Reflection and forcing in E-recursion theory

E-recursive enumerability is compared via forcing to Σ₁ definability. It is shown that for every countable E-closed ordinal κ there is a set of reals, X, so that L_κ[X] is the least E-closed structure over X.     

Generalized Frattini subgroups as coradicals of groups

The paper deals with finite solvable groups only. It is established that the class of all regular subgroup m-functors coincides with the class of all X-abnormal m-functors, where X ranges over all subclasses of the class of all primitive groups. The properties of the lattice of all regular subgroup m-functors are studied and the atoms and coatoms of this lattice are described. It is proved that the generalized Frattini subgroup of G corresponding to a regularm-functor coincides with the X-coradical of G for some R ₀-closed class X.     

A generalisation of the functional approach to compactness

A topological space X is compact iff the projection π:X×Y→Y is closed for any space Y. Taking this as a definition and then asking that π maps α-closed subspaces of X×Y onto β-closed subspaces of Y, for different closures α and β, extends the notion of compactness to include also examples of “asymmetric compactness” pursued in the article.Categorical closure operators and a so-called “functional approach to general topology” are employed to define and prove fundamental properties of compact objects and proper maps in this generalised setting.     

Inadmissible forcing

A structure is E-closed if it is closed under all partial E-recursive functions from V into V, a set theoretic extension of Kleene's partial recursive functions of finite type in the normal case. Let L(κ) be E-closed and ∑₁ inadmissible. Then L(κ) has reflection properties useful in the study of generic extensions of L(κ). Every set generic extension of L(κ) via countably closed forcing conditions is E-closed. A class generic construction shows: if L(κ) is countable, and inside L(κ) the greatest cardinal gc(κ), has uncountable cofinality, then there exists a T ⊆ gc(κ) such that L(κ, T) = E(T), the least E-closed set with T as a member. A partial converse is obtained via a selection theorem that implies E(X) is ∑₁ admissible when X is a set of ordinals and the greatest cardinal in the sense of E(X) has countable cofinality in E(X).     

Finite groups with ℱ-subnormal conditions

Let ? be a subgroup-closed saturated formation. A finite group G is called an ?_pc-group provided that each subgroup X of G is ?-subabnormal in the ?-subnormal closure of X in G. Let ?_pc be the class of all ?_pc-groups. We study some properties of ? _pc-groups and describe the structure of ?_pc-groups when ? is the class of all soluble π-closed groups, where π is a given nonempty set of prime numbers.     

A unified theory of generalized closed sets in weak structures

A new kind of sets called generalized w-closed (briefly gw-closed) sets is introduced and studied in a topological space by using the concept of weak structures introduced by á.?Császár in?[6]. The class of all gw-closed sets is strictly larger than the class of all w-closed sets. Furthermore, g-closed sets (in the sense of N.?Levine?[17]) is a special type of gw-closed sets in a topological space. Some of their properties are investigated. Finally, some characterizations of w-regular and w-normal spaces have been given.     

Unification Approach to the Separation Axioms Between T 0 and Completely Hausdorff

The aim of this paper is to introduce a new weak separation axiom that generalizes the separation properties between T ₁ and completely Hausdorff. We call a topological space (X, τ) a T _κ,ξ-space if every compact subset of X with cardinality ≦ κ is ξ-closed, where ξ is a general closure operator. We concentrate our attention mostly on two new concepts: kd-spaces and T _1/3-spaces.     

On generalized CS-modules

An S-closed submodule of a module M is a submodule N for which M/N is nonsingular. A module M is called a generalized CS-module (or briefly, GCS-module) if any S-closed submodule N of M is a direct summand of M. Any homomorphic image of a GCS-module is also a GCS-module. Any direct sum of a singular (uniform) module and a semi-simple module is a GCS-module. All nonsingular right R-modules are projective if and only if all right R-modules are GCS-modules.     

S(n)-θ-closed spaces

A P-space is called P-θ-closed if it is θ-closed in every P-space in which it is embedded. S(n)-θ-closed spaces are characterized in terms of special covers and filters, and a categorical property of S(n)-θ-closed spaces is established. Examples are given to distinguish S(n)-θ-closed and S(n)-closed spaces and to show that the property of being S(n)-θ-closed is not productive.     

Neighbourhoods of transitive graphs and GRR's

Let X be a vertex-transitive graph with complement $\text{X}$. We show that if both N, the neighbourhood of a vertex in X, and $\text{N}$, the neighbourhood of a vertex in $\text{X}$, are disconnected, then either X is isomorphic to K₃ × K₃ or both N and $\text{N}$ contain isolated vertices. We characterize the graphs which satisfy this last condition and show in consequence that they admit automorphisms of the form (12)(34). It follows that if X is a GRR for some graph G then at least one of N and $\text{N}$ is connected. (X is said to be a graphical regular representation, or GRR, for G if its automorphism group is isomorphic to G and acts regularly on its vertices.) Using this result we determine those groups generated by their involutions which do not have a GRR. The largest such group has order 18. As a corollary we conclude that all non-abelian simple groups have GRR's.     

Nearly <Emphasis Type="Italic">H</Emphasis>-closed spaces

For a subset M of a topological space X, the θ-closure {ie626-01} M is defined as the set of all x ∈ X such that any closed neighborhood of x intersects M. A Urysohn space X is said to be U-closed if, whenever X ∪ {ξ} is a Urysohn space obtained from X by adding one point ξ, the point ξ is isolated in X ∪ {ξ}. The θ-closure operator is applied to study compactness-type properties of (weakly) U-and H-closed and closed-hereditarily U-and H-closed spaces. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 48, General Topology, 2007.     

Camina pairs that are not <Emphasis Type="Italic">p</Emphasis>-closed

We show for every prime p that there exists a Camina pair (G, N), where N is a p-group and G is not p-closed.     

A module as a torsion-free cover

LetR be a ring, and let (ℐ, ℱ) be an hereditary torsion theory of leftR-modules. An epimorphism ψ:M→X is called a torsion-free cover ofX if (1)M∈ ℱ, (2) every homomorphism from a torsion-free module intoX can be factored throughM, and (3) ker ψ contains no nonzero ℐ -closed submodules ofM. Conditions onM andN are studied to determine when the natural mapsM→M/N andQ(M)→Q(M)/N are torsion-free covers, whenQ(M) is the localization ofM with respect to (ℐ, ℱ). IfM→M/N is a torsion-free cover andM is projective, thenN⊆radM. Consequently, the concepts of projective cover and torsion-free cover coincide in some interesting cases.     

On a sequence of integral operators on weighted L p spaces

Continuing some investigations started in previous papers, we introduce and study a sequence of multidimensional positive integral operators which generalize the Gauss-Weierstrass operators. We show that this sequence is an approximation process in some classes of weighted L ^p spaces on ? ^N , N ≥ 1. Estimates of the rate of convergence are also obtained. Our mean tool is a Korovkin-type theorem which we establish in the context of L ^p (X, µ) spaces, X being a locally compact Hausdorff space and µ a regular positive Borel measure on X. Several examples are explicitly indicated as well.     

Banach Spaces with the PC Property

A Banach space X possesses the PC (point of continuity) property if for any w-closed bounded subset A ? X the identity map (A,w)→(A, ∥ ? ∥) has a point of continuity (w is the weak topology in X). We deduce some criteria for Banach spaces to have the PC property and describe (for dual Banach spaces) relationships between spaces possessing the PC property and spaces possessing the RN or the WRN property.

     

Upper semicontinuous fuzzy sets and applications

Let (X, τ) be a generalized topological space of type $\text{V}$_α (see A. Appert and K. Fan, “Espaces topologiques intermédiares,” Herman, Paris, 1951) and (L, ?) be a complete Brouwerian lattice such that the dual lattice of (L, ?) is also Brouwerian. We prove that every upper semicontinuous L-fuzzy subset of X can be represented by a τ-closed random set. As an important application we obtain a fuzzification of measurable spaces as well as of topological spaces. In particular a concept of measurable (open) L-fuzzy sets is developed.     

On*-operfect sets and α-*-closed sets

The main purpose of this paper is to introduce *-operfect, τ*-clopen, α-*-closed, strongly α-*-closed and pre-*-closed sets. We compare them and obtain a diagram to show their relationships among these sets and related sets.     

Relative ranks of Lipschitz mappings on countable discrete metric spaces

Let X be a countable discrete metric space and let XX denote the family of all functions on X. In this article, we consider the problem of finding the least cardinality of a subset A of XX such that every element of XX is a finite composition of elements of A and Lipschitz functions on X. It follows from a classical theorem of Sierpiński that such an A either has size at most 2 or is uncountable.We show that if X contains a Cauchy sequence or a sufficiently separated, in some sense, subspace, then |A|≤1. On the other hand, we give several results relating |A| to the cardinal d; defined as the minimum cardinality of a dominating family for NN. In particular, we give a condition on the metric of X under which |A|≥d holds and a further condition that implies |A|≤d. Examples satisfying both of these conditions include all subsets of Nk and the sequence of partial sums of the harmonic series with the usual euclidean metric.To conclude, we show that if X is any countable discrete subset of the real numbers R with the usual euclidean metric, then |A|=1 or almost always, in the sense of Baire category, |A|=d.