Giles Richardson

Lattice-valued spaces: Fuzzy convergence

Abstract An alternative set of axioms is given for the study of lattice-valued convergence spaces. These axioms extend the notion of a probabilistic convergence space in the special case when the lattice is L = [ 0 , 1 ] . The category described here is shown to be topological, cartesian closed, and extensional.

Regularity: Lattice-valued Cauchy spaces

A category of lattice-valued Cauchy spaces is defined, and its properties are investigated. The relationship between this category and the recent work by Jager is presented. The notion of regularity in the Cauchy space context is the primary emphasis here. In particular, the category consisting of the regular Cauchy spaces is shown to be bireflective in the category of all lattice-valued Cauchy spaces having Cauchy-continuous maps as its morphisms. Moreover, completions are also investigated.

Lattice-valued convergence: Diagonal axioms

Diagonal axioms are defined in the category whose objects consist of all the lattice-valued convergence spaces. When the latter is linearly ordered, a diagonal condition is given which characterizes those objects in the category that are determined by a probabilistic convergence space which is topological.

Subcategories of Filter Tower Spaces

The concept of a convergence tower space, or equivalently, a convergence approach space is formulated here in the context of a Cauchy setting in order to include a completion theory. Subcategories of filter tower spaces are defined in terms of axioms involving a general t-norm, T, in order to include a broad range of spaces. A T-regular sequence for a filter tower space is defined and, moreover, it is shown that the category of T-regular objects is a bireflective subcategory of all filter tower spaces. A completion theory for subcategories of filter tower spaces is given.

Modifications: Lattice-valued structures

Several important topological concepts such as regularity, local compactness, and local boundedness are investigated in the category of lattice-valued convergence spaces. These definitions coincide with the usual ones in familiar categories.

Regularity in fuzzy convergence spaces

The notions of regularity and weak regularity are introduced in the category of fuzzy convergence spaces, and each is shown to be an extension of regularity with respect to the embedding functor from the category of convergence spaces. Moreover, the category whose objects are weakly regular fuzzy spaces is topological and thus initial and final regular structures exist. The relationship between regularity and a diagonal condition is given.

Extensions for Filter Spaces

A family of extensions for a filter space is defined and its properties, including completeness and total boundedness, are investigated.

Lattice-valued fuzzy interior operators

A category of lattice-valued fuzzy interior operator spaces is defined and studied. Axioms are given in order for this category to be isomorphic to the category whose objects consist of all the stratified, lattice-valued, pretopological convergence spaces.

T -REGULAR PROBABILISTIC CONVERGENCE SPACES

A probabilistic convergence structure assigns a probability that a given filter converges to a given element of the space. The role of the t -norm (triangle norm) in the study of regularity of probabilistic convergence spaces is investigated. Given a probabilistic convergence space, there exists a finest T -regular space which is coarser than the given space, and is referred to as the ‘ T -regular modification’. Moreover, for each probabilistic convergence space, there is a sequence of spaces, indexed by nonnegative ordinals, whose first term is the given space and whose last term is its T -regular modification. The T -regular modification is illustrated in the example involving ‘convergence with probability λ’ for several t -norms. Suitable function space structures in terms of a given t -norm are also considered.

Convergence Approach Spaces : Actions

Properties of continuous actions and pseudoquotients are studied in the category of convergence approach spaces. Invariance properties of continuous actions on convergence approach spaces are given. It is shown that the formation of pseudoquotient spaces is idempotent. Function space actions are also investigated.