Defining algebraic elements

1973 ◽  
Vol 38 (1) ◽  
pp. 93-101 ◽  
Author(s):  
Paul D. Bacsich
Keyword(s):  
Algebraic Closure ◽  
Amalgamation Property ◽  
Injective Hull ◽  
Universal Theory ◽  
Closed Set ◽  
Algebraic Element ◽  
Algebraic Elements ◽  
Natural Way

This paper is a survey and a synthesis of the various approaches to defining algebraic elements. Most of it is devoted to proving the following result.Let T be a universal theory with the Amalgamation Property. Then for T the notions of algebraic element introduced by Robinson, Jónsson, and Morley are identical. Furthermore, they extend in a natural way the notion of algebraic element introduced by Park, and used by Lachlan and Baldwin and by Kueker.In the course of proving this we shall construct the algebraic closure as a suitable injective hull and prove a unique factorisation theorem for algebraic predicates.We shall also show (in §3) that if T is closed under products then algebraic elements all have degree 1. Thus in algebra, algebraic elements reduce to epimorphisms.To demonstrate the remarkable stability of the notion we shall show (at the end of §5) that defining algebraic elements by infinitary formulas yields no new ones.Let L be a language. The cardinality of the set of formulas of L is denoted by ∣L∣. An L-theory is a deductively closed set of L-sentences. We let denote the category of models of T and (L-structure) homomorphisms between them. If A is a substructure of B we write A ≤ B. We call u: A → B an injection if u is an isomorphism of A with a substructure of B, and let denote the subcategory of consisting of all injections.

An Epi-Reflector for Universal Theories

1973 ◽  
Vol 16 (2) ◽  
pp. 167-171 ◽  
Author(s):  
Paul D. Bacsich
Keyword(s):  
Algebraic Closure ◽  
Amalgamation Property ◽  
Injective Hull ◽  
Universal Theory ◽  
Quotient Field ◽  
Injective Hulls

A construction of an epi-reflector by injective hull techniques is given which applies to the class of models of any universal theory with the Amalgamation Property and there yields a weak but functorial type of algebraic closure. Various completions such as the boolean envelope and quotient field constructions are identified as such injective hulls over epimorphic injections. Forms of the Amalgamation Property are also shown to eliminate various pathologies of epimorphisms and equalizers.

An expansion of F̃p

1989 ◽  
Vol 54 (2) ◽  
pp. 512-521
Author(s):  
Zoé Chatzidakis
Keyword(s):  
Galois Group ◽  
Cross Section ◽  
Algebraic Closure ◽  
Quantifier Elimination ◽  
Constant Term ◽  
Solutions To Equations ◽  
Galois Extensions ◽  
Valued Fields ◽  
Additive Endomorphism ◽  
Natural Way

Let K be a field of characteristic p. The map τ(X) = Xp − X is an additive endomorphism of K, with kernel Fp. The Galois extensions of K of order p are obtained by adjoining to K solutions to equations of the form Xp − X = a for some a in K. These extensions are called the Artin-Schreier extensions of K and have a cyclic Galois group.The study of Artin-Schreier extensions is very important for studying fields of characteristic p, in particular for studying valued fields of the form K((t)). An attempt at getting quantifier elimination for those fields would necessitate the adjunction to the language of fields of a cross-section for the function τ, i.e. a function σ such that τ ∘ σ is the identity on the image of τ. When K = Fp, such a cross-section is in fact definable in K((t)): it associates to τ(x) the element of {x, x + 1, …, x + p – 1} whose constant term is 0 (see [2]). When K is infinite, such a cross-section is usually not definable.The results presented in this paper originate from a question of L. van den Dries: is there a natural way of defining a cross-section σ for τ on F̃p, and is the theory of (F̃p, σ) decidable? (F̃p is the algebraic closure of Fp.)

Norm and spectral radius for algebraic elements of a Banach algebra

1980 ◽  
Vol 88 (1) ◽  
pp. 129-133 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Upper Bound ◽  
Good Deal ◽  
Additional Information ◽  
Algebraic Element ◽  
Algebraic Elements ◽  
Image Position ◽  
Spectral Radius Formula ◽  
Unit Norm

The purpose of this note is to show that, for any algebraic element a of a Banach algebra and certain analytic functions f, one can give an upper bound for ‖f(a)‖ in terms of ‖a‖ and the spectral radius ρ(a) of a. To illustrate the nature of the result, consider the norms of powers of an element a of unit norm. In general, the spectral radius formulacontains all that can be said (that is, the limit ρ(a) can be approached arbitrarily slowly). If we have the additional information that a is algebraic of degree n we can say a good deal more. In the case of a C*-algebra we have the neat result that, if ‖a‖ ≤ 1,(see Theorem 2), while for a general Banach algebra we have at least

Galois Theory of Essential Extensions of Modules

1972 ◽  
Vol 24 (4) ◽  
pp. 573-579 ◽  
Author(s):  
Sylvia Wiegand
Keyword(s):  
Galois Theory ◽  
Algebraic Closure ◽  
Essential Extension ◽  
Algebraic Extension ◽  
Injective Hull ◽  
Algebraic Systems ◽  
Image Position ◽  
Essential Extensions

The purpose of this paper is to exploit an analogy between algebraic extensions of fields and essential extensions of modules, in which the role of the algebraic closure of a field F is played by the injective hull H(M) of a unitary left R-module M. (The notion of * ‘algebraic’ extensions of general algebraic systems has been studied by Shoda; see, for example [5].)In this analogy, the role of a polynomial p(x) is played by a homomorphism of R-modules(1)which will be called an ideal homomorphism into M. The process of solving the equation p(x) = 0 in F, or in an algebraic extension of F, will be replaced by the process of extending an ideal homomorphism (1) to a homomorphism F* from R into M, or into an essential extension of M.

On Limits of Sequences of Algebraic Elements over a Complete Field

2005 ◽  
Vol 12 (04) ◽  
pp. 617-628
Author(s):  
Saurabh Bhatia ◽  
Sudesh K. Khanduja
Keyword(s):  
Cauchy Sequence ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Cauchy Sequences ◽  
Complete Field ◽  
Algebraic Elements

Let K be a complete field with respect to a real non-trivial valuation v, and [Formula: see text] be the extension of v to an algebraic closure [Formula: see text] of K. A well-known result of Ostrowski asserts that the limit of a Cauchy sequence of elements of [Formula: see text] does not always belong to [Formula: see text] unless [Formula: see text] is a finite extension of K. In this paper, it is shown that when a Cauchy sequence { bn } of elements of [Formula: see text] is such that the sequence { [K(bn) : K] } of degrees of the extensions K(bn) / K does not tend to infinity as n approaches infinity, then { bn } has a limit in [Formula: see text]. We also give a characterization of those Cauchy sequences { bn } of elements of [Formula: see text] whose limit is not in [Formula: see text], which generalizes a result of Alexandru, Popescu and Zaharescu.

scholarly journals Ring and Field Adjunctions, Algebraic Elements and Minimal Polynomials

2020 ◽  
Vol 28 (3) ◽  
pp. 251-261
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Vector Space ◽  
Minimal Polynomial ◽  
Field Extension ◽  
Splitting Field ◽  
Minimal Polynomials ◽  
Algebraic Element ◽  
Algebraic Elements

Summary In [6], [7] we presented a formalization of Kronecker’s construction of a field extension of a field F in which a given polynomial p ∈ F [X]\F has a root [4], [5], [3]. As a consequence for every field F and every polynomial there exists a field extension E of F in which p splits into linear factors. It is well-known that one gets the smallest such field extension – the splitting field of p – by adjoining the roots of p to F. In this article we start the Mizar formalization [1], [2] towards splitting fields: we define ring and field adjunctions, algebraic elements and minimal polynomials and prove a number of facts necessary to develop the theory of splitting fields, in particular that for an algebraic element a over F a basis of the vector space F (a) over F is given by a 0 , . . ., an− 1, where n is the degree of the minimal polynomial of a over F .

INEVITABILITY OF REVIVAL: PRO AND COUNTER ANTINATALISM

2019 ◽  
Vol 19 (1) ◽  
pp. 80-88
Author(s):  
Nikolay S. Savkin
Keyword(s):  
Social Relations ◽  
Systematic Approach ◽  
Human Life ◽  
Laws Of Nature ◽  
Arthur Schopenhauer ◽  
Expected Effect ◽  
Philosophy Of Life ◽  
Active Subject ◽  
Over Time ◽  
Natural Way

Introduction. Radical pessimism and militant anti-natalism of Arthur Schopenhauer and David Benathar create an optimistic philosophy of life, according to which life is not meaningless. It is given by nature in a natural way, and a person lives, studies, works, makes a career, achieves results, grows, develops. Being an active subject of his own social relations, a person does not refuse to continue the race, no matter what difficulties, misfortunes and sufferings would be experienced. Benathar convinces that all life is continuous suffering, and existence is constant dying. Therefore, it is better not to be born. Materials and Methods. As the main theoretical and methodological direction of research, the dialectical materialist and integrative approaches are used, the realization of which, in conjunction with the synergetic technique, provides a certain result: is convinced that the idea of anti-natalism is inadequate, the idea of giving up life. A systematic approach and a comprehensive assessment of the studied processes provide for the disclosure of the contradictory nature of anti-natalism. Results of the study are presented in the form of conclusions that human life is naturally given by nature itself. Instincts, needs, interests embodied in a person, stimulate to active actions, and he lives. But even if we finish off with all of humanity by agreement, then over time, according to the laws of nature and according to evolutionary theory, man will inevitably, objectively, and naturally reappear. Discussion and Conclusion. The expected effect of the idea of inevitability of rebirth can be the formation of an optimistic orientation of a significant part of the youth, the idea of continuing life and building happiness, development. As a social being, man is universal, and the awareness of this universality allows one to understand one’s purpose – continuous versatile development.

Synergetic Theory of Document in Evolution of Document Science

2018 ◽  
pp. 1060-1068
Author(s):  
Galina A. Dvoenosova ◽  
Keyword(s):  
Social Role ◽  
Scientific Study ◽  
Self Organization ◽  
Universal Theory ◽  
Social Self ◽  
Social Goal ◽  
The Social ◽  
Synergetic Theory ◽  
First Time

The article assesses synergetic theory of document as a new development in document science. In information society the social role of document grows, as information involves all members of society in the process of documentation. The transformation of document under the influence of modern information technologies increases its interest to representatives of different sciences. Interdisciplinary nature of document as an object of research leads to an ambiguous interpretation of its nature and social role. The article expresses and contends the author's views on this issue. In her opinion, social role of document is incidental to its being a main social tool regulating the life of civilized society. Thus, the study aims to create a scientific theory of document, explaining its nature and social role as a tool of social (goal-oriented) action and social self-organization. Substantiation of this idea is based on application of synergetics (i.e., universal theory of self-organization) to scientific study of document. In the synergetic paradigm, social and historical development is seen as the change of phases of chaos and order, and document is considered a main tool that regulates social relations. Unlike other theories of document, synergetic theory studies document not as a carrier and means of information transfer, but as a unique social phenomenon and universal social tool. For the first time, the study of document steps out of traditional frameworks of office, archive, and library. The document is placed on the scales with society as a global social system with its functional subsystems of politics, economy, culture, and personality. For the first time, the methods of social sciences and modern sociological theories are applied to scientific study of document. This methodology provided a basis for theoretical vindication of nature and social role of document as a tool of social (goal-oriented) action and social self-organization. The study frames a synergetic theory of document with methodological foundations and basic concepts, synergetic model of document, laws of development and effectiveness of document in the social continuum. At the present stage of development of science, it can be considered the highest form of theoretical knowledge of document and its scientific explanatory theory.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

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