A note on partial numberings

The different behaviour of total and partial numberings with respect to the reducibility preorder is investigated. Partial numberings appear quite naturally in computability studies for topological spaces. The degrees of partial numberings form a distributive lattice which in the case of an infinite numbered set is neither complete nor contains a least element. Friedberg numberings are no longer minimal in this situation. Indeed, there is an infinite descending chain of non‐equivalent Friedberg numberings below every given numbering, as well as an uncountable antichain. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Positive undecidable numberings in the Ershov hierarchy

A sufficient condition is given under which an infinite computable family of Σ^-1 _a -sets has computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved for finite levels of the Ershov hierarchy in [1]. As a consequence, it is stated that the family of all Σ^-1 _a -sets has a computable positive undecidable numbering. In addition, for every ordinal notation a > 1, an infinite family of Σ^-1 _a -sets is constructed which possesses a computable positive numbering but has no computable Friedberg numberings. This answers the question of whether such families exist at any—finite or infinite—level of the Ershov hierarchy, which was originally raised by Badaev and Goncharov only for the finite levels bigger than 1.     

On the existence of universal numberings for finite families of d.c.e. sets

We investigate properties of universal numberings of finite families of d.c.e. sets. We show different cases of finite families of d.c.e. sets for which there is a universal numbering and for which there is not.     

Strong reducibility of partial numberings

A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice.In addition, Ershovs completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic.This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices.This research has partially been supported by INTAS under grant 00-499 Computability in Hierarchies and Topological Spaces.Mathematics Subject Classification (2000): 03D45     

Distributive lattices of numberings

We study into a semilattice of numberings generated by a given fixed numbering via operations of completion and taking least upper bounds. It is proved that, except for the trivial cases, this semilattice is an infinite distributive lattice every principal ideal in which is finite. The least upper and the greatest lower bounds in the semilattice are invariant under extensions in the semilattice of all numberings. Isomorphism types for the semilattices in question are in one-to-one correspondence with pairs of cardinals the first component of which is equal to the cardinality of a set of non-special elements, and the second — to the cardinality of a set of special elements, of the initial numbering. Supported by INTAS grant No. 00-429. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 83–102, January–February, 2007.     

Effectively closed sets and enumerations

An effectively closed set, or class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of classes and for the subclasses of decidable and of homogeneous classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends. Research partially supported by National Science Foundation grants DMS 0554841, 0532644 and 0652732.     

Positive Numberings of Families of Sets in the Ershov Hierarchy

It is proved that there exist infinitely many positive undecidable _n ^-1 -computable numberings of every infinite family that admits at least one _n ^-1 -computable numbering and contains either the empty set, for even n, or N for odd n.     

Computable numberings of families of low sets and Turing jumps in the Ershov hierarchy

If ν and μ are some Δ₂⁰-computable numberings of families of sets of the naturals then P(x,y) ⇔ ν(x)′ ≠ μ(y) is a Σ₂⁰-predicate. Deriving corollaries from this result, we obtain a sufficient condition for existence of a Δ₂⁰-computable numbering of the subfamily of all sets in a given family with the Turing jumps belonging to a fixed level of the Ershov hierarchy, and we deduce existence of a Σ_ω⁻¹-computable numbering of the family of all superlow sets.     

Decidable Computable A-Numberings

The article deals in the numbering theory for admissible sets, brought in sight in [1]. For models of two special classes, we resolve the problem of there being 1-1 computable numberings of the families of all computable sets and of all computable functions. In proofs, for the former case the role of finite objects is played by syntactic constructions, and for the latter — by finite subsets on hereditarily finite superstructures.     

The universal Lachlan semilattice without the greatest element

We deal with some upper semilattices of m-degrees and of numberings of finite families. It is proved that the semilattice of all c.e. m-degrees, from which the greatest element is removed, is isomorphic to the semilattice of simple m-degrees, the semilattice of hypersimple m-degrees, and the semilattice of Σ ₂ ⁰ -computable numberings of a finite family of Σ ₂ ⁰ -sets, which contains more than one element and does not contain elements that are comparable w.r.t. inclusion. Supported by the Grant Council (under RF President) for Young Russian Scientists via project MK-1820.2005.1. __________ Translated from Algebra i Logika, Vol. 46, No. 3, pp. 299–345, May–June, 2007.     

Positive Presentations of Families Relative to <Emphasis Type="Italic">e</Emphasis>-Oracles

We introduce the notion of A-numbering which generalizes the classical notion of numbering. All main attributes of classical numberings are carried over to the objects considered here. The problem is investigated of the existence of positive and decidable computable A-numberings for the natural families of sets e-reducible to a fixed set. We prove that, for every computable A-family containing an inclusion-greatest set, there also exists a positive computable A-numbering. Furthermore, for certain families we construct a decidable (and even single-valued) computable total A-numbering when A is a low set; we also consider a relativization containing all cases of total sets (this in fact corresponds to computability with a usual oracle).     

Completely unimodal numberings of a simple polytope

A completely unimodal numbering of the m vertices of a simple d-dimensional polytope is a numbering 0, 1, …,m−1 of the vertices such that on every k-dimensional face (2≤k≤d) there is exactly one local minimum (a vertex with no lower-numbered neighbors on that face). Such numberings are abstract objective functions in the sense of Adler and Saigal [1]. It is shown that a completely unimodal numbering of the vertices of a simple polytope induces a shelling of the facets of the dual simplicial polytope. The h-vector of the dual simplicial polytope is interpreted in terms of the numbering (with respect to using a local-improvement algorithm to locate the vertex numbered 0). In the case that the polytope is combinatorially equivalent to a d-dimensional cube, a ‘successor-tuple’ for each vertex is defined which carries the crucial information of the numbering for local-improvement algorithms. Combinatorial properties of these d-tuples are studied. Finally the running time of one particular local-improvement algorithm, the Random Algorithm, is studied for completely unimodal numberings of the d-cube. It is shown that for a certain class of numberings (which includes the example of Klee and Minty [8] showing that the simplex algorithm is not polynomial and all Hamiltonian saddle-free injective pseudo-Boolean functions [6]) this algorithm has expected running time that is at worst quadratic in the dimension d.     

Generalizations of Cantor's theorem in

A set x is Dedekind infinite if there is an injection from ω into x ; otherwise x is Dedekind finite. A set x is power Dedekind infinite if , the power set of x , is Dedekind infinite; otherwise x is power Dedekind finite. For a set x , let pdfin(x ) be the set of all power Dedekind finite subsets of x . In this paper, we prove in (without the axiom of choice) two generalizations of Cantor's theorem (i.e., the statement that for all sets x , there are no injections from into x ): The first one is that for all power Dedekind infinite sets x , there are no Dedekind finite to one maps from into pdfin(x ). The second one is that for all sets , if x is infinite and there is a power Dedekind finite to one map from y into x , then there are no surjections from y onto . We also obtain some related results.     

On a problem of Ishmukhametov

Given a d.c.e. degree d, consider the d.c.e. sets in d and the corresponding degrees of their Lachlan sets. Ishmukhametov provided a systematic investigation of such degrees, and proved that for a given d.c.e. degree d > 0, the class of its c.e. predecessors in which d is c.e., denoted as R[d], can consist of either just one element, or an interval of c.e. degrees. After this, Ishmukhametov asked whether there exists a d.c.e. degree d for which the class R[d] has no minimal element. We give a positive answer to this question.     

Some applications of computable one-one numberings

We present a simple proof of a Theorem of Khutoretskij on the number of incomparable one-one numberings of an r.e. family of r.e. sets. The proof directly generalizes to effective domains. In the second part, applying a Theorem of Goncharov, we show that for anyk there exist total recursive functions having exactlyk recursive isomorphism classes. Using a Theorem of Selivanov, it is shown that a certain notion of computability via gödelization is different from Lacombe's notion ofV-recursiveness. Finally, we discuss the complexity (w.r.t.T-degrees) of translating a Gödelnumbering into a direct sum of Friedbergnumberings.     

A colorful theorem on transversal lines to plane convex sets

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line. All authors are partially supported by CONACYT research grant 5040017.     

Density of the Medvedev lattice of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

The partial ordering of Medvedev reducibility restricted to the family of ⁰₁ classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a ⁰₁ class, which we call a ``c.e. separating class'. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes. Mathematics Subject Classification (2000): 03D30, 03D25     

Non‐discrete metrics in and some notions of finiteness

We show that (i) it is consistent with that there are infinite sets X on which every metric is discrete; (ii) the notion of real infinite is strictly stronger than that of metrically infinite; (iii) a set X is metrically infinite if and only if it is weakly Dedekind‐infinite if and only if the cardinality of the set of all metrically finite subsets of X is strictly less than the size of ; and (iv) an infinite set X is weakly Dedekind‐infinite if and only if has infinite towers if and only if X has countable partitions.     

Effective embeddings into strong degree structures

We show that any partial order with a Σ₃ enumeration can be effectively embedded into any partial order obtained by imposing a strong reducibility such as ≤_tt on the c. e. sets. As a consequence, we obtain that the partial orders that result from imposing a strong reducibility on the sets in a level of the Ershov hiearchy below ω + 1 are co‐embeddable.     

Menger's theorem for infinite graphs with ends

A well‐known conjecture of Erd?s states that given an infinite graph G and sets A, ? V(G), there exists a family of disjoint A ? B paths ?? together with an A ? B separator X consisting of a choice of one vertex from each path in ??. There is a natural extension of this conjecture in which A, B, and X may contain ends as well as vertices. We prove this extension by reducing it to the vertex version, which was recently proved by Aharoni and Berger. © 2005 Wiley Periodicals, Inc. J Graph Theory 50: 199–211, 2005