Showing papers on "Disjoint sets published in 1968"

On the lattice of recursively enumerable sets

Abstract: This paper presents some new theorems concerning recursively enumerable (r.e.) sets. The aim of the paper is to advance the search for a decision procedure for the elementary theory of r.e. sets. More precisely, an effective method is sought for deciding whether or not an arbitrary sentence formulated in the lower predicate calculus with sole relative symbol c is true of the r.e. sets. The main achievement of the paper is the characterisation of the hh-simple sets as those coinfinite r.e. sets whose r.e. supersets form a Boolean algebra. The reader is referred to Davis's book [1] for basic information about the partial recursive (p.r.) functions and about r.e. sets. Other background material required for a proper understanding of the present paper consists of [8], [3, Theorem 2], [10, Introduction and ?4], and [5] where the contributions have been listed in their natural order. We take the formulation of the lower predicate calculus given in Abraham Robinson [9]. Natural numbers are denoted by lower case Roman letters and sets of them by lower case Greek letters. The empty set is denoted by 0 and the set of all natural numbers by v. The complement of any set a is denoted by a'; a is called cofinite or coinfinite just if a' is finite or infinite respectively. For sets a, / we write aoc3 just if the set (a -3 u (3 a) is finite; otherwise we write aZ 3. By function we mean a map of some subset of v x v x **. x v into v; functions will be denoted by upper case Roman letters as will relations on the natural numbers. The informal logical signs used are v , &, ->, , (x), (Ex), -,> which are to be read as " or ", " and ", " implies ", "not", "for all x", "there exists x", "is equivalent to" respectively. Let ,' be a finite class of propositions; then s otherwise sup a is to be oo. The plan of the paper is as follows. In the first section we give a brief discussion of the elementary theory of r.e. sets and prove that its decision problem is of the same degree as that of the elementary theory of the lattice obtained by taking the equivalence classes of r.e. sets with respect to -. In ?2 we prove the main theorem which states: if a is an r.e. subset of an r.e. set / then either there exists a recursive subset 8 of:/ such that a u 8 = 3 or there exists a recursive sequence {8il of disjoint finite subsets of / such that 8i a is nonempty for all i. This theorem was inspired by

The point-arboricity of a graph

Abstract: The point-arboricity ρ(G) of a graphG is defined as the minimum number of subsets in a partition of the point set ofG so that each subset induces an acyclic subgraph Dually, the tuleity τ(G) is the maximum number of disjoint, point-induced, non-acyclic subgraphs contained inG Several results concerning these numbers are presented, among which are formulas for the point arboricity and tulgeity of the class of completen-partite graphs

Maximal separation theorems for convex sets

Abstract: : Most previous applications of separation theorems for convex sets have depended on rather crude separation results. Recent developments, however, indicate a need for more refined theorems, particularly for ones involving stronger types of separation. The stronger types of separation considered in this study include nice, open, closed, strict, and strong. The attempt to obtain separation theorems under minimal hypotheses suggests a search for maximal theorems, since each such theorem is, in a sense, a best possible result. Eighteen maximal theorems for disjoint nonempty convex subsets are derived, involving open, nice, strong and strict separation. (Author)

On sets of almost disjoint subsets of a set

Abstract: disjoint denumerable sets. In view of recent axiomatic results 4Pis independent of the usual axioms of set theory and the generalized continuumhypothesis. In § 4 we show that YE implies a certain unsolved problem of [3].In §5 we consider another question about sets of almost disjoint subsets of aset which was raised by F.

A note on compactifications and semi-normal spaces

Abstract: Recently Orrin Frink (see [2]) gave a neat internal characterization of Tychonoff or completely regular T spaces. This characterization was given in terms of the notion of a normal base for the closed sets of a space X . A normal base for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of . In a normal space the ring of closed sets is a normal base.

On equations with sets as unknowns.

Abstract: We shall present here a number of results in set theory concerning the decomposition of a set E in various ways as sum (union) of its subsets . These results have connection with problems on countably additive measure functions in abstract sets, but they may also bear on the problems of the axiomatics of set theory and generally on foundations of set theory itself . Some of these results employ the continuum hypothesis or the generalized continuum hypothesis. The several _problems which will be presented also put these hypotheses in a certain limelight . The impossibility of defining a countably additive measure for all subsets of a set of power of continuum (a measure which would vanish for subsets consisting of any single point) was first established with the use of the continuum hypothesis by Banach and Kuratowski .l Very shortly afterwards, one of us showed the impossibility of such a measure for subsets of a set of power NI without the use of any hypothesis.' The same result was shown there to hold for sets of higher powers, in fact, for all the accessible alephs . More recently, these results have been extended to a large class of inaccessibles as well . These results show that this "problem of measure" is closely related to fundamental problems concerning the role of axioms of set theory . Recent developments have further clarified these relations . Important results have been obtained by Scott, Solovay, Martin, and others . The proofs of these relations make use of the methods introduced by Paul Cohen in proving the independence of the continuum hypothesis. Both the results of Banach and Kuratowski and the stronger result of Ulam are obtained by exhibiting purely combinatorial schemata of decomposition of abstract sets with certain properties : B • and K • show a countable sequence of decompositions of a set of power of the continuum, each into countably many disjoint subsets so that, no matter how one takes a finite number of sets from each of these decompositions, the intersection of all these finite unions contains, at most, countably many points . Sierpinski3 generalized the B • and K • schema in the following way. There exists a sequence of decompositions into aleph disjoint sets, each so that if one is selected from any countably many of these (not necessarily all), the union of the selected sets gives the whole of the space, except perhaps for countably many points . Decomposition given by U • show, without the use of the continuum hypothesis, the following phenomenon . A set E of power NI can be decomposed countably many times into NI disjoint sets in the following way : A "matrix" of sets cam be constructed such that we have countably many rows and noncountably many columns . Sets in each row are disjoint. The anion of sets in any column gives the whole set E except for possibly countably many points. As is easy to see, the existence of such a decomposition (a sequence

The Algebra of Sets of Trees, k-Trees, and Other Configurations

Abstract: In linear graphs a commutative ring (Wang algebra) yields relations between sets of partial graphs such as trees, k -trees, cut sets, circuits, and paths. This algebra is defined, explored, and applied, resulting in a unified approach by which theorems long connected with Wang algebra are rederived and new theorems are obtained. Some scattered relations, previously found by the method of "derivatives," appear as natural and special results. Special stress is put on the generation of sets of partial graphs in graphs compounded by interconnecting disjoint graphs, or by methods of cutting up the given graph. Many new theorems are derived which simplify computations by splitting a given problem into several of smaller dimension.

On The Determination of Sets by Sets of Sums of Fixed Order

Abstract: The present investigation is based on two papers: “On the determination of numbers by their sums of a fixed order,” by J. L. Self ridge and E. G. Straus (4), and “On the determination of sets by the sets of sums of a certain order,” by B. Gordon, A. S. Fraenkel, and E. G. Straus (2). First of all, we explain the terms implicit in the above titles. Throughout these considerations we use the term “set” to mean “a totality having possible multiplicities,” so that two sets will be counted as equal if, and only if, they have the same elements with identical multiplicities. In the most general sense the term “numbers” of (4) can be replaced by “elements of any given torsioniree Abelian group.”

Norm-compact sets of representing measures

Abstract: Let K be a compact subset of the complex plane C and let p be a point of KO, the interior of K. Let R(K) be the uniform closure on K of the rational functions with poles off K and let Mp be the set of all positive measures X on AK, the topological boundary of K, with the property that f8K4dX =4)(p) for all 4) in R(K). (Such a measure is called a representing measure for evaluation at p.) In an effort to cast some light on the problem of putting analytic structure in the maximal ideal space of a function algebra, Bishop has conjectured in [1, problem 8, p. 347] that Mp is compact in the norm topology as a subset of the space measures on AK. Theorem 1 of this paper shows that this is indeed the case for many compact sets; however, Theorem 2 gives some necessary conditions that Mp be norm-compact and consequently provides a number of counterexamples to Bishop's conjecture. If the compact set has only a finite number of components in its complement, then Mp is norm-compact. This follows immediately from the fact that the linear span of the real measures that annihilate R(K) is finite-dimensional. (This is a consequence of a classical theorem of Walsh [6, p. 518].) However, when K has infinitely many complementary components this argument is no longer valid, for both Mp and the space of real annihilating measures maycontain infinitely many linearly independent elements. For example, let K be the set obtained by deleting from the closed unit disc a sequence Ci } of open subdiscs with disjoint closures whose centers lie on the positive real axis and increase to 1 and whose radii decrease to 0. Let ,u be harmonic measure on AK for p and let fi be the element of LZ(OK, ,u) such that for each g in L1(OK, ,u), faKgf,dg is the period about C, of the harmonic conjugate of the harmonic extension of g to KO. It is easily seen thatf1,f2, * * * are linearly independent and consequently that the representing measures X= (1 (filc)),g are linearly independent, where c,= jlfiljl. Nevertheless, Mp is normcompact for this and other compact sets. In order to prove this we must make an assumption about two subsets of OK; these two subsets are defined below. Let K be compact and let Co, Ci, * * * be the components of C-K. For each integer n, n=O, 1, 2, . . . , let F,, consist of those points of

Maximal fields disjoint from finite sets

Abstract: Let F be a subfield of an algebraically closed field A of characteristic 0, S a finite subset of A disjoint from F, K a subfield of A containing F and maximal with respect to disjointness from S, L a finite extension of K, and G=G(L/K) the group of automorphisms of L/K. Quigley [4] and McCarthy [1] obtained precise information about G in the case where S has one or two elements, respectively (they handled the characteristic p case also). Theorem 1 of this paper shows that there is some restriction on G in the general case. In particular (Theorem 2), G is solvable if S has at most twenty elements.

An integral representation for the product of spectral measures

Abstract: Let be a Hilbert space with inner product (•, •) and let E(•) and E0(•) be spectral measures in corresponding to self-adjoint operators and . In this paper we consider the set function ƒ(I × J) = E(I)E0(J) defined on the semiring of bounded rectangles, and obtain an integral representation for this set function for disjoint I, J under the hypotheses that H — H0 is a type of Carleman operator.

Harmonic differential with prescribed singularities

Abstract: 1. Throughout this paper we denote by R an open Riemann surface and by Ro a relatively compact subdomain of R with the relative boundary aRo consisting of a finite number of mutually disjoint closed analytic Jordan curves. The open set R, = R Po can be considered to be a neighborhood of the ideal boundary 3 of R. For the sake of simplicity, we denote by a the common relative boundary aRo = aR and we fix the orientation of a positively with respect to the domain Ro. A harmonic differential a defined on R1 = R1 u a is called a harmonic singularity

Nonlinear pseudo-boolean equations and inequalities

Abstract: Pseudo-Boolean equations, that is equations of the form $$ f\left( {{x_{\text{1}}},{\text{ }}.{\text{ }}..,{x_n}} \right) = 0$$ (1) , where f is a pseudo-Boolean function, have been studied by R. Fortet [1–3], P. Camion [2], and P. L. Hammer [3, 4]. These authors have suggested various methods for expressing the general solution in a parametric form.