scholarly journals On the continuous action of enriched lattice-valued convergence groups: Some examples

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3045-3064
Author(s):  
T.M.G. Ahsanullah ◽  
Fawzi Al-Thukair ◽  
Jawaher Al-Mufarrij
Keyword(s):  
Transformation Group ◽  
Transformation Groups ◽  
Triangular Norm ◽  
Topological Transformation ◽  
Continuous Action ◽  
Convergence Spaces ◽  
Group A ◽  
Convergence Approach Spaces ◽  
Cartesian Closed ◽  
Approach Spaces

Starting with a category SL-CONVGRP, of stratified enriched cl-premonoid-valued convergence groups as introduced earlier, we present a category SL-CONVTGRP, of stratified enriched cl-premonoid-valued convergence transformation groups, the idea behind this category is crept in the notion of convergence transformation group - a generalization of topological transformation group. In this respect, we are able to provide natural examples in support to our endeavor; these examples, however, stem from the action of convergence approach groups on convergence approach spaces, and the action of probabilistic convergence groups under triangular norm on probabilistic convergence spaces. Based on the category of enriched lattice-valued convergence spaces, a Cartesian closed category that enjoys lattice-valued convergence structure on function space, we look into among others, the lattice-valued convergence structures on the group of homeomorphisms of enriched lattice-valued convergence spaces, generalizing a concept of convergence transformation groups on convergence spaces, obtaining a characterization.

Discontinuity Conditions on Transformation Groups

1972 ◽  
Vol 15 (3) ◽  
pp. 417-419
Author(s):  
D. V. Thompson
Keyword(s):  
Metric Space ◽  
Transformation Group ◽  
Convergence Condition ◽  
Transformation Groups ◽  
Compact Metric Space ◽  
Topological Transformation ◽  
Discontinuity Conditions ◽  
Group 1

Throughout this paper, (X, T, π) is a topological transformation group [1], L={x∊X:xt=x for some t∊{e}} and 0=X—L is nonempty; standard topological concepts are used as defined in [2].The problem to be considered here has been studied in [3] and [6]. In [3], X is assumed to be a compact metric space, and each t e T satisfies a convergence condition on certain subsets of X. Under these conditions, Kaul proved that if T is equicontinuous on 0, then the group properties of discontinuity, proper discontinuity, and Sperner's condition (see Definition 1) are equivalent.

scholarly journals Suspensions of topological transformation groups

1990 ◽  
Vol 10 (1) ◽  
pp. 101-117
Author(s):  
David B. Ellis
Keyword(s):  
Topological Group ◽  
Transformation Group ◽  
Main Tool ◽  
Transformation Groups ◽  
Minimal Flow ◽  
Topological Transformation ◽  
Enveloping Semigroup ◽  
Dynamical Properties ◽  
The Relationship

AbstractLet S be a subgroup of a topological group T, and suppose that S acts on a space X. One can form a T-transformation group (X ×sT, T) called the suspension of the S-transformation group (X, S). In this paper we study the relationship between the dynamical properties of (X, S) and those of its suspension when S is syndetic in T. The main tool used in this study is a notion of the group of a minimal flow (X, T) which is sensitive to the topology on the group T. We are able, using this group and the enveloping semigroup to obtain results on which T-transformation groups can be realized as suspensions of S-transformation groups, and give conditions under which the suspension of an equicontinuous S-flow is an equicontinuous T-flow.

Equicontinuity and -recursive transformation groups

1965 ◽  
Vol 61 (2) ◽  
pp. 333-336 ◽  
Author(s):  
Janet Allsbrook ◽  
R. W. Bagley
Keyword(s):  
Phase Space ◽  
Transformation Group ◽  
Transformation Groups ◽  
Almost Periodic ◽  
Convergence Space ◽  
Topological Transformation ◽  
Uniform Convergence Space ◽  
Left And Right ◽  
Uniformly Continuous ◽  
Special Case

In this paper we obtain results on equicontinuity and apply them to certain recursive properties of topological transformation groups (X, T, π) with uniform phase space X. For example, in the special case that each transition πt is uniformly continuous we consider the transformation group (X,Ψ,ρ), where Ψ is the closure of {πt|t∈T} in the space of all unimorphisms of X onto X with the topology of uniform convergence (space index topology) and p(x, φ) = φ(x) for (x, φ)∈ X × Ψ. (See (1), page 94, 11·18.) If π is a mapping on X × T we usually write ‘xt’ for ‘π(x, t)’ and ‘AT’ for ‘π(A × T)’ where A ⊂ X. In this case we obtain the following results:I. If (X, T, π) is almost periodic [regularly almost periodic] and πxis equicontinuous, then (X,Φ,ρ) is almost periodic [regularly almost periodic]II. Let A be a compact subset of X such that. If the left and right uniformities of T are equal and (X, T, π) is almost periodic [regularly almost periodic], then (X, T, π) is almost periodic [regularly almost periodic].

IMBEDDING OF TRANSFORMATION GROUPS IN AFFINE ONES AND RENORMALIZATION OF TWO-DIMENSIONAL HOMOGENEOUS σ-MODELS

1993 ◽  
Vol 08 (31) ◽  
pp. 2937-2942
Author(s):  
A. V. BRATCHIKOV
Keyword(s):  
Homogeneous Space ◽  
Transformation Group ◽  
Affine Group ◽  
Coupling Constants ◽  
Transformation Groups ◽  
Two Dimensional

The BLZ method for the analysis of renormalizability of the O(N)/O(N − 1) model is extended to the σ-model built on an arbitrary homogeneous space G/H and in arbitrary coordinates. For deriving Ward-Takahashi (WT) identities an imbedding of the transformation group G in an affine group is used. The structure of the renormalized action is found. All the infinities can be absorbed in a coupling constants renormalization and in a renormalization of auxiliary constants which are related to the imbedding.

A Mapping Problem and Jp-Index. I

1970 ◽  
Vol 22 (4) ◽  
pp. 705-712 ◽  
Author(s):  
Masami Wakae ◽  
Oma Hamara
Keyword(s):  
Cyclic Group ◽  
Prime Order ◽  
Transformation Group ◽  
Classical Group ◽  
Cohomology Ring ◽  
Transformation Groups ◽  
Normal Space ◽  
Mapping Problem ◽  
Countable Basis ◽  
Kakutani Theorem

Indices of normal spaces with countable basis for equivariant mappings have been investigated by Bourgin [4; 6] and by Wu [11; 12] in the case where the transformation groups are of prime order p. One of us has extended the concept to the case where the transformation group is a cyclic group of order pt and discussed its applications to the Kakutani Theorem (see [10]). In this paper we will define the Jp-index of a normal space with countable basis in the case where the transformation group is a cyclic group of order n, where n is divisible by p. We will decide, by means of the spectral sequence technique of Borel [1; 2], the Jp-index of SO(n) where n is an odd integer divisible by p. The method used in this paper can be applied to find the Jp-index of a classical group G whose cohomology ring over Jp has a system of universally transgressive generators of odd degrees.

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