scholarly journals Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups

2021 ◽  
Author(s):  
Mikhail Tkachenko
Keyword(s):  
Typical Result ◽  
Topological Groups ◽  
Dense Subgroup ◽  
Finite Set ◽  
Continuous Character

We study factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let $D=\prod_{i\in I}D_i$ be a product of paratopological groups, $S$ be a dense subgroup of $D$, and $\chi$ a continuous character of $S$. Then one can find a finite set $E\subset I$ and continuous characters $\chi_i$ of $D_i$, for $i\in E$, such that $\chi=\big(\prod_{i\in E} \chi_i\circ p_i\big)\hs1\res\hs1 S$, where $p_i\colon D\to D_i$ is the projection.

scholarly journals Factoring Continuous Characters Defined on Subgroups of Products of Topological Groups †

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 167
Author(s):  
Mikhail G. Tkachenko
Keyword(s):  
Typical Result ◽  
Topological Groups ◽  
Dense Subgroup ◽  
Finite Set ◽  
Continuous Character

This study is on the factorization properties of continuous homomorphisms defined on subgroups (or submonoids) of products of (para)topological groups (or monoids). A typical result is the following one: Let D=∏i∈IDi be a product of paratopological groups, S be a dense subgroup of D, and χ a continuous character of S. Then one can find a finite set E⊂I and continuous characters χi of Di, for i∈E, such that χ=∏i∈Eχi∘piS, where pi:D→Di is the projection.

scholarly journals Subgroups of direct products closely approximated by direct sums

2017 ◽  
Vol 29 (5) ◽  
pp. 1125-1144 ◽  
Author(s):  
Maria Ferrer ◽  
Salvador Hernández ◽  
Dmitri Shakhmatov
Keyword(s):  
Direct Product ◽  
Coding Theory ◽  
List Type ◽  
Direct Products ◽  
Topological Groups ◽  
Product Topology ◽  
Direct Sums ◽  
Cyclic Groups ◽  
Finite Set ◽  
Infinite Set

AbstractLet I be an infinite set, let {\{G_{i}:i\in I\}} be a family of (topological) groups and let {G=\prod_{i\in I}G_{i}} be its direct product. For {J\subseteq I}, {p_{J}:G\to\prod_{j\in J}G_{j}} denotes the projection. We say that a subgroup H of G is(i)uniformly controllable in G provided that for every finite set {J\subseteq I} there exists a finite set {K\subseteq I} such that {p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in K}G_{i})}, (ii)controllable in G provided that {p_{J}(H)=p_{J}(H\cap\bigoplus_{i\in I}G_{i})} for every finite set {J\subseteq I},(iii)weakly controllable in G if {H\cap\bigoplus_{i\in I}G_{i}} is dense in H, when G is equipped with the Tychonoff product topology.One easily proves that (i) {\Rightarrow} (ii) {\Rightarrow} (iii). We thoroughly investigate the question as to when these two arrows can be reversed. We prove that the first arrow can be reversed when H is compact, but the second arrow cannot be reversed even when H is compact. Both arrows can be reversed if all groups {G_{i}} are finite. When {G_{i}=A} for all {i\in I}, where A is an abelian group, we show that the first arrow can be reversed for all subgroups H of G if and only if A is finitely generated. We also describe compact groups topologically isomorphic to a direct product of countably many cyclic groups. Connections with coding theory are highlighted.

Automorphisms of the completion of relatively free Lie algebras

2018 ◽  
Vol 28 (06) ◽  
pp. 1091-1100
Author(s):  
C. E. Kofinas
Keyword(s):  
Automorphism Group ◽  
Power Series ◽  
Lie Algebras ◽  
Lower Central Series ◽  
Free Lie Algebra ◽  
Central Series ◽  
Dense Subgroup ◽  
Free Lie Algebras ◽  
Finite Set ◽  
Formal Power

Let [Formula: see text] be a relatively free Lie algebra of finite rank [Formula: see text], with [Formula: see text], [Formula: see text] be the completion of [Formula: see text] with respect to the topology defined by the lower central series [Formula: see text] of [Formula: see text] and [Formula: see text], with [Formula: see text]. We prove that, with respect to the formal power series topology, the automorphism group [Formula: see text] of [Formula: see text] is dense in the automorphism group [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is nilpotent. Furthermore, we show that there exists a dense subgroup of [Formula: see text] generated by [Formula: see text] and a finite set of IA-automorphisms if and only if [Formula: see text] is generated by [Formula: see text] and a finite set of IA-automorphisms independent upon [Formula: see text] for all [Formula: see text]. We apply our study to several varieties of Lie algebras.

scholarly journals Topological groups with dense compactly generated subgroups

2002 ◽  
Vol 3 (1) ◽  
pp. 85 ◽  
Author(s):  
Hiroshi Fujita ◽  
Dimitri Shakhmatov
Keyword(s):  
Topological Group ◽  
Compact Subset ◽  
Open Subset ◽  
Identity Element ◽  
Topological Groups ◽  
Finitely Generated ◽  
Finite Set ◽  
Metrizable Topological Group ◽  
Compactly Generated ◽  
Compact Subsets

<p>A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) -compact if G is a union of countably many compact subsets, (iii) <sub>0</sub>-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F  U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both <sub>0</sub>-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both <sub>0</sub>-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is -compact and finitely generated modulo open sets.</p>

scholarly journals Group reflection and precompact paratopological groups

2013 ◽  
Vol 1 ◽  
pp. 22-30 ◽  
Author(s):  
Mikhail Tkachenko
Keyword(s):  
Abelian Group ◽  
Topological Groups ◽  
Dense Subgroup ◽  
Completely Regular ◽  
Paratopological Group

AbstractWe construct a precompact completely regular paratopological Abelian group G of size (2ω)+ such that all subsets of G of cardinality ≤ 2ω are closed. This shows that Protasov’s theorem on non-closed discrete subsets of precompact topological groups cannot be extended to paratopological groups. We also prove that the group reflection of the product of an arbitrary family of paratopological (even semitopological) groups is topologically isomorphic to the product of the group reflections of the factors, and that the group reflection, H*, of a dense subgroup G of a paratopological group G is topologically isomorphic to a subgroup of G*.

SOME PROPERTIES OF G-SPACES ACTED WITH TOPOLOGICAL GROUPS

2020 ◽  
Vol 9 (7) ◽  
pp. 4917-4922
Author(s):  
S. Sivakumar ◽  
D. Lohanayaki
Keyword(s):  
Topological Groups

High-speed isotachophoresis. Analytical aspects of the steady-state longitudinal temperature profiles

1979 ◽  
Vol 44 (3) ◽  
pp. 841-853 ◽  
Author(s):  
Zbyněk Ryšlavý ◽  
Petr Boček ◽  
Miroslav Deml ◽  
Jaroslav Janák
Keyword(s):  
Steady State ◽  
Temperature Distribution ◽  
Electric Field ◽  
Field Intensity ◽  
Electric Field Intensity ◽  
Minor Component ◽  
Physical Models ◽  
Temperature Profiles ◽  
Longitudinal Temperature ◽  
Continuous Character

The problem of the longitudinal temperature distribution was solved and the bearing of the temperature profiles on the qualitative characteristics of the zones and on the interpretation of the record of the separation obtained from a universal detector was considered. Two approximative physical models were applied to the solution: in the first model, the temperature dependences of the mobilities are taken into account, the continuous character of the electric field intensity at the boundary being neglected; in the other model, the continuous character of the electric field intensity is allowed for. From a comparison of the two models it follows that in practice, the variations of the mobilities with the temperature are the principal factor affecting the shape of the temperature profiles, the assumption of a discontinuous jump of the electric field intensity at the boundary being a good approximation to the reality. It was deduced theoretically and verified experimentally that the longitudinal profiles can appreciably affect the longitudinal variation of the effective mobilities in the zone, with an infavourable influence upon the qualitative interpretation of the record. Pronounced effects can appear during the analyses of the minor components, where in the corresponding short zone a temperature distribution occurs due to the influence of the temperatures of the neighbouring zones such that the temperature in the zone of interest in fact does not attain a constant value in axial direction. The minor component does not possess the steady-state mobility throughout the zone, which makes the identification of the zone rather difficult.

The Defense of Justice and the Teleology of Action

2017 ◽  
Author(s):  
Andrew Payne
Keyword(s):  
Political Community ◽  
Typical Result

This chapter discusses Socrates’ defense of justice in Republic 4, which draws upon Plato’s functional teleology of action. The activity of engaging in partnerships is the function linked to just actions, as the just person acts with the intention of maintaining harmony and friendship between the three parts of the soul. David Sachs’ claim that Socrates attempts to reconcile psychic justice and conventional justice is criticized. Socrates does not attempt to show that acting from psychic justice will lead to conventional justice. For Socrates the typical result of just action is maintaining just partnerships with other citizens in a political community and thus promoting the good of others. Socrates’ just person does not act with the intention of promoting the good of others, but this is one end of her actions in the sense specified by the functional teleology of action.

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