The order indiscernibles of divisible ordered abelian groups

1984 ◽  
Vol 49 (1) ◽  
pp. 151-160
Author(s):  
David Rosenthal
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Simple Condition ◽  
Algebraic Invariants ◽  
Image Position ◽  
And Algebra ◽  
Simplified Form

There has been much work in developing the interconnections between model theory and algebra. Here we look at a particular example, the divisible ordered abelian groups, and show how the indiscernibles are related to the algebraic structure. Now a divisible ordered abelian group is a model of Th (Q, +, 0, <) and so is linearly ordered by <. Thus the theory is unstable and has a large number of models. It is therefore unrealistic to expect that a simple condition will completely determine a model. Instead we would just like to obtain nice algebraic invariants.Definition. A subset C of a divisible ordered abelian group is a set of (order) indiscernibles iff for every sequence of integers n1,…,nk and for every c1 < … < ck and d1 < … < dk in CNote that this simplified form of indiscernibility is an immediate consequence of quantifier elimination for the theory. The above definition could be formulated in the language of +, 0, < but we have used subtraction as a matter of convenience. Similarly we may also use rational coefficients. Also note that a set of order indiscernibles is usually defined with respect to some external order. But in this case there are only two possibilities: the ordering inherited from or its reverse. So we will always assume that a set of order indiscernibles has the ordering inherited from . We may sometimes refer to a set of indiscernibles as a sequence of indiscernibles if we want to explicitly mention the ordering associated with the set.

scholarly journals A Study on Finite Abelian Groups: Sylow’s Theorems Based

2017 ◽  
Vol 1 (3) ◽  
Author(s):  
Amaira Moaitiq Mohammed Al-Johani
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Commutative Rings ◽  
Abstract Algebra ◽  
Vector Spaces ◽  
Finite Abelian Groups ◽  
Study Objective ◽  
Group Operation ◽  
Basic Concepts

In abstract algebra, an algebraic structure is a set with one or more finitary operations defined on it that satisfies a list of axioms. Algebraic structures include groups, rings, fields, and lattices, etc. A group is an algebraic structure (????, ∗), which satisfies associative, identity and inverse laws. An Abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutatively. The concept of an Abelian group is one of the first concepts encountered in abstract algebra, from which many other basic concepts, such as rings, commutative rings, modules and vector spaces are developed. This study sheds the light on the structure of the finite abelian groups, basis theorem, Sylow’s theorem and factoring finite abelian groups. In addition, it discusses some properties related to these groups. The researcher followed the exploratory and comparative approaches to achieve the study objective. The study has shown that the theory of Abelian groups is generally simpler than that of their non-abelian counter parts, and finite Abelian groups are very well understood.  

scholarly journals Some projective representations of finite abelian groups

1987 ◽  
Vol 29 (2) ◽  
pp. 197-203 ◽  
Author(s):  
A. O. Morris ◽  
M. Saeed-Ul-Islam ◽  
E. Thomas
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Groups ◽  
Projective Representations ◽  
Complete Sets ◽  
Image Position

In this paper, we continue the work initiated by Morris [5] and Saeed-ul-Islam [6,7] and determine complete sets of inequivalent irreducible projective representations (which we shall write as i.p.r.) of finite Abelian groups with respect to some additional factor sets.We consider an Abelian groupwhich will be referred to as an Abelian group of type (a1, …, am).

A Spectral Sequence for Cohomotopy

1969 ◽  
Vol 21 ◽  
pp. 712-729 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Abelian Group ◽  
Spectral Sequence ◽  
Prime Number ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Image Position

For a prime number p let be the class of finite abelian groups whose orders are prime to p. For a finitely generated abelian group G, let Gp be the sum of the free and p-primary components of G. Our aim in this paper is to prove the following theorem.Theorem. Suppose that(i) Hi(X;Z) = 0 for i > k,(ii) for i > k – dThen there exists a spectral sequence withand the differential is given by

Elimination of quantifiers for ordered valuation rings

1987 ◽  
Vol 52 (1) ◽  
pp. 116-128 ◽  
Author(s):  
M. A. Dickmann
Keyword(s):  
Model Theory ◽  
Maximal Ideal ◽  
Valuation Ring ◽  
Quantifier Elimination ◽  
Point Of View ◽  
List Type ◽  
Divisibility Property ◽  
Valuation Rings ◽  
Commutative Domain ◽  
Image Position

Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result:Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that:(1) The symbol “∣” is interpreted as the honest divisibility relation: (2) The following divisibility property holds in T:If T admits q.e. in ℒ, then T = RCVR.We do not know at present whether the restriction imposed by condition (2) can be weakened.The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings, for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings:Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent:(a) A is a convexly ordered valuation ring.(b) Every ideal (or, equivalently, principal ideal) is convex in A.(c) A is a valuation ring convex in its field of fractions quot(A).(d) A is a valuation ring and its maximal ideal MA is convex (in A or, equivalently, in quot (A)).(e) A is a valuation ring and its maximal ideal is bounded by ± 1.

A result concerning cardinalities of ultraproducts

1974 ◽  
Vol 39 (1) ◽  
pp. 43-48 ◽  
Author(s):  
H. Jerome Keisler ◽  
Karel Prikry
Keyword(s):  
Model Theory ◽  
Abelian Groups ◽  
Image Position

The cardinality problem for ultraproducts is as follows: Given an ultrafilter over a set I and cardinals αi, i ∈ I, what is the cardinality of the ultraproduct ? Although many special results are known, several problems remain open (see [5] for a survey). For example, consider a uniform ultrafilter over a set I of power κ (uniform means that all elements of have power κ). It is open whether every countably incomplete has the property that, for all infinite α, the ultra-power has power ακ. However, it is shown in [4] that certain countably incomplete , namely the κ-regular , have this property.This paper is about another cardinality property of ultrafilters which was introduced by Eklof [1] to study ultraproducts of abelian groups. It is open whether every countably incomplete ultrafilter has the Eklof property. We shall show that certain countably incomplete ultrafilters, the κ-good ultrafilters, do have this property. The κ-good ultrafilters are important in model theory because they are exactly the ultrafilters such that every ultraproduct modulo is κ-saturated (see [5]).Let be an ultrafilter on a set I. Let αi, n, i ∈ I, n ∈ ω, be cardinals and αi, n, ≥ αi, m if n < m. Let.Then ρn are nonincreasing and therefore there is some m and ρ such that ρn = ρ if n ≥ m. We call ρ the eventual value (abbreviated ev val) of ρn.

scholarly journals Extension of partial endomorphisms of abelian groups

1963 ◽  
Vol 6 (1) ◽  
pp. 45-48 ◽  
Author(s):  
C. G. Chehata
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
Normal Subgroups ◽  
Extension Group ◽  
Image Position ◽  
Necessary And Sufficient

It is known [1] that for a partial endomorphism μ of a group G that maps the subgroup A ⊆ G onto B ⊆ G. G to be extendable to a total endomorphism μ* of a supergroup G* ⊆ G such that μ an isomorphism on G*(μ*)m for some positive integer m, it is necessary and sufficient that there exist in G a sequence of normal subgroupssuch that L1 ƞA is the kernel of μ andfor ι = 1, 2,…, m–1.The question then arises whether these conditions could be simplified when the group G is abelian. In this paper it is shown not only that the conditions are simplified when Gis abelian but also that the extension group G*⊇G can be chosen as an abelian group.

scholarly journals Truncations of ordered abelian groups

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Paola D’Aquino ◽  
Jamshid Derakhshan ◽  
Angus Macintyre
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Abelian Groups ◽  
Forthcoming Paper ◽  
Peano Arithmetic ◽  
Local Rings ◽  
Principal Ideals ◽  
Valuation Rings ◽  
Nonstandard Models ◽  
Quotient Rings

AbstractWe give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$ + , but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$ [ 0 , τ ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$ Z or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

scholarly journals Galkin Quandles, Pointed Abelian Groups, and Sequence A000712

10.37236/2676 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
William E. Clark ◽  
Xiang-dong Hou
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Knot Invariants ◽  
Group A

For each pointed abelian group $(A,c)$, there is an associated Galkin quandle $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order $q^n$ ($q$ prime) is $\sum_{0\le m\le n}p(m)p(n-m)$, where $p(m)$ is the number of partitions of integer $m$.

Harmonic conjugation in L1 on compact abelian groups

1992 ◽  
Vol 111 (1) ◽  
pp. 113-126
Author(s):  
Nakhl Asmar ◽  
Saleem Watson ◽  
Saleem Watson
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Abelian Groups ◽  
Duality Theorems ◽  
Character Group ◽  
Compact Abelian Groups ◽  
Image Position ◽  
Torsion Free ◽  
Harmonic Conjugation

Let G denote a compact connected abelian group with character group and normalized Haar measure . As a consequence of the duality theorems (11, theorem 2518), is torsion-free and hence can be ordered. That is, there is a sub-semigroup P of such that

scholarly journals Representations of finite abelian groups Cnm,p

1985 ◽  
Vol 26 (2) ◽  
pp. 133-140 ◽  
Author(s):  
M. Saeed-ul-Islam
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Projective Representation ◽  
Complex Numbers ◽  
Finite Abelian Groups ◽  
Image Position ◽  
Factor Set

Let C= CmXCmX…XCm be the finite abelian group of order mn generated by n elements w1…,wn of order m. Let C be the field of complex numbers and P a projective representation of G with factor set α over C (see Morris [2]). Further letand

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