Binary Operations, Finite Arithmetic and Groups

1979 ◽  
pp. 82-92
Author(s):  
Owen Perry ◽  
Joyce Perry
Keyword(s):  
Binary Operations ◽  
Finite Arithmetic

Pembentukan Lapangan Faktor dari Suatu Daerah Integral

2012 ◽  
Vol 8 (2) ◽  
Author(s):  
Tri Widjajanti ◽  
Dahlia Ramlan ◽  
Rium Hilum
Keyword(s):  
Polynomial Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Ring Of Integers ◽  
Binary Operations

<em>Ring of integers under the addition and multiplication as integral domain can be imbedded to the field of rational numbers. In this paper we make&nbsp; a construction such that any integral domain can be&nbsp; a field of quotient. The construction contains three steps. First, we define element of field F from elements of integral domain D. Secondly, we show that the binary operations in fare well-defined. Finally, we prove that </em><em>&nbsp;</em><em>f</em><em> </em><em>:</em><em> </em><em>D </em><em>&reg;</em><em> </em><em>F is an isomorphisma. In this case, the polynomial ring F[x] as the integral domain can be imbedded to the field of quotient.</em>

Generalized interval-valued hesitant intuitionistic fuzzy soft sets

2021 ◽  
pp. 1-12
Author(s):  
Admi Nazra ◽  
Yudiantri Asdi ◽  
Sisri Wahyuni ◽  
Hafizah Ramadhani ◽  
Zulvera
Keyword(s):  
Hesitant Fuzzy Set ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Binary Operations ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Soft Sets ◽  
Comparison Table ◽  
Definition Of ◽  
Interval Valued ◽  
Intuitionistic Fuzzy Soft Sets

This paper aims to extend the Interval-valued Intuitionistic Hesitant Fuzzy Set to a Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS). Definition of a GIVHIFSS and some of their operations are defined, and some of their properties are studied. In these GIVHIFSSs, the authors have defined complement, null, and absolute. Soft binary operations like operations union, intersection, a subset are also defined. Here is also verified De Morgan’s laws and the algebraic structure of GIVHIFSSs. Finally, by using the comparison table, a different approach to GIVHIFSS based decision-making is presented.

scholarly journals Priestley duality for MV-algebras and beyond

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wesley Fussner ◽  
Mai Gehrke ◽  
Samuel J. van Gool ◽  
Vincenzo Marra
Keyword(s):  
Large Class ◽  
Order Conditions ◽  
Enriched Environment ◽  
Priestley Duality ◽  
Distributive Lattices ◽  
First Order ◽  
Dual Spaces ◽  
Binary Operations ◽  
New Perspective ◽  
Mv Algebras

Abstract We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.

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 A Set of Integral Grid-Coding Algebraic Operations Based on GeoSOT-3D

2021 ◽  
Vol 10 (7) ◽  
pp. 489
Author(s):  
Kaihua Hou ◽  
Chengqi Cheng ◽  
Bo Chen ◽  
Chi Zhang ◽  
Liesong He ◽  
...  
Keyword(s):  
Real Time ◽  
Spatial Information ◽  
Data Retrieval ◽  
Geospatial Data ◽  
Algebraic Operation ◽  
Real Time Processing ◽  
C Language ◽  
Time Processing ◽  
Binary Operations ◽  
Set Operations

As the amount of collected spatial information (2D/3D) increases, the real-time processing of these massive data is among the urgent issues that need to be dealt with. Discretizing the physical earth into a digital gridded earth and assigning an integral computable code to each grid has become an effective way to accelerate real-time processing. Researchers have proposed optimization algorithms for spatial calculations in specific scenarios. However, a complete set of algorithms for real-time processing using grid coding is still lacking. To address this issue, a carefully designed, integral grid-coding algebraic operation framework for GeoSOT-3D (a multilayer latitude and longitude grid model) is proposed. By converting traditional floating-point calculations based on latitude and longitude into binary operations, the complexity of the algorithm is greatly reduced. We then present the detailed algorithms that were designed, including basic operations, vector operations, code conversion operations, spatial operations, metric operations, topological relation operations, and set operations. To verify the feasibility and efficiency of the above algorithms, we developed an experimental platform using C++ language (including major algorithms, and more algorithms may be expanded in the future). Then, we generated random data and conducted experiments. The experimental results show that the computing framework is feasible and can significantly improve the efficiency of spatial processing. The algebraic operation framework is expected to support large geospatial data retrieval and analysis, and experience a revival, on top of parallel and distributed computing, in an era of large geospatial data.

scholarly journals Finite Arithmetic Axiomatization for the Basis of Hyperrational Non-Standard Analysis

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 263
Author(s):  
Yuri N. Lovyagin ◽  
Nikita Yu. Lovyagin
Keyword(s):  
Set Theory ◽  
Mathematical Analysis ◽  
Axiomatic System ◽  
Elementary Number Theory ◽  
Mathematical Application ◽  
First Order ◽  
Logic Approach ◽  
Order Arithmetic ◽  
Finite Arithmetic ◽  
Non Standard Analysis

The standard elementary number theory is not a finite axiomatic system due to the presence of the induction axiom scheme. Absence of a finite axiomatic system is not an obstacle for most tasks, but may be considered as imperfect since the induction is strongly associated with the presence of set theory external to the axiomatic system. Also in the case of logic approach to the artificial intelligence problems presence of a finite number of basic axioms and states is important. Axiomatic hyperrational analysis is the axiomatic system of hyperrational number field. The properties of hyperrational numbers and functions allow them to be used to model real numbers and functions of classical elementary mathematical analysis. However hyperrational analysis is based on well-known non-finite hyperarithmetic axiomatics. In the article we present a new finite first-order arithmetic theory designed to be the basis of the axiomatic hyperrational analysis and, as a consequence, mathematical analysis in general as a basis for all mathematical application including AI problems. It is shown that this axiomatics meet the requirements, i.e., it could be used as the basis of an axiomatic hyperrational analysis. The article in effect completes the foundation of axiomatic hyperrational analysis without calling in an arithmetic extension, since in the framework of the presented theory infinite numbers arise without invoking any new constants. The proposed system describes a class of numbers in which infinite numbers exist as natural objects of the theory itself. We also do not appeal to any “enveloping” set theory.

scholarly journals On Topologies Induced by Graphs Under Some Unary and Binary Operations

2019 ◽  
Vol 12 (2) ◽  
pp. 499-505
Author(s):  
Caen Grace Sarona Nianga ◽  
Sergio R. Canoy Jr.
Keyword(s):  
Undirected Graph ◽  
Cartesian Product ◽  
Binary Operations ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

Let G = (V (G),E(G)) be any simple undirected graph. The open hop neighborhood of v ϵ V(G) is the set 𝑁_𝐺^2(𝑣) = {u ϵ V(G):  𝑑_𝐺 (u,v) = 2}. Then G induces a topology τ_G on V (G) with base consisting of sets of the form F_G^2[A] = V(G) \ N_G^2 [A] where N_G^2 [A] = A ∪ {v ϵ V(G):  𝑁_𝐺^2(𝑣) ∩ A ≠ ∅ } and A ranges over all subsets of V (G). In this paper, we describe the topologies induced by the complement of a graph, the join, the corona, the composition and the Cartesian product of graphs.

scholarly journals How the Brain might use Division

2020 ◽  
Vol 8 ◽  
pp. 126-137
Author(s):  
Kieran Greer
Keyword(s):  
Artificial Intelligence ◽  
Binary System ◽  
Problem Size ◽  
Symbol Manipulation ◽  
Mathematical Functions ◽  
Binary Operations ◽  
Neural Architecture ◽  
Large Numbers ◽  
The Brain

One of the most fundamental questions in Biology or Artificial Intelligence is how the human brainperforms mathematical functions. How does a neural architecture that may organise itself mostly throughstatistics, know what to do? One possibility is to extract the problem to something more abstract. This becomesclear when thinking about how the brain handles large numbers, for example to the power of something, whensimply summing to an answer is not feasible. In this paper, the author suggests that the maths question can beanswered more easily if the problem is changed into one of symbol manipulation and not just number counting.If symbols can be compared and manipulated, maybe without understanding completely what they are, then themathematical operations become relative and some of them might even be rote learned. The proposed systemmay also be suggested as an alternative to the traditional computer binary system. Any of the actual maths stillbreaks down into binary operations, while a more symbolic level above that can manipulate the numbers andreduce the problem size, thus making the binary operations simpler. An interesting result of looking at this is thepossibility of a new fractal equation resulting from division, that can be used as a measure of good fit and wouldhelp the brain decide how to solve something through self-replacement and a comparison with this good fit.

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