scholarly journals New construction of algebras as quotients

2020 ◽  
Vol 5 (2) ◽  
pp. 53
Author(s):  
José Ramón Játem Lásser
Keyword(s):  
Vector Space ◽  
Natural Number ◽  
Clifford Algebras ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Natural Numbers ◽  
New Approach ◽  
Binary Operations ◽  
Direct Sum ◽  
New Construction

  In this article we have presented a new approach to define algebras using for a natural number k ≥ 2, the set of natural numbers in base k, none of their digits equal to zero. The study was developed in the context of vector R -spaces and the vector space definitions of the formal multiples of any element x of the field R, of the direct sum of vector spaces and binary operations on vector spaces were used. The results obtained were the construction of a vector space denoted by V, on the basis of the particular set of natural numbers in base k mentioned, which allowed novel ways of defining the well-known and very important algebras of complex numbers and that of quaternions on R as quotients of ideals of V, for suitably chosen ideals I. With this new approach and with the help of the vector spaces V, known algebras can be presented in a different way than those found up to now, by using certain ideals of those spaces in their quotient form. The spaces V can be over any field K and other algebras such as Clifford algebras can be constructed using this procedure.   Keywords: Algebras, Quotients in algebras, Complex numbers and quaternions as quotients of algebras.   Abstract En este artículo se ha presentado un nuevo enfoque para definir álgebras usando para un número natural k ≥ 2, el conjunto de números naturales en base k, ninguno de sus dígitos iguales a cero. El estudio se desarrolló en el contexto de los R-espacios vectoriales y se usaron las definiciones de espacio vectorial de los múltiplos formales de un elemento cualquiera x del cuerpo R, de la suma directa de espacios vectoriales y operaciones binarias sobre espacios vectoriales. Los resultados obtenidos fueron la construcción de un espacio vectorial denotado por V, sobre la base del particular conjunto de números naturales en base k mencionado, que permitió novedosas formas de definir las conocidas y muy importantes álgebras de los números complejos y la de los cuaterniones sobre R como cocientes de ideales de V, para ideales I convenientemente elegidos. Con este nuevo enfoque y con la ayuda de los espacios vectoriales V se pueden presentar álgebras conocidas de manera distinta a las encontradas hasta ahora, al usar en su forma de cociente ciertos ideales de esos espacios V. Los espacios V pueden ser sobre cualquier cuerpo K y otras álgebras como las álgebras de Clifford se pueden construir usando este procedimiento.   Palabras claves: Algebras, cocientes en álgebras, Números complejos y quaterniones como cocientes en álgebras.  

On r.e. and co-r.e. vector spaces with nonextendible bases

1980 ◽  
Vol 45 (1) ◽  
pp. 20-34 ◽  
Author(s):  
J. Remmel
Keyword(s):  
Vector Space ◽  
Quotient Space ◽  
Independent Set ◽  
Effective Procedure ◽  
Vector Spaces ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Vector Addition ◽  
Zero Vector

The concern of this paper is with recursively enumerable and co-recursively enumerable subspaces of a recursively presented vector spaceV∞ over a (finite or infinite) recursive field F which is defined in [6] to consist of a recursive subset U of the natural numbers N and operations of vector addition and scalar multiplication which are partial recursive and under which V∞ becomes a vector space. Throughout this paper, we will identify V∞ with N, say via some fixed Gödel numbering, and assume V∞ is infinite dimensional and has a dependence algorithm, i.e., there is a uniform effective procedure which determines whether or not any given n-tuple v0, …, vn−1 from V∞ is linearly dependent. Various properties of V∞ and its sub-spaces have been studied by Dekker [1], Guhl [3], Metakides and Nerode [6], Kalantari and Retzlaff [4], and the author [7].Given a subspace W of V∞, we say W is r.e. (co-r.e.) if W(V∞ − W) is an r.e. subset of N and write dim(V) for the dimension of V. Given subspaces V, W of V∞, V + W will denote the weak sum of V and W and if V ⋂ M = {0} (where 0 is the zero vector of V∞), we write V ⊕ Winstead of V + W. If W ⊇ V, we write Wmod V for the quotient space. An independent set A ⊆ V∞ is extendible if there is an r.e. independent set I ⊇ A such that I − A is infinite and A is nonextendible if it is not the case An is extendible. A r.e. subspace M ⊇ V∞ is maximal if dim(V∞ mod M) = ∞ and for any r.e. subspace W ⊇ Meither dim(W mod M) < ∞ or dim(V∞ mod W) < ∞.

scholarly journals Recursive equivalence types of vector spaces

1974 ◽  
Vol 18 (3) ◽  
pp. 376-384 ◽  
Author(s):  
Alan G. Hamilton
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Countably Infinite

We consider subspaces of a vector space UF, which is countably infinite dimensional over a recursively enumerable field F with recursive operations, where the operations in UF are also recursive, and where, of course, F and UF are sets of natural numbers. It is the object of this paper to investigate recursive equivalence types of such vector spaces and the ways in which their properties are analogous to and depend on properties of recursive equivalence types of sets.

scholarly journals Countable vector lattices

1974 ◽  
Vol 10 (3) ◽  
pp. 371-376 ◽  
Author(s):  
Paul F. Conrad
Keyword(s):  
Vector Space ◽  
Vector Lattice ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Real Numbers ◽  
Vector Lattices ◽  
Direct Sum ◽  
Ordered Vector Spaces ◽  
Countable Dimension

In his paper “On the structure of ordered real vector spaces” (Publ. Math. Debrecen 4 (1955–56), 334–343), Erdös shows that a totally ordered real vector space of countable dimension is order isomorphic to a lexicographic direct sum of copies of the group of real numbers. Brown, in “Valued vector spaces of countable dimension” (Publ. Math. Debrecen 18 (1971), 149–151), extends the result to a valued vector space of countable dimension and greatly simplifies the proof. In this note it is shown that a finite valued vector lattice of countable dimension is order isomorphic to a direct sum of o–simple totally ordered vector spaces. One obtains as corollaries the result of Erdös and the applications that Brown makes to totally ordered spaces.

scholarly journals A New Approach on Proving Collatz Conjecture

2019 ◽  
Vol 2019 ◽  
pp. 1-12 ◽  
Author(s):  
Wei Ren
Keyword(s):  
Natural Number ◽  
Natural Numbers ◽  
Induction Method ◽  
New Approach ◽  
Collatz Conjecture ◽  
The Future ◽  
Future Work ◽  
Left Portion

Collatz Conjecture (3x+1 problem) states any natural number x will return to 1 after 3⁎x+1 computation (when x is odd) and x/2 computation (when x is even). In this paper, we propose a new approach for possibly proving Collatz Conjecture (CC). We propose Reduced Collatz Conjecture (RCC)—any natural number x will return to an integer that is less than x. We prove that RCC is equivalent to CC. For proving RCC, we propose exploring laws of Reduced Collatz Dynamics (RCD), i.e., from a starting integer to the first integer less than the starting integer. RCC can also be stated as follows: RCD of any natural number exists. We prove that RCD is the components of original Collatz dynamics (from a starting integer to 1); i.e., RCD is more primitive and presents better properties. We prove that RCD presents unified structure in terms of (3⁎x+1)/2 and x/2, because 3⁎x+1 is always followed by x/2. The number of forthcoming (3⁎x+1)/2 computations can be determined directly by inputting x. We propose an induction method for proving RCC. We also discover that some starting integers present RCD with short lengths no more than 7. Hence, partial natural numbers are proved to guarantee RCC in this paper, e.g., 0 module 2; 1 module 4; 3 module 16; 11 or 23 module 32; 7, 15, or 59 module 128. The future work for proving CC can follow this direction, to prove that RCD of left portion of natural numbers exists.

Bases and α-dimensions of countable vector spaces with recursive operations

1970 ◽  
Vol 35 (1) ◽  
pp. 85-96
Author(s):  
Alan G. Hamilton
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Natural Numbers ◽  
Infinite Dimensional ◽  
Recursively Enumerable ◽  
Countably Infinite

This paper is based on the notions originally described by Dekker [2], [3], and the reader is referred to these for explanation of notation etc. Briefly, we are concerned with a countably infinite dimensional countable vector space Ū with recursive operations, regarded as being coded as a set of natural numbers. Necessarily, then, Ū must be a vector space over a field which itself is in some sense recursively enumerable and has recursive operations.

scholarly journals On One Sufficient Condition for the Irregularity of Languages

2018 ◽  
pp. 1-11 ◽  
Author(s):  
A. I. Belousov ◽  
R. S. Ismagilov
Keyword(s):  
Vector Space ◽  
Formal Languages ◽  
General Characteristic ◽  
Point Of View ◽  
Finite Alphabet ◽  
Sufficient Condition ◽  
Natural Numbers ◽  
Proved Theorem ◽  
Direct Sum ◽  
Distribution Vector

The article deals with a proof of one sufficient condition for the irregularity of languages. This condition is related to the properties of certain relations on the set of natural numbers, namely relations possessing the property, referred to as strong separability. In turn, this property is related to the possibility of decomposition of an arithmetic vector space into a direct sum of subspaces. We specify languages in some finite alphabet through the properties of a vector that shows the number of occurrences of each letter of the alphabet in the language words and is called the word distribution vector in the word. The main result of the paper is the proof of the theorem according to which a language given in such a way that the vector of distribution of letters in each word of the language belongs to a strongly separable relation on the set of natural numbers is not regular. Such an approach to the proof of irregularity is based on the Myhill-Nerode theorem known in the theory of formal languages, according to which the necessary and sufficient condition for the regularity of a language consists in the finiteness of the index of some equivalence relation defined by the language.The article gives a definition of a strongly separable relation on the set of natural numbers and examines examples of such relations. Also describes a construction covering a considerably wide class of strongly separable relations and connected with decomposition of the even-dimensional vector space into a direct sum of subspaces of the same dimension. Gives the proof of the lemma to assert an availability of an infinite sequence of vectors, any two terms of which are pairwise disjoint, i.e. one belongs to some strongly separable relation, and the other does not. Based on this lemma, there is a proof of the main theorem on the irregularity of a language defined by a strongly separable relation.This result sheds additional light on the effectiveness of regularity / irregularity analysis tools based on the Myhill-Neroud theorem. In addition, the proved theorem and analysis of some examples of strongly separable relations allows us to establish non-trivial connections between the theory of formal languages and the theory of linear spaces, which, as analysis of sources shows, is relevant.In terms of development of the obtained results, the problem of the general characteristic of strongly separable relations is of interest, as well as the analysis of other properties of numerical sets that are important from the point of view of regularity / irregularity analysis of languages.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

scholarly journals BIM for Existing Construction: A Different Logic Scheme and an Alternative Semantic to Enhance the Interoperabilty

2021 ◽  
Vol 11 (4) ◽  
pp. 1855
Author(s):  
Franco Guzzetti ◽  
Karen Lara Ngozi Anyabolu ◽  
Francesca Biolo ◽  
Lara D’Ambrosio
Keyword(s):  
Project Management ◽  
Building Information Modeling ◽  
Information Modeling ◽  
Historical Buildings ◽  
Architectural Heritage ◽  
New Approach ◽  
Building Information ◽  
Heritage Building ◽  
New Construction

In the construction field, the Building Information Modeling (BIM) methodology is becoming increasingly predominant and the standardization of its use is now an essential operation. This method has become widespread in recent years, thanks to the advantages provided in the framework of project management and interoperability. Hoping for its complete dissemination, it is unthinkable to use it only for new construction interventions. Many are experiencing what happens with the so-called Heritage Building Information Modeling (HBIM); that is, how BIM interfaces with Architectural Heritage or simply with historical buildings. This article aims to deal with the principles and working methodologies behind BIM/HBIM and modeling. The aim is to outline the themes on which to base a new approach to the instrument. In this way, it can be adapted to the needs and characteristics of each type of building. Going into the detail of standards, the text also contains a first study regarding the classification of moldable elements. This proposal is based on current regulations and it can provide flexible, expandable, and unambiguous language. Therefore, the content of the article focuses on a revision of the thinking underlying the process, also providing a more practical track on communication and interoperability.

scholarly journals M-Hazy Vector Spaces over M-Hazy Field

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1118
Author(s):  
Faisal Mehmood ◽  
Fu-Gui Shi
Keyword(s):  
Vector Space ◽  
Linear Transformation ◽  
Binary Operation ◽  
Vector Spaces ◽  
Fundamental Properties ◽  
Classical Algebra ◽  
Important Development ◽  
Fuzzy Algebra ◽  
Convex Spaces

The generalization of binary operation in the classical algebra to fuzzy binary operation is an important development in the field of fuzzy algebra. The paper proposes a new generalization of vector spaces over field, which is called M-hazy vector spaces over M-hazy field. Some fundamental properties of M-hazy field, M-hazy vector spaces, and M-hazy subspaces are studied, and some important results are also proved. Furthermore, the linear transformation of M-hazy vector spaces is studied and their important results are also proved. Finally, it is shown that M-fuzzifying convex spaces are induced by an M-hazy subspace of M-hazy vector space.

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