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.

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.

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

When a locally compact monothetic semigroup is compact

2020 ◽  
Vol 23 (6) ◽  
pp. 983-989
Author(s):  
Yevhen Zelenyuk ◽  
Yuliya Zelenyuk
Keyword(s):  
Topological Group ◽  
Closed Ideal ◽  
Semitopological Semigroup ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Limit Points ◽  
Quasitopological Group

AbstractA semigroup endowed with a topology is monothetic if it contains a dense monogenic subsemigroup. A semigroup (group) endowed with a topology is semitopological (quasitopological) if the translations (the translations and the inversion) are continuous. If S is a nondiscrete monothetic semitopological semigroup, then the set {S^{\prime}} of all limit points of S is a closed ideal of S. Let S be a locally compact nondiscrete monothetic semitopological semigroup. We show that (1) if the translations of {S^{\prime}} are open, then {S^{\prime}} is compact, and (2) if {S^{\prime}} can be topologically and algebraically embedded in a quasitopological group, then {S^{\prime}} is a compact topological group.

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.

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.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

Measure in Semigroups

1952 ◽  
Vol 4 ◽  
pp. 396-406 ◽  
Author(s):  
B. R. Gelbaum ◽  
G. K. Kalisch
Keyword(s):  
Topological Group ◽  
Invariant Measure ◽  
Commutative Semigroup ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Counter Example ◽  
Compact Semigroups ◽  
Topological Measures ◽  
Weakened Form ◽  
Locally Compact Semigroups

The major portion of this paper is devoted to an investigation of the conditions which imply that a semigroup (no identity or commutativity assumed) with a bounded invariant measure is a group. We find in §3 that a weakened form of “shearing” is sufficient and a counter-example (§5) shows that “shearing” may not be dispensed with entirely. In §4 we discuss topological measures in locally compact semigroups and find that shearing may be dropped without affecting the results of the earlier sections (Theorem 2). The next two theorems show that under certain circumstances (shearing or commutativity) the topology of the semigroup (already known to be a group by virtue of earlier results) can be weakened so that the structure becomes a separated compact topological group. The last section treats the problem of extending an invariant measure on a commutative semigroup to an invariant measure on its quotient structure.

scholarly journals On topological groups with remainder of character k

2016 ◽  
Vol 17 (1) ◽  
pp. 51
Author(s):  
Maddalena Bonanzinga ◽  
Maria Vittoria Cuzzupè
Keyword(s):  
Topological Group ◽  
Infinite Cardinal ◽  
Topological Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Locally Compact Topological Group

<p style="margin: 0px;">In [A.V. Arhangel'skii and J. van Mill, On topological groups with a first-countable remainder, Top. Proc. <span id="OBJ_PREFIX_DWT1099_com_zimbra_phone" class="Object">42 (2013), 157-163</span>] it is proved that the character of a non-locally compact topological group with a first countable remainder doesn't exceed $\omega_1$ and a non-locally compact topological group of character $\omega_1$ having a compactification whose reminder is first countable is given. We generalize these results in the general case of an arbitrary infinite cardinal k.</p><p style="margin: 0px;"> </p>

scholarly journals On One-parameter Subgroups in Finite Dimensional Locally Compact Group with No Small Subgroups

1952 ◽  
Vol 4 ◽  
pp. 89-96
Author(s):  
Masatake Kuranishi
Keyword(s):  
Topological Group ◽  
Compact Group ◽  
Lie Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Finite Dimensional ◽  
Locally Compact Topological Group

Let G be a locally compact topological group and let U be a neighborhood of the identity in G. A curve g(λ) (|λ| ≦ 1) in G, which satisfies the conditions, g(s)g(t) = g(s + t) (|s|, |f|, |s + t| ≦ l),is called a one-parameter subgroup of G. If there exists a neighborhood U1 of the identity in G such that for every element x of U1 there exists a unique one-parameter subgroup g(λ) which is contained in U and g(1) =x, we shall call, for the sake of simplicity, that U has the property (S). It is well known that the neighborhoods of the identity in a Lie group have the property (S). More generally it is proved that if G is finite dimensional, locally connected, and is without small subgroups, G has the same property. In this note, these theorems will be generalized to the case when G is unite dimensional and without small subgroups.

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