scholarly journals An Ulm-type classification theorem for equivalence relations in Solovay model

1997 ◽  
Vol 62 (4) ◽  
pp. 1333-1351 ◽  
Author(s):  
Vladimir Kanovei
Keyword(s):  
Equivalence Relation ◽  
Binary Sequences ◽  
Classification Theorem ◽  
Equivalence Relations ◽  
Equality Relation ◽  
Solovay Model ◽  
Type Classification

AbstractWe prove that in the Solovay model, every OD equivalence relation, Ε, over the reals, either admits an OD reduction to the equality relation on the set of all countable (of length < ω1) binary sequences, or continuously embeds Ε0, the Vitali equivalence.If Ε is a (resp. ) relation then the reduction above can be chosen in the class of all Δ1 (resp. Δ2) functions.The proofs are based on a topology generated by OD sets.

Classifying positive equivalence relations

1983 ◽  
Vol 48 (3) ◽  
pp. 529-538 ◽  
Author(s):  
Claudio Bernardi ◽  
Andrea Sorbi
Keyword(s):  
Topological Space ◽  
Equivalence Relation ◽  
Recursive Function ◽  
Equivalence Classes ◽  
Point Of View ◽  
Classification Theorem ◽  
Equivalence Relations ◽  
Natural Numbers ◽  
Uniformity Property

AbstractGiven two (positive) equivalence relations ~1, ~2 on the set ω of natural numbers, we say that ~1 is m-reducible to ~2 if there exists a total recursive function h such that for every x, y ∈ ω, we have x ~1y iff hx ~2hy. We prove that the equivalence relation induced in ω by a positive precomplete numeration is complete with respect to this reducibility (and, moreover, a “uniformity property” holds). This result allows us to state a classification theorem for positive equivalence relations (Theorem 2). We show that there exist nonisomorphic positive equivalence relations which are complete with respect to the above reducibility; in particular, we discuss the provable equivalence of a strong enough theory: this relation is complete with respect to reducibility but it does not correspond to a precomplete numeration.From this fact we deduce that an equivalence relation on ω can be strongly represented by a formula (see Definition 8) iff it is positive. At last, we interpret the situation from a topological point of view. Among other things, we generalize a result of Visser by showing that the topological space corresponding to a partition in e.i. sets is irreducible and we prove that the set of equivalence classes of true sentences is dense in the Lindenbaum algebra of the theory.

Fundamental relations and identities of fuzzy hyperalgebras

2021 ◽  
pp. 1-10
Author(s):  
Narjes Firouzkouhi ◽  
Abbas Amini ◽  
Chun Cheng ◽  
Mehdi Soleymani ◽  
Bijan Davvaz
Keyword(s):  
Universal Algebra ◽  
Equivalence Relation ◽  
Polynomial Function ◽  
Equivalence Relations ◽  
Fundamental Relation ◽  
Term Function ◽  
Biological Phenomena ◽  
Definition Of

Inspired by fuzzy hyperalgebras and fuzzy polynomial function (term function), some homomorphism properties of fundamental relation on fuzzy hyperalgebras are conveyed. The obtained relations of fuzzy hyperalgebra are utilized for certain applications, i.e., biological phenomena and genetics along with some elucidatory examples presenting various aspects of fuzzy hyperalgebras. Then, by considering the definition of identities (weak and strong) as a class of fuzzy polynomial function, the smallest equivalence relation (fundamental relation) is obtained which is an important tool for fuzzy hyperalgebraic systems. Through the characterization of these equivalence relations of a fuzzy hyperalgebra, we assign the smallest equivalence relation α i 1 i 2 ∗ on a fuzzy hyperalgebra via identities where the factor hyperalgebra is a universal algebra. We extend and improve the identities on fuzzy hyperalgebras and characterize the smallest equivalence relation α J ∗ on the set of strong identities.

scholarly journals The absorption theorem for affable equivalence relations

2008 ◽  
Vol 28 (5) ◽  
pp. 1509-1531 ◽  
Author(s):  
THIERRY GIORDANO ◽  
HIROKI MATUI ◽  
IAN F. PUTNAM ◽  
CHRISTIAN F. SKAU
Keyword(s):  
Dynamical Systems ◽  
Equivalence Relation ◽  
Closed Subset ◽  
Cantor Set ◽  
Equivalence Relations ◽  
Orbit Structure ◽  
Orbit Equivalent ◽  
Finite Set ◽  
Significant Generalization

AbstractWe prove a result about extension of a minimal AF-equivalence relation R on the Cantor set X, the extension being ‘small’ in the sense that we modify R on a thin closed subset Y of X. We show that the resulting extended equivalence relation S is orbit equivalent to the original R, and so, in particular, S is affable. Even in the simplest case—when Y is a finite set—this result is highly non-trivial. The result itself—called the absorption theorem—is a powerful and crucial tool for the study of the orbit structure of minimal ℤn-actions on the Cantor set, see Remark 4.8. The absorption theorem is a significant generalization of the main theorem proved in Giordano et al [Affable equivalence relations and orbit structure of Cantor dynamical systems. Ergod. Th. & Dynam. Sys.24 (2004), 441–475] . However, we shall need a few key results from the above paper in order to prove the absorption theorem.

On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type

2013 ◽  
Vol 56 (1) ◽  
pp. 136-147
Author(s):  
Radu-Bogdan Munteanu
Keyword(s):  
Equivalence Relation ◽  
Product Type ◽  
Bratteli Diagram ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Property A

AbstractProduct type equivalence relations are hyperfinitemeasured equivalence relations, which, up to orbit equivalence, are generated by product type odometer actions. We give a concrete example of a hyperfinite equivalence relation of non-product type, which is the tail equivalence on a Bratteli diagram. In order to show that the equivalence relation constructed is not of product type we will use a criterion called property A. This property, introduced by Krieger for non-singular transformations, is defined directly for hyperfinite equivalence relations in this paper.

An axiomatization of the logic with the rough quantifier

1991 ◽  
Vol 56 (2) ◽  
pp. 608-617 ◽  
Author(s):  
Michał Krynicki ◽  
Hans-Peter Tuschik
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Order Structure ◽  
Partition Model ◽  
Weak Model ◽  
First Order ◽  
Basic Properties ◽  
Generalized Quantifier ◽  
Order Language ◽  
The Right

We consider the language L(Q), where L is a countable first-order language and Q is an additional generalized quantifier. A weak model for L(Q) is a pair 〈, q〉 where is a first-order structure for L and q is a family of subsets of its universe. In case that q is the set of classes of some equivalence relation the weak model 〈, q〉 is called a partition model. The interpretation of Q in partition models was studied by Szczerba [3], who was inspired by Pawlak's paper [2]. The corresponding set of tautologies in L(Q) is called rough logic. In the following we will give a set of axioms of rough logic and prove its completeness. Rough logic is designed for creating partition models.The partition models are the weak models arising from equivalence relations. For the basic properties of the logic of weak models the reader is referred to Keisler's paper [1]. In a weak model 〈, q〉 the formulas of L(Q) are interpreted as usual with the additional clause for the quantifier Q: 〈, q〉 ⊨ Qx φ(x) iff there is some X ∊ q such that 〈, q〉 ⊨ φ(a) for all a ∊ X.In case X satisfies the right side of the above equivalence we say that X is contained in φ(x) or, equivalently, φ(x) contains X.

Glimm-Effros for coanalytic equivalence relations

2009 ◽  
Vol 74 (2) ◽  
pp. 402-422 ◽  
Author(s):  
Greg Hjorth
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations

AbstractAssuming every real has a sharp, we prove that for any equivalence relation either Borel reduces E0 or in a manner allows the assignment of bounded subsets of ω1 as complete invariants.

scholarly journals Stability of products of equivalence relations

2018 ◽  
Vol 154 (9) ◽  
pp. 2005-2019 ◽  
Author(s):  
Amine Marrakchi
Keyword(s):  
Probability Measure ◽  
Direct Product ◽  
Equivalence Relation ◽  
Equivalence Relations ◽  
Similar Question ◽  
Stable Equivalence ◽  
Local Characterization

An ergodic probability measure preserving (p.m.p.) equivalence relation ${\mathcal{R}}$ is said to be stable if ${\mathcal{R}}\cong {\mathcal{R}}\times {\mathcal{R}}_{0}$ where ${\mathcal{R}}_{0}$ is the unique hyperfinite ergodic type $\text{II}_{1}$ equivalence relation. We prove that a direct product ${\mathcal{R}}\times {\mathcal{S}}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components ${\mathcal{R}}$ or ${\mathcal{S}}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\text{II}_{1}$ factors is also discussed and some partial results are given.

scholarly journals FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (4) ◽  
pp. 1225-1254 ◽  
Author(s):  
RUSSELL MILLER ◽  
KENG MENG NG
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Arithmetical Hierarchy ◽  
Natural Equivalence ◽  
Image Position ◽  
The Way

AbstractWe introduce the notion of finitary computable reducibility on equivalence relations on the domainω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be${\rm{\Pi }}_{n + 2}^0$-complete under computable reducibility, we show that, for everyn, there does exist a natural equivalence relation which is${\rm{\Pi }}_{n + 2}^0$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.

scholarly journals Actions of amenable equivalence relations on CAT(0) fields

2012 ◽  
Vol 34 (1) ◽  
pp. 21-54 ◽  
Author(s):  
MARTIN ANDEREGG ◽  
PHILIPPE HENRY
Keyword(s):  
Equivalence Relation ◽  
Metric Spaces ◽  
Amenable Group ◽  
Equivalence Relations ◽  
General Notion ◽  
Rigidity Result ◽  
Finite Dimensional ◽  
Borel Field

AbstractWe present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

Using Rough Set to Develop the Building Materials Management System

2013 ◽  
Vol 281 ◽  
pp. 658-663
Author(s):  
Jian Xu
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Equivalence Relation ◽  
Management System ◽  
Building Materials ◽  
Rough Set Theory ◽  
Equivalence Relations ◽  
Materials Management ◽  
Cpu Time ◽  
Materials Used

Materials used in buildings are collectively referred to building materials. Building materials can be divided into structural materials, decoration materials and some special materials. Rough set theory is built on the basis of the classification mechanism, it will be classified understand in a particular space on the equivalence relation, equivalence relations constitute the division of space. The paper puts forward using rough set to develop the building materials management system. The experiment shows the CPU Time in the attribute numbers, indicating that rough set is superior to FCA in building materials management system.

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