f I-sets and decomposition of R I C-continuity

First, we introduce the notion of f _I-sets and investigate their properties in ideal topological spaces. Then, we also introduce the notions of R _I C-continuous, f _I-continuous and contra*-continuous functions and we show that a function f: (X,τ,I) to (Y,φ) is R _I C -continuous if and only if it is f _I-continuous and contra*-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

On γ-semi-open sets

Summary We introduce the concepts of γ-semi-open set, γ_s-set, γ^s-set, generalized γ_s-set, generalized γ^s-set, semi-T₁^γspace and semi-R₀^γspace by using γ-open-sets.     

Idealization of a decomposition theorem

In 1986, Tong [13] proved that a function f : (X,τ)→(Y,φ) is continuous if and only if it is α-continuous and A-continuous. We extend this decomposition of continuity in terms of ideals. First, we introduce the notions of regular-I-closed sets, A _I-sets and A _I -continuous functions in ideal topological spaces and investigate their properties. Then, we show that a function f : (X,τ,I)→(Y, φ) is continuous if and only if it is α-I-continuous and A _I-continuous. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Notion of Rank for Right Congruences on Semigroups

ABSTRACT

We introduce a new notion of rank for a semigroup S. The rank is associated with pairs (I,ρ), where ρ is a right congruence and I is a ρ-saturated right ideal. We allow I to be the empty set; in this case the rank of (?, ρ) is the Cantor-Bendixson rank of ρ in the lattice of right congruences of S, with respect to a topology we title the finite type topology. If all pairs have rank, then we say that S is ranked. Our notion of rank is intimately connected with chain conditions: every right Noetherian semigroup is ranked, and every ranked inverse semigroup is weakly right Noetherian.

Our interest in ranked semigroups stems from the study of the class ± b? _S of existentially closed S-sets over a right coherent monoid S. It is known that for such S the set of sentences in the language of S-sets that are true in every existentially closed S-set, that is, the theory T _S of ± b? _S , has the model theoretic property of being stable. Moreover, T _S is superstable if and only if S is weakly right Noetherian. In the present article, we show that T _S satisfies the stronger property of being totally transcendental if and only if S is ranked and weakly right Noetherian.     

Exact Pairs for Abstract Bounded Reducibilities

In an attempt to give a unified account of common properties of various resource bounded reducibilities, we introduce conditions on a binary relation ≤_r between subsets of the natural numbers, where ≤_r is meant as a resource bounded reducibility. The conditions are a formalization of basic features shared by most resource bounded reducibilities which can be found in the literature. As our main technical result, we show that these conditions imply a result about exact pairs which has been previously shown by Ambos-Spies [2] in a setting of polynomial time bounds: given some recursively presentable ≤_r-ideal I and some recursive ≤_r-hard set B for I which is not contained in I, there is some recursive set C, where B and C are an exact pair for I, that is, I is equal to the intersection of the lower ≤_r-cones of B and C, where C is not in I. In particular, if the relation ≤_r is in addition transitive and there are least sets, then every recursive set which is not in the least degree is half of a minimal pair of recursive sets.     

A construction of a perfect set of Euler tours of K2k + I

In this article we define a perfect set of Euler tours of K_2k + I, I a 1-factor of K_2k, to be a set of Euler tours of K_2k + I that partition the 2-paths of K_2k, with the added condition that if ab ∈ I, then each Euler tour contains either the digon a b a or b a b. We prove for all k > 1 that K_2k + I has a perfect set of Euler tours, and, as a corollary, that L(K_2k) has a Hamilton decomposition. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 183–211, 1998     

On δ-I- open sets and decomposition of α-I-continuity

We introduce the notions of δ-I- open sets and semi δ-I-continuous functions in ideal topological spaces and investigate some of their properties. Additionally, we obtain decompositions of semi-I-continuous functions and α-I-continuous functions by using δ-I-open sets. This revised version was published online in June 2006 with corrections to the Cover Date.     

I m -Quasi Upward Sets,with Their Best Approximations

The main purpose of this article is to introduce particular subsets of R ^I , which are not necessarily convex, and we call them I _m -quasi upward, or I _m -quasi downward. We show that these sets can be translated to downward or upward sets. We introduce the connection of these sets with downward and upward subsets of R ^I , and discuss the best approximation of these sets. Also we introduce embedded I _m -quasi upward and embedded I _m -quasi downward subsets of a normed space X.     

Structure of stability sets and asymptotic stability sets of families of linear differential systems with parameter multiplying the derivative: I

We consider families of linear differential systems continuously depending on a real parameter. The stability (respectively, asymptotic stability) set of such a family is defined as the set of all values of the parameter for which the corresponding systems in the family are stable (respectively, asymptotically stable). We show that a set on the real axis is the stability (respectively, asymptotic stability) set of some family of this kind if and only if it is an F _σ -set (respectively, an F _σδ -set). For families in which the parameter occurs only as a factor multiplying the matrix of the system, their stability sets are exactly F _σ -sets containing zero on the real line. The asymptotic stability sets of such families will be described in the second part of the present paper.     

Some new sets and decompositions of A I-R-continuity, α-I-continuity, continuity via idealization

The concepts of I-R closed set, A _I-R-set, α _I M ₁-set, α _I M ₂-set, α _I N ₁-set, α _I N ₂-set, α _I N ₃-set, α _I N ₄-set and α _I N ₅-set are introduced via idealization. New decompositions of some weaker forms of continuity are obtained by using these sets.     

S-Regular and S-Normal Spaces

In this paper s-regular and s-normal spaces are characterized using semi-T₀-identification spaces, topological sums of s-regular and s-normal spaces are examined, and the relationships between s-regular, s-normal, and other separation axioms are further examined.     

Exact equiconsistency results for Δ 3 1 -sets of reals

Summary We improve a theorem of Raisonnier by showing that Cons(ZFC+every ₂ ¹ -set of reals in Lebesgue measurable+every ₂ ¹ -set of reals isK -regular) implies Cons(ZFC+there exists an inaccessible cardinal). We construct, fromL, a model where every ₃ ¹ -sets of reals is Lebesgue measurable, has the property of Baire, and every ₂ ¹ -set of reals isK -regular. We prove that if there exists a _n+1 ¹ unbounded filter on , then there exists a nonK -regular ₂ ¹ -subset.The author would like to thank the Basic Research Foundation (Israel Academy of Science) for partially supporting this research     

Further remarks on enlargements of generalized topologies

The concepts of κ _μ -open set and enlargement κ of a generalized topology μ were introduced by Császár?[5]. In this paper, we introduce the concept of weak κ _μ -open sets induced by an enlargement κ of a generalized topology μ, and study some basic properties. We also introduce the concept of κ-regular open sets, and investigate the characterizations of weak κ _μ -open sets in terms of κ-regular open sets.     

Embeddings with multiple regularity

We introduce (k,l)-regular maps, which generalize two previously studied classes of maps: affinely k-regular maps and totally skew embeddings. We exhibit some explicit examples and obtain bounds on the least dimension of a Euclidean space into which a manifold can be embedded by a (k,l)-regular map. The problem can be regarded as an extension of embedding theory to embeddings with certain non-degeneracy conditions imposed, and is related to approximation theory.     

Definability of Geometric Properties in Algebraically Closed Fields

We prove that there exists no sentence F of the language of rings with an extra binary predicat I₂ satisfying the following property: for every definable set X ? ?², X is connected if and only if (?, X) ? F, where I₂ is interpreted by X. We conjecture that the same result holds for closed subset of ?². We prove some results motivated by this conjecture.     

Unified characterizations for modifications ofR 0 andR 1 topological spaces

Cammaroto and Noiri [14] introduced a separation axiom calledm-R ₀ in anm-space (X, m). In this paper, we introduce the notion ofm-R ₁ spaces and offer many characterizations ofm-R ₀ (resp.m-R ₁) spaces which enable us to obtain unified characterizations of separation axiomsR ₀, semi-R ₀, pre-R ₀,α-R ₀,δ-semiR ₀, (δ, p)-R ₀ (resp.R ₁, semi-R ₁, pre-R ₁,α-R ₁,δ-semiR ₁, (δ, p)-R ₁).     

Topological finite-determinacy of functions with non-isolated singularities

We introduce the concept of topological finite-determinacy for germs of analytic functions within a fixed ideal I, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We prove the following statement which generalizes classical results of Thom and Varchenko: let A be the complement in the ideal I of the space of germs whose topological type remains unchanged under a deformation within the ideal that only modifies sufficiently large order terms of the Taylor expansion. Then A has infinite codimension in I in a suitable sense. We also prove the existence of generic topological types of families of germs of I parametrized by an irreducible analytic set.     

Bregman distances without coercive condition: suns,Chebyshev sets and Klee sets

ABSTRACT

In a real Banach space X, we introduce for a non-empty set C in X the notion of suns in the sense of Bregman distances and show that C is such a sun if and only if C is convex. Also, we give some necessary and sufficient conditions for a compact set to be the Klee set, extending corresponding results on the Euclidean space.     

The structure of invertible substitutions on a three-letter alphabet

We study the structure of invertible substitutions on three-letter alphabet. We show that there exists a finite set of invertible substitutions such that any invertible substitution can be written as I_wσ₁σ₂σ_k, where I_w is the inner automorphism associated with w, and for 1jk. As a consequence, M is the matrix of an invertible substitution if and only if it is a finite product of non-negative elementary matrices.