scholarly journals A further study on ordered regular equivalence relations in ordered semihypergroups

2018 ◽  
Vol 16 (1) ◽  
pp. 168-184 ◽  
Author(s):  
Jian Tang ◽  
Xinyang Feng ◽  
Bijan Davvaz ◽  
Xiang-Yun Xie
Keyword(s):  
Equivalence Relation ◽  
Open Problem ◽  
Equivalence Relations ◽  
Ordered Semigroups ◽  
Regular Equivalence

AbstractIn this paper, we study the ordered regular equivalence relations on ordered semihypergroups in detail. To begin with, we introduce the concept of weak pseudoorders on an ordered semihypergroup, and investigate several related properties. In particular, we construct an ordered regular equivalence relation on an ordered semihypergroup by a weak pseudoorder. As an application of the above result, we completely solve the open problem on ordered semihypergroups introduced in [B. Davvaz, P. Corsini and T. Changphas, Relationship between ordered semihypergroups and ordered semigroups by using pseuoorders, European J. Combinatorics 44 (2015), 208–217]. Furthermore, we establish the relationships between ordered regular equivalence relations and weak pseudoorders on an ordered semihypergroup, and give some homomorphism theorems of ordered semihypergroups, which are generalizations of similar results in ordered semigroups.

scholarly journals Labelled seeds and the mutation group

2016 ◽  
Vol 163 (2) ◽  
pp. 193-217
Author(s):  
ALASTAIR KING ◽  
MATTHEW PRESSLAND
Keyword(s):  
Automorphism Group ◽  
Homogeneous Space ◽  
Equivalence Relation ◽  
Cluster Algebra ◽  
Equivalence Relations ◽  
Connected Components ◽  
Model Field ◽  
Regular Equivalence ◽  
Mutation Group

AbstractWe study the set${\mathcal{S}}$of labelled seeds of a cluster algebra of rankninside a field${\mathcal{F}}$as a homogeneous space for the groupMnof (globally defined) mutations and relabellings. Regular equivalence relations on${\mathcal{S}}$are associated to subgroupsWof AutMn(${\mathcal{S}}$), and we thus obtain groupoidsW\${\mathcal{S}}$. We show that for two natural choices of equivalence relation, the corresponding groupsWcandW+act on${\mathcal{F}}$, and the groupoidsWc\${\mathcal{S}}$andW+\${\mathcal{S}}$act on the model field${\mathcal{K}}$=ℚ(x1,. . .,xn). The groupoidW+\${\mathcal{S}}$is equivalent to Fock–Goncharov's cluster modular groupoid. Moreover,Wcis isomorphic to the group of cluster automorphisms, andW+to the subgroup of direct cluster automorphisms, in the sense of Assem–Schiffler–Shramchenko.We also prove that, for mutation classes whose seeds have mutation finite quivers, the stabiliser of a labelled seed underMndetermines the quiver of the seed up to ‘similarity’, meaning up to taking opposites of some of the connected components. Consequently, the subgroupWcis the entire automorphism group of${\mathcal{S}}$in these cases.

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.

Quantumness, generalized spherical 2-design, and symmetric informationally complete POVM

2007 ◽  
Vol 7 (8) ◽  
pp. 730-737
Author(s):  
I.H. Kim
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Equivalence Relation ◽  
Open Problem ◽  
Natural Generalization ◽  
Quantum States ◽  
Minimal Set ◽  
Dimensional Hilbert Space

Fuchs and Sasaki defined the quantumness of a set of quantum states in \cite{Quantumness}, which is related to the fidelity loss in transmission of the quantum states through a classical channel. In \cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum quantumness is $\frac{2}{d+1}$, and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than $d^2$ states can achieve this bound. Recently, in a different context, Scott introduced a concept of generalized $t$-design in \cite{GenSphet}, which is a natural generalization of spherical $t$-design. In this paper, we show that the lower bound on the quantumness can be achieved if and only if the states form a generalized 2-design. As a corollary, we show that this bound can be only achieved if the number of states are larger or equal to $d^2$, answering the open problem. Furthermore, we also show that the minimal set of such ensemble is Symmetric Informationally Complete POVM(SIC-POVM). This leads to an equivalence relation between SIC-POVM and minimal set of ensemble achieving minimal quantumness.

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.

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