scholarly journals Commutative Topological Semigroups Embedded into Topological Abelian Groups

Axioms ◽  
2020 ◽  
Vol 9 (3) ◽  
pp. 87
Author(s):  
Julio César Hernández Arzusa
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Topological Semigroup ◽  
Abelian Groups ◽  
Sufficient Conditions ◽  
Locally Compact ◽  
Topological Abelian Group ◽  
Topological Semigroups ◽  
Topological Monoid

In this paper, we give conditions under which a commutative topological semigroup can be embedded algebraically and topologically into a compact topological Abelian group. We prove that every feebly compact regular first countable cancellative commutative topological semigroup with open shifts is a topological group, as well as every connected locally compact Hausdorff cancellative commutative topological monoid with open shifts. Finally, we use these results to give sufficient conditions on a commutative topological semigroup that guarantee it to have countable cellularity.

Homeomorphism and Isomorphism of Abelian Groups

1974 ◽  
Vol 26 (6) ◽  
pp. 1515-1519 ◽  
Author(s):  
Stephen Scheinberg
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Topological Space ◽  
Abelian Groups ◽  
Group Structure ◽  
Locally Compact ◽  
Abelian Topological Group

An abelian topological group can be considered simply as an abelian group or as a topological space. The question considered in this article is whether the topological group structure is determined by these weaker structures. Denote homeomorphism, isomorphism, and homeomorphic isomorphism by ≈, ≅ , and =, respectively. The principal results are these.Theorem 1. If G1andG2are locally compact and connected, then G1≈ G2implies G1= G2.

scholarly journals Positive answers to Koch’s problem in special cases

2020 ◽  
Vol 8 (1) ◽  
pp. 76-87
Author(s):  
Taras Banakh ◽  
Serhii Bardyla ◽  
Igor Guran ◽  
Oleg Gutik ◽  
Alex Ravsky
Keyword(s):  
Topological Group ◽  
Topological Semigroup ◽  
Negative Answer ◽  
Positive Answer ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Special Cases ◽  
Quasitopological Group ◽  
Topological Monoid ◽  
Special Classes

AbstractA topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.

scholarly journals CONTINUITY OF MEASURABLE HOMOMORPHISMS

2008 ◽  
Vol 78 (1) ◽  
pp. 171-176 ◽  
Author(s):  
JANUSZ BRZDȨK
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Baire Property ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

AbstractWe give some general results concerning continuity of measurable homomorphisms of topological groups. As a consequence we show that a Christensen measurable homomorphism of a Polish abelian group into a locally compact topological group is continuous. We also obtain similar results for the universally measurable homomorphisms and the homomorphisms that have the Baire property.

Semidirect Product Compactifications

1983 ◽  
Vol 35 (1) ◽  
pp. 1-32
Author(s):  
F. Dangello ◽  
R. Lindahl
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Topological Semigroup ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Semigroups ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

The Meet Operator in the Lattice of Group Topologies

1986 ◽  
Vol 29 (4) ◽  
pp. 478-481
Author(s):  
Bradd Clark ◽  
Victor Schneider
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Locally Compact Group ◽  
Group Topology ◽  
Locally Compact ◽  
Hausdorff Topology ◽  
Lattice Of Topologies ◽  
Group Form ◽  
Complemented Lattice ◽  
Infinite Abelian Group

AbstractIt is well known that the lattice of topologies on a set forms a complete complemented lattice. The set of topologies which make G into a topological group form a complete lattice L(G) which is not a sublattice of the lattice of all topologies on G.Let G be an infinite abelian group. No nontrivial Hausdorff topology in L(G) has a complement in L(G). If τ1 and τ2 are locally compact topologies then τ1Λτ2 is also a locally compact group topology. The situation when G is nonabelian is also considered.

Algebras of measures on C-distinguished topological semigroups

1978 ◽  
Vol 84 (2) ◽  
pp. 323-336 ◽  
Author(s):  
H. A. M. Dzinotyiweyi
Keyword(s):  
Total Variation ◽  
Topological Semigroup ◽  
Continuous Mapping ◽  
Continuous Functions ◽  
Compact Set ◽  
Locally Compact ◽  
Radon Measures ◽  
Topological Semigroups ◽  
Bounded Complex ◽  
Complex Valued

Let S be a (jointly continuous) topological semigroup, C(S) the set of all bounded complex-valued continuous functions on S and M (S) the set of all bounded complex-valued Radon measures on S. Let (S) (or (S)) be the set of all µ ∈ M (S) such that x → │µ│ (x-1C) (or x → │µ│(Cx-1), respectively) is a continuous mapping of S into ℝ, for every compact set C ⊆ S, and . (Here │µ│ denotes the measure arising from the total variation of µ and the sets x-1C and Cx-1 are as defined in Section 2.) When S is locally compact the set Ma(S) was studied by A. C. and J. W. Baker in (1) and (2), by Sleijpen in (14), (15) and (16) and by us in (3). In this paper we show that some of the results of (1), (2), (14) and (15) remain valid for certain non-locally compact S and raise some new problems for such S.

Topological semigroups and their prequantale models

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 6205-6210 ◽  
Author(s):  
Bin Zhao ◽  
Changchun Xia ◽  
Kaiyun Wang
Keyword(s):  
Topological Semigroup ◽  
Satisfying Condition ◽  
Locally Compact ◽  
Topological Semigroups

In this paper, we introduce a condition (?) on topological semigroups, and prove that every T1 topological semigroup satisfying condition (?) has a bounded complete algebraic prequantale model. On the basis of this result, we also show that every T0 topological semigroup satisfying condition (?) can be embedded into a compact and locally compact sober topological semigroup.

Actions of a Locally Compact Group with Zero

1971 ◽  
Vol 23 (3) ◽  
pp. 413-420 ◽  
Author(s):  
T. H. McH. Hanson
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Topological Semigroup ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Definition Of ◽  
Isolated Point ◽  
Locally Compact Topological Group

In [2] we find the definition of a locally compact group with zero as a locally compact Hausdorff topological semigroup, S, which contains a non-isolated point, 0, such that G = S – {0} is a group. Hofmann shows in [2] that 0 is indeed a zero for S, G is a locally compact topological group, and the unit, 1, of G is the unit of S. We are to study actions of S and G on spaces, and the reader is referred to [4] for the terminology of actions.If X is a space (all are assumed Hausdorff) and A ⊂ X, A* denotes the closure of A. If {xρ} is a net in X, we say limρxρ = ∞ in X if {xρ} has no subnet which converges in X.

Abelian actions on compact nonorientable Riemann surfaces

2021 ◽  
pp. 1-15
Author(s):  
Jesús Rodríguez
Keyword(s):  
Abelian Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Sufficient Conditions ◽  
New Method ◽  
Order Problem ◽  
Maximum Order ◽  
Necessary And Sufficient

Abstract Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.

On the supports of Gauss measures on algebraic groups

1984 ◽  
Vol 96 (3) ◽  
pp. 437-445 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Topological Semigroup ◽  
Weak Topology ◽  
Algebraic Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Borel Measures ◽  
Image Position ◽  
Locally Compact Topological Group

For any locally compact topological group G let M(G) denote the topological semigroup of all probability (Borel) measures on G, furnished with the weak topology and with convolution as the multiplication. A Gauss semigroup on G is a homomorphism t→ μt of the strictly positive reals (under addition) into M(G) such that(i) no μt is a point mesaure,(ii) for each neighbourhood V of 1 in G we have

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