Semigroup actions on ordered groupoids

2013 ◽  
Vol 63 (1) ◽  
Author(s):  
Niovi Kehayopulu ◽  
Michael Tsingelis
Keyword(s):  
Commutative Semigroup ◽  
Equivalence Relation ◽  
Semigroup Actions ◽  
Ordered Groupoid ◽  
Quasi Order

AbstractIn this paper we prove that if S is a commutative semigroup acting on an ordered groupoid G, then there exists a commutative semigroup S̃ acting on the ordered groupoid G̃:=(G × S)/ρ̄ in such a way that G is embedded in G̃. Moreover, we prove that if a commutative semigroup S acts on an ordered groupoid G, and a commutative semigroup S̄ acts on an ordered groupoid Ḡ in such a way that G is embedded in S̄, then the ordered groupoid G̃ can be also embedded in Ḡ. We denote by ρ̄ the equivalence relation on G × S which is the intersection of the quasi-order ρ (on G × S) and its inverse ρ −1.

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 On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

scholarly journals Multiple Granulation Rough Set Approach to Interval-Valued Intuitionistic Fuzzy Ordered Information Systems

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 949
Author(s):  
Zhen Li ◽  
Xiaoyan Zhang
Keyword(s):  
Information Theory ◽  
Pattern Recognition ◽  
Information System ◽  
Information Systems ◽  
Rough Set ◽  
Fuzzy Set ◽  
Equivalence Relation ◽  
Target Concept ◽  
Actual Case ◽  
Interval Valued

As a further extension of the fuzzy set and the intuitive fuzzy set, the interval-valued intuitive fuzzy set (IIFS) is a more effective tool to deal with uncertain problems. However, the classical rough set is based on the equivalence relation, which do not apply to the IIFS. In this paper, we combine the IIFS with the ordered information system to obtain the interval-valued intuitive fuzzy ordered information system (IIFOIS). On this basis, three types of multiple granulation rough set models based on the dominance relation are established to effectively overcome the limitation mentioned above, which belongs to the interdisciplinary subject of information theory in mathematics and pattern recognition. First, for an IIFOIS, we put forward a multiple granulation rough set (MGRS) model from two completely symmetry positions, which are optimistic and pessimistic, respectively. Furthermore, we discuss the approximation representation and a few essential characteristics for the target concept, besides several significant rough measures about two kinds of MGRS symmetry models are discussed. Furthermore, a more general MGRS model named the generalized MGRS (GMGRS) model is proposed in an IIFOIS, and some important properties and rough measures are also investigated. Finally, the relationships and differences between the single granulation rough set and the three types of MGRS are discussed carefully by comparing the rough measures between them in an IIFOIS. In order to better utilize the theory to realistic problems, an actual case shows the methods of MGRS models in an IIFOIS is given in this paper.

scholarly journals Parametric Matroid of Rough Set

2015 ◽  
Vol 23 (06) ◽  
pp. 893-908 ◽  
Author(s):  
Yanfang Liu ◽  
Hong Zhao ◽  
William Zhu
Keyword(s):  
Rough Set ◽  
Equivalence Relation ◽  
Rough Sets ◽  
Rough Set Theory ◽  
Attribute Reduction ◽  
Independent Set ◽  
Approximation Operator ◽  
Approximation Number ◽  
Lower Approximation ◽  
The One

Rough set is mainly concerned with the approximations of objects through an equivalence relation on a universe. Matroid is a generalization of linear algebra and graph theory. Recently, a matroidal structure of rough sets is established and applied to the problem of attribute reduction which is an important application of rough set theory. In this paper, we propose a new matroidal structure of rough sets and call it a parametric matroid. On the one hand, for an equivalence relation on a universe, a parametric set family, with any subset of the universe as its parameter, is defined through the lower approximation operator. This parametric set family is proved to satisfy the independent set axiom of matroids, therefore a matroid is generated, and we call it a parametric matroid of the rough set. Through the lower approximation operator, three equivalent representations of the parametric set family are obtained. Moreover, the parametric matroid of the rough set is proved to be the direct sum of a partition-circuit matroid and a free matroid. On the other hand, partition-circuit matroids are well studied through the lower approximation number, and then we use it to investigate the parametric matroid of the rough set. Several characteristics of the parametric matroid of the rough set, such as independent sets, bases, circuits, the rank function and the closure operator, are expressed by the lower approximation number.

scholarly journals Aggregation of Indistinguishability Fuzzy Relations Revisited

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1441
Author(s):  
Juan-De-Dios González-Hedström ◽  
Juan-José Miñana ◽  
Oscar Valero
Keyword(s):  
Equivalence Relation ◽  
Fuzzy Relations ◽  
Measure Of Similarity ◽  
Dual Notion ◽  
The Relationship

Indistinguishability fuzzy relations were introduced with the aim of providing a fuzzy notion of equivalence relation. Many works have explored their relation to metrics, since they can be interpreted as a kind of measure of similarity and this is, in fact, a dual notion to dissimilarity. Moreover, the problem of how to construct new indistinguishability fuzzy relations by means of aggregation has been explored in the literature. In this paper, we provide new characterizations of those functions that allow us to merge a collection of indistinguishability fuzzy relations into a new one in terms of triangular triplets and, in addition, we explore the relationship between such functions and those that aggregate extended pseudo-metrics, which are the natural distances associated to indistinguishability fuzzy relations. Our new results extend some already known characterizations which involve only bounded pseudo-metrics. In addition, we provide a completely new description of those indistinguishability fuzzy relations that separate points, and we show that both differ a lot.

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

scholarly journals ON THE STRUCTURE OF GROUPS ENDOWED WITH A COMPATIBLE C-RELATION

2018 ◽  
Vol 83 (3) ◽  
pp. 939-966
Author(s):  
GABRIEL LEHÉRICY
Keyword(s):  
Ordered Groups ◽  
Quasi Order ◽  
Minimal Groups

AbstractWe use quasi-orders to describe the structure of C-groups. We do this by associating a quasi-order to each compatible C-relation of a group, and then give the structure of such quasi-ordered groups. We also reformulate in terms of quasi-orders some results concerning C-minimal groups given in [5].

A Generalization of “Concordance of PL-Homeomorphisms of Sp × Sq

1972 ◽  
Vol 24 (3) ◽  
pp. 426-431 ◽  
Author(s):  
J. P. E. Hodgson
Keyword(s):  
Equivalence Relation ◽  
Cohomology Ring

Let Mm be a closed PL manifold of dimension m. Then a concordance between two PL-homeomorphisms h0, h1:M → M is a PL-homeomorphismH: M × I → M × I such that H|M × 0 = h0 and H|M × 1 = h. Concordance is an equivalence relation and in his paper [2], M. Kato classifies PL-homeomorphisms of Sp × Sq up to concordance. To do this he treats first the problem of classifying those homeomorphisms that induce the identity in homology, and then describes the automorphisms of the cohomology ring that can arise from homeomorphisms of Sp × Sq. In this paper we show that for sufficiently connected PL-manifolds that embed in codimension 1, one can extend Kato's classification of the homeomorphisms that induce the identity in homology.

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