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Journal articles on the topic 'Equivalence relation'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 June 2025

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1

DOUCHA, MICHAL. "Fσ EQUIVALENCE RELATIONS AND LAVER FORCING". Journal of Symbolic Logic 79, № 2 (2014): 644–53. http://dx.doi.org/10.1017/jsl.2013.32.

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AbstractFollowing the topic of the book Canonical Ramsey Theory on Polish Spaces by V. Kanovei, M. Sabok, and J. Zapletal we study Borel equivalences on Laver trees. We prove that equivalence relations Borel reducible to an equivalence relation on 2ω given by some FσP-ideal on ω can be canonized to the full equivalence relation or to the identity relation.This has several consequences, e.g., Silver type dichotomy for the Laver ideal and equivalences Borel reducible to equivalence relations given by FσP-ideals.

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2

Marks, Andrew S. "Uniformity, universality, and computability theory." Journal of Mathematical Logic 17, no. 01 (2017): 1750003. http://dx.doi.org/10.1142/s0219061317500039.

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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.

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3

Molberg, Mats. "AF-equivalence relations." MATHEMATICA SCANDINAVICA 99, no. 2 (2006): 247. http://dx.doi.org/10.7146/math.scand.a-15011.

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Let $R=\lim\limits_{\longrightarrow}R_{n}$ be the inductive limit of an ascending sequence of étale finite equivalence relations $R_n$ on the zero-dimensional space $X$. We prove that $R$ is an AF-equivalence relation.

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4

Solecki, Sławomir. "Actions of non-compact and non-locally compact Polish groups." Journal of Symbolic Logic 65, no. 4 (2000): 1881–94. http://dx.doi.org/10.2307/2695084.

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AbstractWe show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces.

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5

Friedman, Sy-David, and Luca Motto Ros. "Analytic equivalence relations and bi-embeddability." Journal of Symbolic Logic 76, no. 1 (2011): 243–66. http://dx.doi.org/10.2178/jsl/1294170999.

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AbstractLouveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ) is far from complete (see [5, 2]).In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

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6

Munteanu, Radu-Bogdan. "On Constructing Ergodic Hyperfinite Equivalence Relations of Non-Product Type." Canadian Mathematical Bulletin 56, no. 1 (2013): 136–47. http://dx.doi.org/10.4153/cmb-2011-132-4.

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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.

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7

Carroll, Jeffrey S. "Maximal r.e. equivalence relations." Journal of Symbolic Logic 55, no. 3 (1990): 1048–58. http://dx.doi.org/10.2307/2274473.

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AbstractThe lattice of r.e. equivalence relations has not been carefully examined even though r.e. equivalence relations have proved useful in logic. A maximal r.e. equivalence relation has the expected lattice theoretic definition. It is proved that, in every pair of r.e. nonrecursive Turing degrees, there exist maximal r.e. equivalence relations which intersect trivially. This is, so far, unique among r.e. submodel lattices.

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8

Rosendal, Christian. "Cofinal families of Borel equivalence relations and quasiorders." Journal of Symbolic Logic 70, no. 4 (2005): 1325–40. http://dx.doi.org/10.2178/jsl/1129642127.

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AbstractFamilies of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.

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9

Mingyi, Zhang, Li Danning, and Zhang Ying. "An Approach to Generation of Decision Rules." JUCS - Journal of Universal Computer Science 16, no. (1) (2010): 140–58. https://doi.org/10.3217/jucs-016-01-0140.

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Classical classification and clustering based on equivalence relations are very important tools in decision-making. An equivalence relation is usually determined by properties of objects in a given domain. When making decision, anything that can be spoken about in the subject position of a natural sentence is an object, properties of which are fundamental elements of the knowledge of the given domain. This gives the possibility of representing the concept related to a given domain. In general, the information about a set of the objects is uncertain or incomplete. Various approaches representing uncertainty of a concept were proposed. In particular, Zadeh?s fuzzy set theory and Pawlak?s rough set theory have been most influential on this research field. Zadeh characterizes uncertainty of a concept by introducing a membership function and a similarity (fuzzy equivalence) relation of a set of objects. Pawlak then characterizes uncertainty of a concept by union of some equivalence classes of an equivalence relation. As one of particular important and widely used binary relations, equivalence relation plays a fundamental role in classification, clustering, pattern recognition, polling, automata, learning, control inference and natural language understanding, etc. An equivalence relation is a binary relation with reflexivity, symmetry and transitivity. However, in many real situations, it is not sufficient to consider equivalence relations only. In fact, a lot of relations determined by the attributes of objects do not satisfy transitivity. In particular, information obtained from a domain of objects is not transitive, when we make decision based on properties of objects. Moreover, the information about symmetry of a relation is mostly uncertain. So, it is needed to approximately make decision and reasoning by indistinct concepts. This provokes us to explore a new class of relations, so-called class of fuzzy semi-equivalence relations. In this paper we introduce the notion of fuzzy semi-equivalence relations and study its properties. In particular, a constructive method of fuzzy semi-equivalence classes is presented. Applying it we present approaches to the fuzzyfication of indistinct concepts approximated by fuzzy relative and semi-equivalence classes, respectively. And an application of the fuzzy semi-equivalence relation theory to generate decision rules is outlined.

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10

Gupta, K. C., and R. K. Gupta. "Fuzzy equivalence relation redefined." Fuzzy Sets and Systems 79, no. 2 (1996): 227–33. http://dx.doi.org/10.1016/0165-0114(95)00155-7.

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11

Marrakchi, Amine. "Stability of products of equivalence relations." Compositio Mathematica 154, no. 9 (2018): 2005–19. http://dx.doi.org/10.1112/s0010437x18007388.

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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.

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12

MILLER, RUSSELL, and KENG MENG NG. "FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS." Journal of Symbolic Logic 81, no. 4 (2016): 1225–54. http://dx.doi.org/10.1017/jsl.2016.23.

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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.

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13

SHIBUYA, TETSUO, and AKIRA YASUHARA. "SELF Ck-MOVE, QUASI SELF Ck-MOVE AND THE CONWAY POTENTIAL FUNCTION FOR LINKS." Journal of Knot Theory and Its Ramifications 13, no. 07 (2004): 877–93. http://dx.doi.org/10.1142/s0218216504003500.

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Nakanishi and Shibuya gave a relation between link homotopy and quasi self delta-equivalence. And they also gave a necessary condition for two links to be self delta-equivalent by using the multivariable Alexander polynomial. Link homotopy and quasi self delta-equivalence are also called self C1-equivalence and quasi self C2-equivalence respectively. In this paper, we generalize their results. In Sec. 1, we give a relation between self Ck-equivalence and quasi self Ck+1-equivalence. In Secs. 2 and 3, we give necessary conditions for two links to be self Ck-equivalent by using the multivariable Conway potential function and the Conway polynomial respectively.

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14

FOKINA, EKATERINA, BAKHADYR KHOUSSAINOV, PAVEL SEMUKHIN, and DANIEL TURETSKY. "LINEAR ORDERS REALIZED BY C.E. EQUIVALENCE RELATIONS." Journal of Symbolic Logic 81, no. 2 (2016): 463–82. http://dx.doi.org/10.1017/jsl.2015.11.

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AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$(or equivalently, the relationE)realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure ($\omega /E$; ⊴) is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.

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15

Feldman, J., C. E. Sutherland, and R. J. Zimmer. "Subrelations of ergodic equivalence relations." Ergodic Theory and Dynamical Systems 9, no. 2 (1989): 239–69. http://dx.doi.org/10.1017/s0143385700004958.

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AbstractWe introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a groupQwhich serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) ofQon it; in the case whereQis amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

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16

Gao, Su, and Michael Ray Oliver. "Borel complexity of isomorphism between quotient Boolean algebras." Journal of Symbolic Logic 73, no. 4 (2008): 1328–40. http://dx.doi.org/10.2178/jsl/1230396922.

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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.

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17

Buchholz, Peter. "Exact performance equivalence: An equivalence relation for stochastic automata." Theoretical Computer Science 215, no. 1-2 (1999): 263–87. http://dx.doi.org/10.1016/s0304-3975(98)00169-8.

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18

Firouzkouhi, Narjes, Abbas Amini, Chun Cheng, Mehdi Soleymani, and Bijan Davvaz. "Fundamental relations and identities of fuzzy hyperalgebras." Journal of Intelligent & Fuzzy Systems 41, no. 1 (2021): 2265–74. http://dx.doi.org/10.3233/jifs-210994.

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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.

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19

DING, LONGYUN. "ON EQUIVALENCE RELATIONS GENERATED BY SCHAUDER BASES." Journal of Symbolic Logic 82, no. 4 (2017): 1459–81. http://dx.doi.org/10.1017/jsl.2017.67.

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AbstractIn this article, a notion of Schauder equivalence relation ℝℕ/L is introduced, where L is a linear subspace of ℝℕ and the unit vectors form a Schauder basis of L. The main theorem is to show that the following conditions are equivalent:(1) the unit vector basis is boundedly complete;(2) L is a Fσ in ℝℕ;(3) ℝℕ/L is Borel reducible to ℓ∞.We show that any Schauder equivalence relation generalized by a basis of ℓ2 is Borel bireducible to ℝℕ/ℓ2 itself, but it is not true for bases of c0 or ℓ1. Furthermore, among all Schauder equivalence relations generated by sequences in c0, we find the minimum and the maximum elements with respect to Borel reducibility.

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20

Pedersen, Pernille Ladegaard, and Mette Bjerre. "Two conceptions of fraction equivalence." Educational Studies in Mathematics 107, no. 1 (2021): 135–57. http://dx.doi.org/10.1007/s10649-021-10030-7.

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AbstractIn this study, we present a mathematical analysis distinguishing two conceptions of equivalence: proportional equivalence and unit equivalence. These two conceptions have distinct meanings in relation to equivalent fractions: one is grounded in proportionality, while the other is grounded in equal wholes. We argue that (a) the distinction of equivalence gives a unified framework of equal fractions that has not previously been described in the literature; (b) a conceptual understanding of both fraction equivalences is integral to understanding rational numbers; and (c) knowledge of both conceptions of equivalence is important for developing a conceptual understanding of fraction arithmetic. Past research has largely overlooked the distinction between the two types of equivalence. However, this may provide an important foundation for central topics that build on equivalence, and a better understanding of these two types of equivalence may support a more flexible understanding of fractions. Last, we propose future directions for teaching equivalence in mathematics.

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21

Austin*, Homer W., and Harel Barzilai. "Classroom note: An equivalence relation." International Journal of Mathematical Education in Science and Technology 34, no. 6 (2003): 919–22. http://dx.doi.org/10.1080/00207390310001595528.

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22

Andretta, Alessandro, Riccardo Camerlo, and Greg Hjorth. "Conjugacy equivalence relation on subgroups." Fundamenta Mathematicae 167, no. 3 (2001): 189–212. http://dx.doi.org/10.4064/fm167-3-1.

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23

Krupski, Mikołaj. "On the t-equivalence relation." Topology and its Applications 160, no. 2 (2013): 368–73. http://dx.doi.org/10.1016/j.topol.2012.11.015.

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24

Liu, Andy. "Is Parallelism an Equivalence Relation?" College Mathematics Journal 42, no. 5 (2011): 372. http://dx.doi.org/10.4169/college.math.j.42.5.372.

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25

Bacher, Roland. "An Equivalence Relation on {0,1}*." European Journal of Combinatorics 21, no. 7 (2000): 853–64. http://dx.doi.org/10.1006/eujc.2000.0394.

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26

Kim, Ki Hang, and Fred W. Roush. "On strong shift equivalence over a Boolean semiring." Ergodic Theory and Dynamical Systems 6, no. 1 (1986): 81–97. http://dx.doi.org/10.1017/s0143385700003308.

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AbstractShift equivalence is the relation between A, B that there exists S, R, n > 0 with RA = BR, AS = SB, SR = An, RS = Bn. Strong shift equivalence is the equivalence relation generated by these equations with n = 1. We prove that for many Boolean matrices strong shift equivalence is characterized by shift equivalence and a trace condition. However, we also show that if A is strongly shift equivalent to B, then there exists a homomorphism from an iterated directed edge graph of A to the graph of B preserving the traces of powers. This yields results on colourings of iterated directed edge graphs and might distinguish new strong equivalence classes.

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27

Bataray, Büşra, and Çetin Camcı. "Applications of Equivalent Curves to Ruled Surfaces." International Electronic Journal of Geometry 18, no. 1 (2025): 135–42. https://doi.org/10.36890/iejg.1528951.

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In this paper, the characterization of equivalent curves in E^3 is used to define ruled surfaces whose base curves are equivalent curves and to examine the relationships between them. At the same time, an equivalence relation for ruled surfaces is obtained. The equivalence classes resulting from this relation are studied. It is concluded that all ruled surfaces in the equivalence class of a developable surface are developable. Thus, a method is established to obtain an infinite ruled surface from a ruled surface. Finally, a new method is given to obtain a developable surface.

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28

Lee, Jeong-Gon, and Kul Hur. "Bipolar Fuzzy Relations." Mathematics 7, no. 11 (2019): 1044. http://dx.doi.org/10.3390/math7111044.

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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.

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29

Adams, Scot. "Trees and amenable equivalence relations." Ergodic Theory and Dynamical Systems 10, no. 1 (1990): 1–14. http://dx.doi.org/10.1017/s0143385700005368.

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AbstractLet R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a ‘treeing’ of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.

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Eric, M.Gitonga, W. Musundi Sammy, and M. Nzimbi Benerd. "METRIC EQUIVALENCE AS AN ALMOST SIMILARITY PROPERTY." Journal of Progressive Research in Mathematics 12, no. 4 (2017): 2030–38. https://doi.org/10.5281/zenodo.3974906.

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Various results that relate to almost similarity and other classes of operators such as isometry, normal, unitary and compact operators have been extensively discussed. It has been shown that if operators S and T are unitarily equivalent, then S is almost similar to T. Similarly, it has been shown that if operators A and B are such that A is almost similar to B and if A is Hermitian, then A and B are said to be unitarily equivalent. Metric equivalence property which is a new relation in operator theory has drawn much attention from mathematicians in the recent past. Two operators S and T are unitarily equivalent if they are metrically equivalent projections. It has been shown that if operators S and T are unitarily equivalent, then S is metrically equivalent to T. However, there is no literature that has been shown for the conditions under which metric equivalence and almost similarity coincide. In this paper we will therefore strive to establish the equivalence relation between metric equivalence property and almost similarity relation. To achieve this, properties of invertible operators, normal operators, similar operators, unitarily operators as well as projection and selfadjoint operators will be employed.

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31

Farzan, Azadeh, and Umang Mathur. "Coarser Equivalences for Causal Concurrency." Proceedings of the ACM on Programming Languages 8, POPL (2024): 911–41. http://dx.doi.org/10.1145/3632873.

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Trace theory (formulated by Mazurkiewicz in 1987) is a principled framework for defining equivalence relations for concurrent program runs based on a commutativity relation over the set of atomic steps taken by individual program threads. Its simplicity, elegance, and algorithmic efficiency makes it useful in many different contexts including program verification and testing. It is well-understood that the larger the equivalence classes are, the more benefits they would bring to the algorithms and applications that use them. In this paper, we study relaxations of trace equivalence with the goal of maintaining its algorithmic advantages. We first prove that the largest appropriate relaxation of trace equivalence, an equivalence relation that preserves the order of steps taken by each thread and what write operation each read operation observes, does not yield efficient algorithms. Specifically, we prove a linear space lower bound for the problem of checking, in a streaming setting, if two arbitrary steps of a concurrent program run are causally concurrent (i.e. they can be reordered in an equivalent run) or causally ordered (i.e. they always appear in the same order in all equivalent runs). The same problem can be decided in constant space for trace equivalence. Next, we propose a new commutativity-based notion of equivalence called grain equivalence that is strictly more relaxed than trace equivalence, and yet yields a constant space algorithm for the same problem. This notion of equivalence uses commutativity of grains , which are sequences of atomic steps, in addition to the standard commutativity from trace theory. We study the two distinct cases when the grains are contiguous subwords of the input program run and when they are not, formulate the precise definition of causal concurrency in each case, and show that they can be decided in constant space , despite being strict relaxations of the notion of causal concurrency based on trace equivalence.

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32

Levitt, Gilbert. "On the cost of generating an equivalence relation." Ergodic Theory and Dynamical Systems 15, no. 6 (1995): 1173–81. http://dx.doi.org/10.1017/s0143385700009846.

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AbstractGiven a measure-preserving equivalence relation R with countable classes, we study relations between the properties of R and metric invariants. We give applications to pseudogroups of measure-preserving homeomorphisms.

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33

Sakai, Hiroshi. "Effective Procedures for Handling Possible Equivalence Relations in Non-deterministic Information Systems." Fundamenta Informaticae 48, no. 4 (2001): 343–62. https://doi.org/10.3233/fun-2001-48403.

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This paper presents a tool for handling equivalence relations in non-deterministic information systems. Some applications of equivalence relations are also shown. In a deterministic information system, it is possible to define an equivalence relation for any set of attributes. However, in a non-deterministic information system, some kinds of equivalence relations which we call possible equivalence relations are definable. This paper proposes two effective procedures producing all possible equivalence relations for any non-deterministic information system and any set of attributes. The details of algorithms, the implementation of algorithms and applications to the rule extraction, etc. are presented.

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34

Rasuli, Rasul. "Norms over intuitionistic fuzzy congruence relations on rings." Notes on Intuitionistic Fuzzy Sets 27, no. 3 (2021): 51–68. http://dx.doi.org/10.7546/nifs.2021.27.3.51-68.

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In this paper, by using norms, we define the concept of intuitionistic fuzzy equivalence relations and intuitionistic fuzzy congruence relations on ring R and we investigate some assertions. Also we define intuitionistic fuzzy ideals of ring R under norms and compare this with fuzzy equivalence relation and fuzzy congruence relation on ring R such that we define new introduced ring.

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35

Ko, Sang-Ki, and Yo-Sub Han. "Left is Better Than Right for Reducing Nondeterminism of NFAs." International Journal of Foundations of Computer Science 32, no. 05 (2021): 531–50. http://dx.doi.org/10.1142/s0129054121410069.

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We study the NFA reductions by invariant equivalences and preorders. It is well-known that the NFA minimization problem is PSPACE-complete. Therefore, there have been many approaches to reduce the size of NFAs in low polynomial time by computing invariant equivalence or preorder relation and merging the states within same equivalence class. Here we consider the nondeterminism reduction of NFAs by invariant equivalences and preorders. We, in particular, show that computing equivalence and preorder relation from the left is more useful than the right for reducing the degree of nondeterminism in NFAs. We also present experimental evidence for showing that NFA reduction from the left achieves the better reduction of nondeterminism than reduction from the right.

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Shavarovskii, B. Z. "Toeplitz Matrices in the Problem of Semiscalar Equivalence of Second-Order Polynomial Matrices." International Journal of Analysis 2017 (October 26, 2017): 1–14. http://dx.doi.org/10.1155/2017/6701078.

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We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices.

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Panasiuk, Igor. "Problematyka tłumaczenia w aspekcie komunikacji interkulturowej." Język. Religia. Tożsamość. 1, no. 25 (2022): 85–101. http://dx.doi.org/10.5604/01.3001.0016.0995.

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This paper deals with the positioning Lacunae Theory in translation theory from the point of views of Russian ethnopsycholinguistics, as well as at applying the lacunae model in the process of translation. The application of Lacunae Theory in translation theory consists in its complementing of the contentious Theory of Equivalence. The equivalence relation in the framework of a context is specified by its approximal character (Wills, Albrecht). What is approximal here is merely the semantic dimension of a lacuna. Such circumstance leads to the establishment of the relation between the two inversely proportional terms: the greater the semantic extent of an equivalence relation between the lexical meanings to be compared or translated, the lower is the semantic intensity of the lacuna frequently used to express in context the approximal similarity (equality) of such lexical meanings. On the other hand, greater semantic intensity of the lacuna indicates lower equality of the equivalents to be compared. Lacuna and equivalence mutually complement one another. Here, lacuna and equivalence are two sides of the same coin, with the lacuna serving as the starting point for the semantic assessment of an equivalence relation.

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Hjorth, Greg. "Bi-Borel reducibility of essentially countable Borel equivalence relations." Journal of Symbolic Logic 70, no. 3 (2005): 979–92. http://dx.doi.org/10.2178/jsl/1122038924.

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This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we haveThe proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the followingMessy Fact 0.2. There are countable Borel equivalence relationssuch that:(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;(iii) if f: Xx → Xxis a measurable reduction ofExto itself then(iv)is a standard Borel space on which the projection functionis Borel and the equivalence relation Ê given byif and only ifx = x′ andzExz′ is Borel;(V)is Borel.We first prove the theorem granted this messy fact. We then prove the fact.(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.

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Eshkaftaki, Ali, and Noha Eftekhari. "Characterization of two-sided order preserving of convex majorization on lp(I)." Filomat 31, no. 18 (2017): 5703–10. http://dx.doi.org/10.2298/fil1718703b.

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In this paper, we consider an equivalence relation ~c on lp(I), which is said to be ?convex equivalent? for p ? [1,+?) and a nonempty set I. We characterize the structure of all bounded linear operators T : lp(I) ? lp(I) that strongly preserve the convex equivalence relation. We prove that the rows of the operator which preserve convex equivalent, belong to l1(I): Also, we show that any bounded linear operators T : lp(I) ? lp(I) which preserve convex equivalent, also preserve convex majorization.

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HOUDAYER, CYRIL, AMINE MARRAKCHI, and PETER VERRAEDT. "Strongly ergodic equivalence relations: spectral gap and type III invariants." Ergodic Theory and Dynamical Systems 39, no. 7 (2017): 1904–35. http://dx.doi.org/10.1017/etds.2017.108.

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We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$-cocycles with values in locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\unicode[STIX]{x1D70F}$ invariants for type $\text{III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type $\text{III}_{1}$ ergodic equivalence relation ${\mathcal{R}}$, the Maharam extension $\text{c}({\mathcal{R}})$ is strongly ergodic if and only if ${\mathcal{R}}$ is strongly ergodic and the invariant $\unicode[STIX]{x1D70F}({\mathcal{R}})$ is the usual topology on $\mathbb{R}$. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes’ structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and $\unicode[STIX]{x1D70F}$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

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Kechris, Alexander S. "Amenable versus hyperfinite Borel equivalence relations." Journal of Symbolic Logic 58, no. 3 (1993): 894–907. http://dx.doi.org/10.2307/2275102.

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LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

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GIORDANO, THIERRY, HIROKI MATUI, IAN F. PUTNAM, and CHRISTIAN F. SKAU. "The absorption theorem for affable equivalence relations." Ergodic Theory and Dynamical Systems 28, no. 5 (2008): 1509–31. http://dx.doi.org/10.1017/s0143385707000946.

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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.

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Angguntur, Wayan, Rudi Hartono, and Bambang Purwanto. "Strategies Used to Translate Idiomatic Expressions in 'A Walk to Remember' into Indonesian and The Resulted Equivalence." Rainbow: Journal of Literature, Linguistics and Cultural Studies 9, no. 1 (2020): 19–26. http://dx.doi.org/10.15294/rainbow.v9i1.37922.

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This study is conducted to analyze the translation strategies used to translate the idiomatic expressions in the novel “A Walk to Remember” into Indonesian, to analyze the degree of equivalence of the Indonesian translation, and to find out the relation between the strategy used and the degree of equivalence. The translation strategies used in this study are proposed by Baker while the degrees of equivalence are analyzed by using Bell’s theory. This research belongs to descriptive research using qualitative method to describe the objectives of this study. The data are analyzed by comparing the idioms in the English version to the Indonesian translation. After the data are identified, they are examined to find out the relation between the strategies used and the degree of equivalence. The findings of the analysis are: (1) four strategies were used to translate idioms in the novel A Walk to Remember into Indonesian, (2) in terms of degree of equivalence, it was found that most of the data were identified into fully equivalent translation, (3) in the relation between translation strategies used and the degree of equivalence, it was revealed that the translation strategies used mostly produce fully equivalent with complete meaning. Keywords: a walk to remember, degree of meaning equivalance, idioms translation strategies

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44

Kuo, T. C. "A natural equivalence relation on singularities." Banach Center Publications 20, no. 1 (1988): 239–43. http://dx.doi.org/10.4064/-20-1-239-243.

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A. Salih, Yassin. "On Equivalence Relation in Factor Algebras." JOURNAL OF EDUCATION AND SCIENCE 26, no. 4 (2013): 107–11. http://dx.doi.org/10.33899/edusj.2013.89983.

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Parhi, Sarat K. "Modification of Generalization of Equivalence Relation." IOSR Journal of Mathematics 9, no. 1 (2013): 70–73. http://dx.doi.org/10.9790/5728-0917073.

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D., Sri lakshmi sudha rani, and Kavitha K. "Rough topological space using equivalence relation." International Research Journal on Advanced Science Hub 2, Special Issue ICAET 11S (2020): 53–55. http://dx.doi.org/10.47392/irjash.2020.233.

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Coskey, Samuel, and Joel David Hamkins. "Infinite Time Decidable Equivalence Relation Theory." Notre Dame Journal of Formal Logic 52, no. 2 (2011): 203–28. http://dx.doi.org/10.1215/00294527-1306199.

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Efrat, Ido. "A generalization of Marshall’s equivalence relation." Transactions of the American Mathematical Society 358, no. 6 (2005): 2561–77. http://dx.doi.org/10.1090/s0002-9947-05-03776-1.

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Karzel, Helmut, and Sayed-Ghahreman Taherian. "Groups with a ternary equivalence relation." Aequationes mathematicae 92, no. 3 (2018): 415–23. http://dx.doi.org/10.1007/s00010-018-0543-x.

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