Hereditary Coreflective Subcategories in Certain Categories of Abelian Semitopological Groups

Let A be an epireflective subcategory of the category of all semitopological groups that consists only of abelian groups. We describe maximal hereditary coreflective subcategories of A that are not bicoreflective in A in the case that the A -reflection of the discrete group of integers is a finite cyclic group, the group of integers with a topology that is not T 0 , or the group of integers with the topology generated by its subgroups of the form p n , where n ∈ N , p ∈ P and P is a given set of prime numbers.

Probability Integral as a Linearization

In fuzzified probability theory, a classical probability space (Ω, A, p) is replaced by a generalized probability space (Ω, ℳ(A), ∫(.) dp), where ℳ(A) is the set of all measurable functions into [0,1] and ∫(.)dp is the probability integral with respect to p. Our paper is devoted to the transition from p to ∫(.) dp. The transition is supported by the following categorical argument: there is a minimal category and its epireflective subcategory such that A and ℳ(A) are objects, probability measures and probability integrals are morphisms, ℳ(A) is the epireflection of A, ∫(.) dp is the corresponding unique extension of p, and ℳ(A) carries the initial structure with respect to probability integrals. We discuss reasons why the fuzzy random events are modeled by ℳ(A) equipped with pointwise partial order, pointwise Łukasiewicz operations (logic) and pointwise sequential convergence. Each probability measure induces on classical random events an additive linear preorder which helps making decisions. We show that probability integrals can be characterized as the additive linearizations on fuzzy random events, i.e., sequentially continuous maps, preserving order, top and bottom elements.

On D-posets of fuzzy sets

D-posets of fuzzy sets constitute a natural simple mathematical structure in which relevant notions of generalized probability theory can be formalized. We present a classification of D-posets leading to a hierarchy of distinguished subcategories of D-posets related to probability and study their relationships. This contributes to a better understanding of the transition from classical probability theory to fuzzy probability theory. In particular, we describe the transition from the Boolean cogenerator {0, 1} to the fuzzy cogenerator [0, 1] and prove that the generated Łukasiewicz tribes form an epireflective subcategory of the bold algebras.

Certain extensions and factorizations of α-complete homomorphisms in archimedean lattice-ordered groups

As a consequence of general principles, we add to the array of ‘hulls’ in the category Arch (of archimedean ℓ-groups with ℓ-homomorphisms) and in its non-full subcategory W (whose objects have distinguished weak order unit, whose morphisms preserve the unit). The following discussion refers to either Arch or W. Let α be an infinite cardinal number or ∞, let Homα; denote the class of α-complete homomorphisms, and let R be a full epireflective subcategory with reflections denoted rG: G → rG. Then for each G, there is rαG ∈ Homα (G, R) such that for each ϕ ∈ Homα (G, R), there is unique with . Moreover if every rG is an essential embedding, then, for every α and every G, rαG = rG, and every Homα. If and R consists of all epicomplete objects, then every Homw1. For α = ∞, and for any R, every Hom∞.

Simplicity of Categories Defined by Symmetry Axioms

We consider two generalizations R0w and R0 of the usual symmetry axiom for topological spaces to arbitrary closure spaces and convergence spaces. It is known that the two properties coincide on Top and define a non-simple subcategory. We show that R0W defines a simple subcategory of closure spaces and R0 a non-simple one. The last negative result follows from the stronger statement that every epireflective subcategory of R0 Conv containing all T1 regular topological spaces is not simple. Similar theorems are shown for the topological categories Fil and Mer.