smallest σ-sublocale of measu...