Π11 Borel sets

1989 ◽  
Vol 54 (3) ◽  
pp. 915-920 ◽  
Author(s):  
Alexander S. Kechris ◽  
David Marker ◽  
Ramez L. Sami
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Final Section ◽  
Borel Sets ◽  
Abstract Setting ◽  
Recursive Theory ◽  
Scott Rank ◽  
Countable Models

The results in this paper were motivated by the following question of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal α such that all models of T have Scott rank below α? If Martin's conjecture is true for T then α ≤ ω · 2.Our goal was to look at this problem in a more abstract setting. Let E be a equivalence relation on ωω with countably many classes each of which is Borel. What can you say about the least α such that each equivalence class is ? This problem is closely related to the following question. Suppose X ⊆ ωω is and Borel. What can you say about the least α such that X is ?In §1 we answer these questions in ZFC. In §2 we give more informative answers under the added assumptions V = L or -determinacy. The final section contains related results on the separation of sets by Borel sets.Our notation is standard. The reader may consult Moschovakis [5] for undefined terms.Some of these results were proved first by Sami and rediscovered by Kechris and Marker.

Diameter of a (0, 1)-Matrix*

1968 ◽  
Vol 11 (2) ◽  
pp. 285-288
Author(s):  
U.S.R. Murty
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Inner Product ◽  
A Chain

Let A be an m×n (0, l )-matrix. Let C1, C2, …, Cn denote its columns. A sequence of distinct columns is said to form a chain if the inner product of and (for 1 ≤ t ≤ k-l) is at least one. k-1 is called the length of the chain and this chain is said to connect are said to be connected. As can be easily seen, connectedness is an equivalence relation on the set of columns. A matrix is called connected if all its columns belong to the same equivalence class. If Ci and Cj belong to the same equivalence class, then s(Ci, Cj) will denote the length of the shortest chain between Ci and Cj We define the distance between any two columns Ci and Cj to be denoted by d(Ci, Cj), in the following manner.

scholarly journals Bipolar Fuzzy Relations

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1044 ◽  
Author(s):  
Jeong-Gon Lee ◽  
Kul Hur
Keyword(s):  
Equivalence Class ◽  
Level Set ◽  
Equivalence Relation ◽  
Level Sets ◽  
Equivalence Classes ◽  
Fuzzy Relation ◽  
Fuzzy Relations ◽  
Transitive Relation ◽  
Fuzzy Partition

We introduce the concepts of a bipolar fuzzy reflexive, symmetric, and transitive relation. We study bipolar fuzzy analogues of many results concerning relationships between ordinary reflexive, symmetric, and transitive relations. Next, we define the concepts of a bipolar fuzzy equivalence class and a bipolar fuzzy partition, and we prove that the set of all bipolar fuzzy equivalence classes is a bipolar fuzzy partition and that the bipolar fuzzy equivalence relation is induced by a bipolar fuzzy partition. Finally, we define an ( a , b ) -level set of a bipolar fuzzy relation and investigate some relationships between bipolar fuzzy relations and their ( a , b ) -level sets.

scholarly journals Uniformity, universality, and computability theory

2017 ◽  
Vol 17 (01) ◽  
pp. 1750003
Author(s):  
Andrew S. Marks
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Technical Tool ◽  
Free Products ◽  
Borel Sets ◽  
Borel Equivalence Relations ◽  
Strengthened Form ◽  
Shift Action ◽  
Natural Classes ◽  
Main Technical Tool

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.

scholarly journals Periodic Coefficients and Random Fibonacci Sequences

10.37236/3204 ◽  
2013 ◽  
Vol 20 (4) ◽  
Author(s):  
Karyn McLellan
Keyword(s):  
Growth Rate ◽  
Equivalence Class ◽  
Equivalence Relation ◽  
Cycle Length ◽  
Fibonacci Sequence ◽  
Periodic Coefficients ◽  
Matrix Products ◽  
Almost All ◽  
Fibonacci Sequences ◽  
Periodic Cycles

The random Fibonacci sequence is defined by $t_1 = t_2 = 1$ and $t_n = \pm t_{n-1} + t_{n-2}$, for $n \geq 3$, where each $\pm$ sign is chosen at random with probability $P(+) = P(-) = \frac{1}{2}$. Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate $1.13198824\ldots$. We will consider what happens to random Fibonacci sequences when we remove the randomness; specifically, we will choose coefficients which belong to the set $\{1, -1\}$ and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length.

scholarly journals Dual skew Heyting almost distributive lattices

2019 ◽  
Vol 4 (1) ◽  
pp. 151-162 ◽  
Author(s):  
Berhanu Assaye ◽  
Mihret Alamneh ◽  
Lakshmi Narayan Mishra ◽  
Yeshiwas Mebrat
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Congruence Relation ◽  
Heyting Algebra ◽  
Congruence Class ◽  
Distributive Lattices ◽  
Lattice Image ◽  
Principal Ideals ◽  
Maximal Lattice

AbstractIn this paper, we introduce the concept of dual skew Heyting almost distributive lattices (dual skew HADLs) and characterise it in terms of dual HADL. We define an equivalence relation θ on a dual skew HADL L and prove that θ is a congruence relation on the equivalence class [x]θ so that each congruence class is a maximal rectangular subalgebra and the quotient [y]θ/θ is a maximal lattice image of [x]θ for any y ∈ [x]θ. Moreover, we show that if the set PI (L) of all the principal ideals of an ADL L with 0 is a dual skew Heyting algebra then L becomes a dual skew HADL. Further we present different conditions on which an ADL with 0 becomes a dual skew HADL.

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

On an Equivalence Relation for Free Mappings Embeddeable in a Flow

2003 ◽  
Vol 13 (07) ◽  
pp. 1911-1915 ◽  
Author(s):  
Z. Leśniak
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Equivalence Classes

We consider an equivalence relation for a given free mapping f of the plane. Under the assumption that f is embeddable in a flow {ft : t ∈ R} we describe the structure of equivalence classes of the relation. Finally, we prove that f restricted to each equivalence class is a Sperner homeomorphism.

Sequential discreteness and clopen-I-Boolean classes

1987 ◽  
Vol 52 (1) ◽  
pp. 232-242
Author(s):  
Randall Dougherty
Keyword(s):  
Final Section ◽  
Baire Space ◽  
Boolean Operations ◽  
Index Set ◽  
Borel Sets ◽  
Borel Hierarchy ◽  
Wadge Reducibility ◽  
Analytic Sets ◽  
Converse Results

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean.The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.

Vanishing Borel sets

1998 ◽  
Vol 63 (1) ◽  
pp. 262-268
Author(s):  
Kenneth Schilling
Keyword(s):  
Final Section ◽  
Borel Function ◽  
The Other ◽  
Counting Measure ◽  
Borel Sets ◽  
Zero Measure

Henson and Ross [1] answered the question of when two hyperfinite sets A, B in an ℵ1-saturated nonstandard universe are bijective by a Borel function: precisely when ∣A∣/∣B∣ ≈ 1. Živaljević [5] generalized this result to nonvanishing Borel sets. He defined a set to be nonvanishing if it is Loeb-measurable and has finite, non-zero measure with respect to some Loeb counting measure. He then showed that two nonvanishing Borel sets are Borel bijective just in case they have the same finite, non-zero measure with respect to some Loeb counting measure.Here we shall complete the cycle, for Borel sets at least. For N ∈ *N, let λN be the internal counting measure given by λN(A) = ∣A∣/N for A internal. Then for vanishing Loeb-measurable sets B, it is natural to consider the Dedekind cut (BL, BR) on *N consisting of those N for which B has 0λN-measure infinity and zero, respectively. We show that, for all vanishing Borel sets B, B and BL are Borel bijective. It follows that vanishing Borel sets B and C are Borel bijective if, and only if, BL = CL. Combined with Živaljević's result, we can characterize when arbitrary Borel sets are Borel bijective: precisely when they have the same measure with respect to all Loeb counting measures.In the final section, we generalize in a similar way results of [2] and [5] to characterize when two Borel sets are bijective by a countably determined function: precisely when, for all N, one has 0λN-measure 0 if and only if the other also has 0λN-measure 0.

Borel complexity of isomorphism between quotient Boolean algebras

2008 ◽  
Vol 73 (4) ◽  
pp. 1328-1340
Author(s):  
Su Gao ◽  
Michael Ray Oliver
Keyword(s):  
Equivalence Relation ◽  
Boolean Algebras ◽  
Equivalence Relations ◽  
Borel Set ◽  
Borel Sets ◽  
Opposite Case ◽  
Borel Complexity ◽  
Borel Ideal ◽  
The One ◽  
Ergodic Equivalence Relations

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

Sign in / Sign up

Export Citation Format

Share Document