scholarly journals Differential Geometry and Binary Operations

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1525 ◽  
Author(s):  
Nikita E. Barabanov ◽  
Abraham A. Ungar
Keyword(s):  
Gaussian Curvature ◽  
Binary Operation ◽  
Sufficient Conditions ◽  
Canonical Representation ◽  
Large Set ◽  
Disk Model ◽  
Binary Operations ◽  
Möbius Addition ◽  
Necessary And Sufficient ◽  
Metric Tensors

We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero.

scholarly journals Isomorphism of Binary Operations in Differential Geometry

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1634
Author(s):  
Nikita E. Barabanov
Keyword(s):  
Sufficient Conditions ◽  
Canonical Representation ◽  
Necessary And Sufficient Conditions ◽  
Function Parameter ◽  
Binary Operations ◽  
Möbius Addition ◽  
Algebraic Automorphism ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Metric Tensors

We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.

scholarly journals Binary Operations in the Unit Ball: A Differential Geometry Approach

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1178 ◽  
Author(s):  
Nikita E. Barabanov ◽  
Abraham A. Ungar
Keyword(s):  
Differential Geometry ◽  
Unit Ball ◽  
Hyperbolic Geometry ◽  
Binary Operation ◽  
Open Unit ◽  
Large Set ◽  
Open Unit Ball ◽  
Binary Operations ◽  
Möbius Addition ◽  
Metric Tensors

Within the framework of differential geometry, we study binary operations in the open, unit ball of the Euclidean n-space R n , n ∈ N , and discover the properties that qualify these operations to the title addition despite the fact that, in general, these binary operations are neither commutative nor associative. The binary operation of the Beltrami-Klein ball model of hyperbolic geometry, known as Einstein addition, and the binary operation of the Beltrami-Poincaré ball model of hyperbolic geometry, known as Möbius addition, determine corresponding metric tensors in the unit ball. For a variety of metric tensors, including these two, we show how binary operations can be recovered from metric tensors. We define corresponding scalar multiplications, which give rise to gyrovector spaces, and to norms in these spaces. We introduce a large set of binary operations that are algebraically equivalent to Einstein addition and satisfy a number of nice properties of this addition. For such operations we define sets of gyrolines and co-gyrolines. The sets of co-gyrolines are sets of geodesics of Riemannian manifolds with zero Gaussian curvatures. We also obtain a special binary operation in the ball, which is isomorphic to the Euclidean addition in the Euclidean n-space.

Using binary operations to construct a transitive set of block transformations

2020 ◽  
Vol 30 (6) ◽  
pp. 375-389
Author(s):  
Igor V. Cherednik
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
The Family ◽  
Transitive Set ◽  
Necessary And Sufficient

AbstractWe study the set of transformations {ΣF : F∈ 𝓑∗(Ω)} implemented by a network Σ with a single binary operation F, where 𝓑∗(Ω) is the set of all binary operations on Ω that are invertible as function of the second variable. We state a criterion of bijectivity of all transformations from the family {ΣF : F∈ 𝓑∗(Ω)} in terms of the structure of the network Σ, identify necessary and sufficient conditions of transitivity of the set of transformations {ΣF : F∈ 𝓑∗(Ω)}, and propose an efficient way of verifying these conditions. We also describe an algorithm for construction of networks Σ with transitive sets of transformations {ΣF : F∈ 𝓑∗(Ω)}.

scholarly journals Strictly Increasing Additive Generators of the Second Kind of Associative Binary Operations

2019 ◽  
Vol 74 (1) ◽  
pp. 159-176
Author(s):  
Peter Viceník
Keyword(s):  
Binary Operation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Operations ◽  
Additive Generator ◽  
Necessary And Sufficient

Abstract The class of strictly increasing additive generators of the second kind is defined and analyzed. Necessary and sufficient conditions for a binary operation generated by a strictly increasing additive generator of the second kind to be associative are introduced. The relation between the class of strictly increasing additive generators of the second kind of associative binary operations and the class of discrete upper additive generators of associative binary operations is revealed.

scholarly journals Clifford semigroups and monotonicity

1985 ◽  
Vol 32 (1) ◽  
pp. 83-92
Author(s):  
T.E. Hays
Keyword(s):  
Algebraic Structure ◽  
Binary Operation ◽  
Monotone Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Clifford Semigroups ◽  
Necessary And Sufficient

A semigroup S is said to be monotone if its binary operation is a monotone function from S × S into S. This paper utilizes some of the known algebraic structure of Clifford semigroups, semigroups which are unions of groups, to study topological Clifford semigroups which are monotone. It is shown that such semigroups are preserved under products, homomorphisms, and, under certain conditions, closures. Necessary and sufficient conditions for monotonicity of groups, paragroups, bands, compact orthodox Clifford semigroups, and compact bands of groups are developed.

scholarly journals Solvability of a Bounded Parametric System in Max-Łukasiewicz Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1026 ◽  
Author(s):  
Martin Gavalec ◽  
Zuzana Němcová
Keyword(s):  
Linear System ◽  
Fuzzy Systems ◽  
Polynomial Algorithm ◽  
Sufficient Conditions ◽  
Recognition Algorithm ◽  
Triangular Norm ◽  
Binary Operations ◽  
Interval Coefficients ◽  
Necessary And Sufficient ◽  
The Given

The max-Łukasiewicz algebra describes fuzzy systems working in discrete time which are based on two binary operations: the maximum and the Łukasiewicz triangular norm. The behavior of such a system in time depends on the solvability of the corresponding bounded parametric max-linear system. The aim of this study is to describe an algorithm recognizing for which values of the parameter the given bounded parametric max-linear system has a solution—represented by an appropriate state of the fuzzy system in consideration. Necessary and sufficient conditions of the solvability have been found and a polynomial recognition algorithm has been described. The correctness of the algorithm has been verified. The presented polynomial algorithm consists of three parts depending on the entries of the transition matrix and the required state vector. The results are illustrated by numerical examples. The presented results can be also applied in the study of the max-Łukasiewicz systems with interval coefficients. Furthermore, Łukasiewicz arithmetical conjunction can be used in various types of models, for example, in cash-flow system.

scholarly journals Strong Tolerance and Strong Universality of Interval Eigenvectors in a Max-Łukasiewicz Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1504
Author(s):  
Martin Gavalec ◽  
Zuzana Němcová ◽  
Ján Plavka
Keyword(s):  
Set Theory ◽  
Binary Operation ◽  
Sufficient Conditions ◽  
Discrete Events ◽  
Linear Combinations ◽  
Main Method ◽  
Max Algebra ◽  
Discrete Events Systems ◽  
Necessary And Sufficient ◽  
Theoretical Results

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.

scholarly journals Extending abelian groups to rings

2007 ◽  
Vol 82 (3) ◽  
pp. 297-314 ◽  
Author(s):  
Lynn M. Batten ◽  
Robert S. Coulter ◽  
Marie Henderson
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Equivalence Relation ◽  
Binary Operation ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Necessary And Sufficient Conditions ◽  
Odd Order ◽  
Weak Isomorphism ◽  
Necessary And Sufficient

AbstractFor any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

scholarly journals Classification of Finite Rings of Orderp6by Generators and Relations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
P. Karimi Beiranvand ◽  
R. Beyranvand ◽  
M. Gholami
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Binary Operation ◽  
Finite Abelian Group ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Finite Rings ◽  
Generators And Relations ◽  
Necessary And Sufficient

For any finite abelian group(R,+), we define a binary operation or “multiplication” onRand give necessary and sufficient conditions on this multiplication forRto extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there aren+1rings of orderpnwith characteristicpn, wherepis a prime number. Also, all finite rings of orderp6are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.

PARTIAL REDUNDANCY OF MOMENT CONDITIONS

2002 ◽  
Vol 18 (2) ◽  
pp. 531-539 ◽  
Author(s):  
Hailong Qian
Keyword(s):  
Method Of Moments ◽  
Generalized Method Of Moments ◽  
Sufficient Conditions ◽  
The Other ◽  
Large Set ◽  
Parameter Vector ◽  
Moment Conditions ◽  
Method Of Moments Estimation ◽  
Necessary And Sufficient ◽  
Partial Redundancy

In this paper, we first transform a set of moment conditions into a set of transformed moment conditions, based on which the efficient partial generalized method of moments estimation for part of a parameter vector is defined. Given the set of transformed moment conditions, we then show that the conditions for partial redundancy of an additional set of moment conditions given an original set of moment conditions simply become the conditions for full redundancy of the second subset of transformed moment conditions given the first subset of transformed moment conditions. Thus the transformed moment conditions proposed in this paper unify partial redundancy of moment conditions with full redundancy of moment conditions. Using transformed moment conditions, we then straightforwardly derive necessary and sufficient conditions for partial redundancy of one or two subset(s) of moment conditions given the other when the large set of moment conditions consists of three subsets of moment conditions. The paper also provides several easily checkable sufficient conditions for partial redundancy of one set of moment conditions given other sets of moment conditions.

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