scholarly journals Cauchy completion of Abelian tight Riesz groups

1975 ◽  
Vol 19 (1) ◽  
pp. 62-73 ◽  
Author(s):  
B. F. Sherman
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Important Case ◽  
Product Theorem ◽  
Real Vector ◽  
Partially Ordered ◽  
Word Group ◽  
Central Product ◽  
Ordered Groups ◽  
Cauchy Completion

This paper concerns the completions of partially ordered groups introduced by Fuchs (1965a) and the author (to appear); the p.o. groups under consideration are, generally, abelian tight Riesz groups, and so, throughout, the word “group” will refer to an abelian group.In section 3 we meet the cornerstone of the work, the central product theorem, by means of which we can interpret the Cauchy completion of a tight Riesz group in terms of the completion of any of its o-ideals; one particularly important case arises when the group has a minimal o-ideal. Such a minimal o-ideal is o-simple, and in section 6 the completion of an isolated o-simple tight Riesz group is shown to be a tight Riesz real vector space.

scholarly journals Cauchy completion of partially ordered groups

1974 ◽  
Vol 18 (2) ◽  
pp. 222-229 ◽  
Author(s):  
B. F. Sherman
Keyword(s):  
Partial Order ◽  
Open Interval ◽  
Macneille Completion ◽  
Partially Ordered ◽  
Interval Topology ◽  
Original Order ◽  
Ordered Groups ◽  
Cauchy Completion ◽  
Original Group ◽  
Topological Completion

A number of completions have been applied to p.o.-groups — the Dede kind-Macneille completion of archimedean l.o. groups; the lateral completion of l.o. groups (Conrad [2]); and the orthocompletion of l.o. groups (Bernau [1]). Fuchs in [3] has considered a completion of p.o. groups having a non-trivial open interval topology — the only l.o. groups of this form being fully ordered. He applies an ordering, which arises from the original partial order, to the group of round Cauchy filters over this topology; Kowaisky in [6] has shown that group, imbued with a suitable topology, is in fact the topological completion of the original group under its open interval topology. In this paper a slightly different ordering, also arising from the original order, is proposed for the group of round Cauchy filters; Fuchs' ordering can be obtained from this one as the associated order.

A theorem on cardinal numbers associated with a locally compact Abelian group

1964 ◽  
Vol 60 (4) ◽  
pp. 701-704 ◽  
Author(s):  
N. Th. Varopoulos
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Compact Abelian Group ◽  
Cardinal Number ◽  
Real Vector Space ◽  
Locally Compact Abelian Group ◽  
Dimensional Torus ◽  
Real Vector ◽  
Locally Compact ◽  
Cardinal Numbers

1. Notation. For an arbitrary set X, |X| will denote its cardinal number. Rn and Tn will denote the n-dimensional real vector space and the n-dimensional torus respectively with the usual topologies.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

Vertex Subgroups of Irreducible Representations of Solvable Groups

1971 ◽  
Vol 23 (1) ◽  
pp. 12-21
Author(s):  
J. Malzan
Keyword(s):  
Vector Space ◽  
Unit Sphere ◽  
Convex Subset ◽  
Orthogonal Group ◽  
Solvable Groups ◽  
Fundamental Region ◽  
Irreducible Representations ◽  
Theoretical Problem ◽  
Real Vector ◽  
The Real

If ρ(G) is a finite, real, orthogonal group of matrices acting on the real vector space V, then there is defined [5], by the action of ρ(G), a convex subset of the unit sphere in V called a fundamental region. When the unit sphere is covered by the images under ρ(G) of a fundamental region, we obtain a semi-regular figure.The group-theoretical problem in this kind of geometry is to find when the fundamental region is unique. In this paper we examine the subgroups, ρ(H), of ρ(G) with a view of finding what subspace, W of V consists of vectors held fixed by all the matrices of ρ(H). Any such subspace lies between two copies of a fundamental region and so contributes to a boundary of both. If enough of these boundaries might be found, the fundamental region would be completely described.

scholarly journals Partially ordered groups which act on oriented order trees

2010 ◽  
Vol 40 (5) ◽  
pp. 1527-1578
Author(s):  
Matthew Horak ◽  
Melanie Stein
Keyword(s):  
Partially Ordered ◽  
Ordered Groups

scholarly journals Cross-Ratio in Real Vector Space

2019 ◽  
Vol 27 (1) ◽  
pp. 47-60
Author(s):  
Roland Coghetto
Keyword(s):  
Vector Space ◽  
Real Line ◽  
Cross Ratio ◽  
Real Vector Space ◽  
Vector Spaces ◽  
Real Vector ◽  
Complex Vector ◽  
The Real ◽  
Mizar Mathematical Library ◽  
The Cross

Summary Using Mizar [1], in the context of a real vector space, we introduce the concept of affine ratio of three aligned points (see [5]). It is also equivalent to the notion of “Mesure algèbrique”1, to the opposite of the notion of Teilverhältnis2 or to the opposite of the ordered length-ratio [9]. In the second part, we introduce the classic notion of “cross-ratio” of 4 points aligned in a real vector space. Finally, we show that if the real vector space is the real line, the notion corresponds to the classical notion3 [9]: The cross-ratio of a quadruple of distinct points on the real line with coordinates x1, x2, x3, x4 is given by: $$({x_1},{x_2};{x_3},{x_4}) = {{{x_3} - {x_1}} \over {{x_3} - {x_2}}}.{{{x_4} - {x_2}} \over {{x_4} - {x_1}}}$$ In the Mizar Mathematical Library, the vector spaces were first defined by Kusak, Leonczuk and Muzalewski in the article [6], while the actual real vector space was defined by Trybulec [10] and the complex vector space was defined by Endou [4]. Nakasho and Shidama have developed a solution to explore the notions introduced by different authors4 [7]. The definitions can be directly linked in the HTMLized version of the Mizar library5. The study of the cross-ratio will continue within the framework of the Klein- Beltrami model [2], [3]. For a generalized cross-ratio, see Papadopoulos [8].

scholarly journals An Algorithm for Recognising the Exterior Square of a Multiset

2000 ◽  
Vol 3 ◽  
pp. 96-116 ◽  
Author(s):  
Catherine Greenhill
Keyword(s):  
Abelian Group ◽  
Vector Space ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Combinatorial Construction ◽  
Exterior Square

AbstractThe exterior square of a multiset is a natural combinatorial construction which is related to the exterior square of a vector space. We consider multisets of elements of an abelian group. Two properties are defined which a multiset may satisfy: recognisability and involution-recognisability. A polynomial-time algorithm is described which takes an input multiset and returns either (a) a multiset which is either recognisable or involution-recognisable and whose exterior square equals the input multiset, or (b) the message that no such multiset exists. The proportion of multisets which are neither recognisable nor involution-recognisable is shown to be small when the abelian group is finite but large. Some further comments are made about the motivating case of multisets of eigenvalues of matrices.

On structure theory of pre-Hilbert algebras

2009 ◽  
Vol 139 (2) ◽  
pp. 303-319 ◽  
Author(s):  
José Antonio Cuenca Mira
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Real Vector Space ◽  
Inner Product ◽  
Structure Theory ◽  
Classical Theorem ◽  
Real Vector ◽  
Algebraic Identity ◽  
Hilbert Algebras

Let A be a real (non-associative) algebra which is normed as real vector space, with a norm ‖·‖ deriving from an inner product and satisfying ‖ac‖ ≤ ‖a‖‖c‖ for any a,c ∈ A. We prove that if the algebraic identity (a((ac)a))a = (a2c)a2 holds in A, then the existence of an idempotent e such that ‖e‖ = 1 and ‖ea‖ = ‖a‖ = ‖ae‖, a ∈ A, implies that A is isometrically isomorphic to ℝ, ℂ, ℍ, $\mathbb{O}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\CC}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{H}}}$,\, $\stackrel{\raisebox{4.5pt}[0pt][0pt]{\fontsize{4pt}{4pt}\selectfont$\star$}}{\smash{\mathbb{O}}}$ or ℙ. This is a non-associative extension of a classical theorem by Ingelstam. Finally, we give some applications of our main result.

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