80 research outputs found
On minimal Hamming compatible distances
No full textA Hamming compatible metric is an integer-valued metric on the words of a finite alphabet
which agrees with the usual Hamming distance for words of equal length. We define a new
Hamming compatible metric and show this metric is minimal in the class of all
“well-behaved” Hamming compatible metrics. This gives a negative answer to a question
stated by Echi in his paper [O. Echi, Appl. Math. Sci. (Ruse) 3
(2009) 813–824.]
Spectral primal spaces
No full text Let [Formula: see text] be a mapping. Consider [Formula: see text] Then, according to Echi, [Formula: see text] is an Alexandroff topology. A topological space [Formula: see text] is called a primal space if its topology coincides with an [Formula: see text] for some mapping [Formula: see text]. We denote by [Formula: see text] the set of all fixed points of [Formula: see text], and [Formula: see text] the set of all periodic points of [Formula: see text]. The topology [Formula: see text] induces a preorder [Formula: see text] defined on [Formula: see text] by: [Formula: see text] if and only if [Formula: see text], for some integer [Formula: see text]. The main purpose of this paper is to provide necessary and sufficient algebraic conditions on the function [Formula: see text] in order to get [Formula: see text] (respectively, the one-point compactification of [Formula: see text]) a spectral topology. More precisely, we show the following results. (1) [Formula: see text] is spectral if and only if [Formula: see text] is a finite set and every chain in the ordered set [Formula: see text] is finite. (2) The one-point(Alexandroff) compactification of [Formula: see text] is a spectral topology if and only if [Formula: see text] and every nonempty chain of [Formula: see text] has a least element. (3) The poset [Formula: see text] is spectral if and only if every chain is finite. As an application the main theorem [12, Theorem 3. 5] of Echi–Naimi may be derived immediately from the general setting of the above results. </jats:p
ON THE SPECTRALIFICATION OF A HEMISPECTRAL SPACE
No full text An open subset U of a topological space X is called intersection compact open, or ICO, if for every compact open set Q of X, U ∩ Q is compact. A continuous map f of topological spaces will be called spectral if f-1 carries ICO sets to ICO sets. Call a topological space Xhemispectral, if the intersection of two ICO sets of X is an ICO. Let HSPEC be the category whose objects are hemispectral spaces and arrows spectral maps. Let SPEC be the full subcategory of HSPEC whose objects are spectral spaces. The main result of this paper proves that SPEC is a reflective subcategory of HSPEC. This gives a complete answer to Problem BST1 of "O. Echi, H. Marzougui and E. Salhi, Problems from the Bizerte–Sfax–Tunis seminar, in Open Problems in Topology II, ed. E. Pearl (Elsevier, 2007), pp. 669–674." </jats:p
The categories of flows of Set and Top
No full textAbstractFollowing John Kennison, a flow (or discrete dynamical system) in a category C is a couple (X,f), where X is an object of C and f:X→X is a morphism, called the iterator. If (A,f) and (B,g) are flows in C, then h:A→B is a morphism of flows from (A,f) to (B,g) if h∘f=g∘h. We let Flow(C) denote the resulting category of flows in C.This paper deals with Flow(Set) and Flow(Top), where Set and Top denote respectively the categories of sets and topological spaces.By a Gottschalk flow, we mean a flow (X,f) in Top satisfying the following conditions:(i)If x∈X is any almost periodic point of f, then the closure Of(x)¯ is a minimal set of f;(ii)All points in any minimal set of f are almost periodic points.As proven by Gottschalk, if X is a compact Hausdorff space and f:X→X is a continuous function, then (X,f) is a Gottschalk flow.In this paper, we prove that for any flow (X,f) of Set, there is a topology P(f) on X for which ((X,P(f)),f) is a Gottschalk flow in Top. This, actually, defines a covariant functor P from Flow(Set) into Flow(Top).The main result of this paper provides a characterization of spaces in the image of the functor P in order-theoretical terms.Some categorical properties of Flow(Set) and Flow(Top) are also given
Non-primitive words of the form
No full textLet p,q be two distinct primitive words. According to Lentin−Schützenberger [
On the product of primal spaces
No full textLet X be a set and f : X →X be a map. We denote by P(f) the topology dened on X whose closed sets are the subsets A of X with f(A) A⊆A topology on X is said to be a primal topology, if it is a P(f) for some map f. Our aim here is to characterize when the product of an arbitrary family of topological spaces is a primal space
The envelope of a subcategory in topology and group theory
No full textA collection of results are presented which are loosely centered
around the notion of reflective subcategory. For example, it is
shown that reflective subcategories are orthogonality classes,
that the morphisms orthogonal to a reflective subcategory are
precisely the morphisms inverted under the reflector, and that
each subcategory has a largest “envelope” in the ambient
category in which it is reflective. Moreover, known results
concerning the envelopes of the category of sober spaces, spectral
spaces, and jacspectral spaces, respectively, are summarized and
reproved. Finally, attention is focused on the envelopes of one-object
subcategories, and examples are considered in the category of groups
Spheres of Strings Under the Levenshtein Distance
No full textLet Σ be a nonempty set of characters, called an alphabet. The run-length encoding (RLE) algorithm processes any nonempty string u over Σ and produces two outputs: a k-tuple (b1,b2,…,bk), where each bi is a character and bi+1≠bi; and a corresponding k-tuple (q1,q2,…,qk) of positive integers, so that the original string can be reconstructed as u=b1q1b2q2…bkqk. The integer k is termed the run-length of u, and symbolized by ρ(u). By convention, we let ρ(ε)=0. In the Euclidean space (Rn,∥·∥2), the volume of a sphere is determined solely by the dimension n and the radius, following well-established formulas. However, for spheres of strings under the edit metric, the situation is more complex, and no general formulas have been identified. This work intended to show that the volume of the sphere SL(u,1), composed of all strings of Levenshtein distance 1 from u, is dependent on the specific structure of the “RLE-decomposition” of u. Notably, this volume equals (2l(u)+1)s−2l(u)−ρ(u), where ρ(u) represents the run-length of u and l(u) denotes its length (i.e., the number of characters in u). Given an integer p≥2, we present a partial result concerning the computation of the volume |SL(u,p)| in the specific case where the run-length ρ(u)=1. More precisely, for a fixed integer n≥1 and a character a∈Σ, we explicitly compute the volume of the Levenshtein sphere of radius p, centered at the string u=an. This case corresponds to the simplest run structure and serves as a foundational step toward understanding the general behavior of Levenshtein spheres
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