scholarly journals On Hypergroups with a β-Class of Finite Height

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 168 ◽  
Author(s):  
Mario De Salvo ◽  
Dario Fasino ◽  
Domenico Freni ◽  
Giovanni Lo Faro
Keyword(s):  
Equivalence Classes ◽  
Fundamental Relation ◽  
Finite Height

In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height of a β -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4 , apart from isomorphisms.

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.

A New Study of Parikh Matrices Restricted to Terms

2020 ◽  
Vol 31 (05) ◽  
pp. 621-638
Author(s):  
Zi Jing Chern ◽  
K. G. Subramanian ◽  
Azhana Ahmad ◽  
Wen Chean Teh
Keyword(s):  
Equivalence Classes ◽  
Graph Distance ◽  
Rewriting Rules

Parikh matrices as an extension of Parikh vectors are useful tools in arithmetizing words by numbers. This paper presents a further study of Parikh matrices by restricting the corresponding words to terms formed over a signature. Some [Formula: see text]-equivalence preserving rewriting rules for such terms are introduced. A characterization of terms that are only [Formula: see text]-equivalent to themselves is studied for binary signatures. Graphs associated to the equivalence classes of [Formula: see text]-equivalent terms are studied with respect to graph distance. Finally, the preservation of [Formula: see text]-equivalence under the term self-shuffle operator is studied.

scholarly journals Free mu-lattices

2000 ◽  
Vol 7 (28) ◽  
Author(s):  
Luigi Santocanale
Keyword(s):  
Ordered Set ◽  
Order Relation ◽  
Equivalence Classes ◽  
Complete Lattices ◽  
Models Of Computation ◽  
Fix Point

A mu-lattice is a lattice with the property that every unary <br />polynomial has both a least and a greatest fix-point. In this paper<br />we define the quasivariety of mu-lattices and, for a given partially<br />ordered set P, we construct a mu-lattice JP whose elements are<br />equivalence classes of games in a preordered class J (P). We prove<br />that the mu-lattice JP is free over the ordered set P and that the<br />order relation of JP is decidable if the order relation of P is <br />decidable. By means of this characterization of free mu-lattices we<br />infer that the class of complete lattices generates the quasivariety<br />of mu-lattices.<br />Keywords: mu-lattices, free mu-lattices, free lattices, bicompletion<br />of categories, models of computation, least and greatest fix-points,<br />mu-calculus, Rabin chain games.

On cyclic hypergroups

2019 ◽  
Vol 18 (11) ◽  
pp. 1950213
Author(s):  
Ze Gu
Keyword(s):  
Index Formula ◽  
Fundamental Relation ◽  
Strongly Regular ◽  
Single Power

In this paper, we introduce the concept of the index of a generator in a cyclic hypergroup, and show that a single power cyclic hypergroup is generated by an element with index [Formula: see text]. Also, a characterization of the fundamental relation on a cyclic hypergroup is given. Finally, we study corresponding quotient structures induced by regular (strongly regular) relations on cyclic hypergroups. As an application, the corresponding results on single power hypergroups are obtained.

scholarly journals An entropic characterization of protein interaction networks and cellular robustness

2006 ◽  
Vol 3 (11) ◽  
pp. 843-850 ◽  
Author(s):  
Thomas Manke ◽  
Lloyd Demetrius ◽  
Martin Vingron
Keyword(s):  
Protein Interaction ◽  
Large Scale ◽  
Global Network ◽  
Molecular Networks ◽  
Local Network ◽  
Large Contribution ◽  
Fundamental Relation ◽  
Cellular Behaviour ◽  
Network Entropy

The structure of molecular networks is believed to determine important aspects of their cellular function, such as the organismal resilience against random perturbations. Ultimately, however, cellular behaviour is determined by the dynamical processes, which are constrained by network topology. The present work is based on a fundamental relation from dynamical systems theory, which states that the macroscopic resilience of a steady state is correlated with the uncertainty in the underlying microscopic processes, a property that can be measured by entropy. Here, we use recent network data from large-scale protein interaction screens to characterize the diversity of possible pathways in terms of network entropy. This measure has its origin in statistical mechanics and amounts to a global characterization of both structural and dynamical resilience in terms of microscopic elements. We demonstrate how this approach can be used to rank network elements according to their contribution to network entropy and also investigate how this suggested ranking reflects on the functional data provided by gene knockouts and RNAi experiments in yeast and Caenorhabditis elegans . Our analysis shows that knockouts of proteins with large contribution to network entropy are preferentially lethal. This observation is robust with respect to several possible errors and biases in the experimental data. It underscores the significance of entropy as a fundamental invariant of the dynamical system, and as a measure of structural and dynamical properties of networks. Our analytical approach goes beyond the phenomenological studies of cellular robustness based on local network observables, such as connectivity. One of its principal achievements is to provide a rationale to study proxies of cellular resilience and rank proteins according to their importance within the global network context.

Solvable groups derived from hypergroups

2016 ◽  
Vol 15 (04) ◽  
pp. 1650067 ◽  
Author(s):  
M. Jafarpour ◽  
H. Aghabozorgi ◽  
B. Davvaz
Keyword(s):  
Solvable Group ◽  
Equivalence Relation ◽  
Solvable Groups ◽  
Equivalence Classes ◽  
Strongly Regular

In this paper, we introduce the smallest equivalence relation [Formula: see text] on a hypergroup [Formula: see text] such that the quotient [Formula: see text], the set of all equivalence classes, is a solvable group. The characterization of solvable groups via strongly regular relations is investigated and several results on the topic are presented.

On Engel Fuzzy Subpolygroups

2017 ◽  
Vol 13 (02) ◽  
pp. 195-206 ◽  
Author(s):  
R. A. Borzooei ◽  
E. Mohammadzadeh ◽  
Violeta Fotea
Keyword(s):  
Equivalence Classes ◽  
Sufficient Condition ◽  
Fundamental Relation ◽  
Necessary And Sufficient Condition ◽  
Engel Group ◽  
Fuzzy Subgroup ◽  
Necessary And Sufficient

In this paper, by considering the notions of polygroup and Engel group, we introduce the concept of Engel fuzzy subpolygroups. In this regard, by a normal Engel fuzzy subpolygroup [Formula: see text] of [Formula: see text] and [Formula: see text], the fundamental relation on a given polygroup [Formula: see text], we construct an Engel fuzzy subgroup [Formula: see text]. We obtain a necessary and sufficient condition between Engel fuzzy subpolygroups and the Engel group [Formula: see text]/[Formula: see text], the group of equivalence classes derived from a fuzzy subpolygroup of [Formula: see text]. Finally, by using these results, we get Zorn’s lemma, in the Engel fuzzy subpolygroups.

scholarly journals On the Relation Between Schmidt Coefficients and Entanglement

2009 ◽  
Vol 16 (02n03) ◽  
pp. 127-143 ◽  
Author(s):  
Paolo Aniello ◽  
Cosmo Lupo
Keyword(s):  
Density Operator ◽  
Equivalence Classes ◽  
Symmetric Polynomials ◽  
Schmidt Decomposition ◽  
Open Problems ◽  
Density Operators ◽  
New Family ◽  
Separability Criteria ◽  
Bipartite States

We consider the Schmidt decomposition of a bipartite density operator induced by the Hilbert-Schmidt scalar product, and we study the relation between the Schmidt coefficients and entanglement. First, we define the Schmidt equivalence classes of bipartite states. Each class consists of all the density operators (in a given bipartite Hilbert space) sharing the same set of Schmidt coefficients. Next, we review the role played by the Schmidt coefficients with respect to the separability criterion known as the 'realignment' or 'computable cross norm' criterion; in particular, we highlight the fact that this criterion relies only on the Schmidt equivalence class of a state. Then, the relevance — with regard to the characterization of entanglement — of the 'symmetric polynomials' in the Schmidt coefficients and a new family of separability criteria that generalize the realignment criterion are discussed. Various interesting open problems are proposed.

scholarly journals MEASUREMENT OF DISTANCE BETWEEN REGULAR EVENTS FOR MULTITAPE AUTOMATA BASED ON A NEW CHARACTERIZATION OF EQUIVALENCE CLASSES

2021 ◽  
Vol 55 (1 (254)) ◽  
pp. 72-80
Author(s):  
Tigran A. Grigoryan ◽  
Murad S. Hayrapetyan
Keyword(s):  
Approximate Calculation ◽  
Finite Automata ◽  
Equivalence Classes ◽  
Regular Expressions

In this paper several problems related to the implementation of the method for the approximate calculation of distance between regular events for multitape finite automata are considered and resolved. An algorithm of matching for the considered regular expressions is suggested and results of the algorithm application to some specific regular expressions are adduced. The proposed method can be used not only for the mentioned implementation, but also separately.

THE CHINESE MONOID

2001 ◽  
Vol 11 (03) ◽  
pp. 301-334 ◽  
Author(s):  
JULIEN CASSAIGNE ◽  
MARC ESPIE ◽  
DANIEL KROB ◽  
JEAN-CHRISTOPHE NOVELLI ◽  
FLORENT HIVERT
Keyword(s):  
Nous Avons ◽  
Equivalence Class ◽  
Cross Section ◽  
Equivalence Classes ◽  
Nous Permet ◽  
Plactic Monoid ◽  
Relation Scheme ◽  
Section Theorem ◽  
Chinese Monoid

Résumé: Cet article présente une étude combinatoire du monoïde Chinois, un monoïde ternaire proche du monoïde plaxique, fondé sur le schéma cba≡bca≡cab. Un algorithme proche de l'algorithme de Schensted nous permet de caractériser les classes d'équivalence et d'exhiber une section du monoïde. Nous énonçons également une correspondance de Robinson–Schensted pour le monoïde Chinois avant de nous intéresser au calcul du cardinal de certaines classes. Ce travail a permis de développer de nouveaux outils combinatoires. Entre autres, nous avons trouvé un plongement de chacune des classes d'équivalence dans la plus grande classe. Quant à la dernière partie de cet article, elle présente l'étude des relations de conjugaison. This paper presents a combinatorial study of the Chinese monoid, a ternary monoid related to the plactic monoid and based on the relation scheme cba≡bca≡cab. An algorithm similar to Schensted's algorithm yields a characterization of the equivalence classes and a cross-section theorem. We also establish a Robinson–Schensted correspondence for the Chinese monoid before computing the order of specific Chinese classes. For this work, we had to develop some new combinatorial tools. Among other things we discovered an embedding of every equivalence class in the largest one. Finally, the end of this paper is devoted to the study of conjugacy classes.

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