A construction of a compact right topological semigroup

1994 ◽  
Vol 116 (2) ◽  
pp. 317-323
Author(s):  
Ahmed El-Mabhouh
Keyword(s):  
Free Group ◽  
Topological Semigroup ◽  
Minimal Ideal ◽  
Discrete Semigroup ◽  
Right Topological Semigroup

It is well-known that the structure of βℕ, the Stone—Čech compactification of the discrete semigroup (ℕ, +), is very complex. For example, it has 2c minimal left ideals and 2c minimal right ideals, its minimal ideal contains 2c copies of the free group on 2c generators, see [5], and Lisan[8] proved the existence of 2c copies of the same group outside the closure of the minimal ideal.

Abelian semigroups whose Stone-Čech compactifications have left ideal decompositions

1985 ◽  
Vol 97 (3) ◽  
pp. 473-479 ◽  
Author(s):  
D. J. Parsons
Keyword(s):  
Left Ideal ◽  
Algebraic Structure ◽  
Topological Semigroup ◽  
Good Deal ◽  
Continuous Extension ◽  
Discrete Semigroup

Stone-Čech compactifications of semigroups have aroused a good deal of interest recently. Several authors, for example Milnes [6], Marcri [5] and Baker and Butcher [1], have concentrated on problems of the existence of continuous extensions to βS of the operation in a topological semigroup S. For a discrete semigroup a separately continuous extension always exists, and others such as Pym and Vasudeva[8] have studied the compactifications of particular classes of semigroups. Further interest has centred on the algebraic structure of these compactifications; see for example Hindman[3].

Compact left ideal groups in semigroup compactification of locally compact groups

1993 ◽  
Vol 113 (3) ◽  
pp. 507-517 ◽  
Author(s):  
J. W. Baker ◽  
A. T. Lau
Keyword(s):  
Minimal Ideal ◽  
Locally Compact Group ◽  
Maximal Subgroups ◽  
Compact Groups ◽  
Locally Compact ◽  
Semigroup Compactification ◽  
Continuous Complex ◽  
Complex Valued ◽  
Uniformly Continuous ◽  
Right Topological Semigroup

Let G be a locally compact group and let UG denote the spectrum of the C*-algebra LUC(G) of bounded left uniformly continuous complex-valued functions on G, with the Gelfand topology. Then there is a multiplication on UG extending the multiplication on G (when naturally embedded in UG) such that UG is a semigroup and for each x ∈ UG, the map y ↦ yx from UG into UG is continuous, i.e. UG is a compact right topological semigroup. Consequently UG has a unique minimal ideal K which is the union of minimal (closed) left ideals UG. Furthermore K is the union of the set of maximal subgroups of K (see [3], theorem 3·ll).

The centre of the second dual of a commutative semigroup algebra

1984 ◽  
Vol 95 (1) ◽  
pp. 71-92 ◽  
Author(s):  
D. J. Parsons
Keyword(s):  
Banach Algebra ◽  
Commutative Semigroup ◽  
Topological Semigroup ◽  
Continuous Functions ◽  
Semigroup Algebra ◽  
Measure Algebra ◽  
Semitopological Semigroup ◽  
Borel Measures ◽  
Second Dual ◽  
Right Topological Semigroup

If S is an infinite, discrete, commutative semigroup then the semigroup algebra l1(S) is a commutative Banach algebra. Its dual is l∞(S), which is isometrically iso-morphic to C(βS), the space of continuous functions on the Stone-Čech compactification of S. This fact enables us to identify the second dual of l1(S) with M(βS), the space of bounded regular Borel measures on βS. Endowed with the Arens product the second dual is also a Banach algebra, so it is natural to ask whether a product may be defined in M(βS) without reference to l1(S). In §4 this is shown to be possible even when S is a non-discrete semitopological semigroup, provided that the operation in S may be extended to make βS into a left-topological semigroup in the manner of, for example, [2] where further references may be found. (Note, however, that the construction there is of a right-topological semigroup.) Having done this we may use results on βS to provide information about the measure algebra.

scholarly journals On the special semigroup “at infinity”

2014 ◽  
Vol 23 (2) ◽  
pp. 131-136
Author(s):  
ABDOL MOHAMMAD AMINPOUR ◽  
◽  
MEHRDAD SEILANI ◽  
Keyword(s):  
Topological Semigroup ◽  
Compact Semigroup ◽  
New Technique ◽  
The One ◽  
One Point Compactification ◽  
Positive Integers ◽  
Quotient Semigroup ◽  
Right Topological Semigroup

This paper presents an important new technique for studying a particular compact semigroup, N∪{∞}, the one-point compactification of positive integers with usual addition, which is an important semigroup. Indeed, the semigroup N ∪ {∞} is constructed as the quotient semigroup of a particular compact right topological semigroup. In the study of such a semigroup, a major role is played by the substructures called standard oids. For instance, some of the already known results on the structure of N ∪ {∞} are obtained as immediate consequences.

scholarly journals Quasiminimal distal function space and its semigroup compactification

1995 ◽  
Vol 18 (3) ◽  
pp. 497-500
Author(s):  
R. D. Pandian
Keyword(s):  
Function Space ◽  
Topological Semigroup ◽  
Mapping Property ◽  
Semitopological Semigroup ◽  
Semigroup Compactification ◽  
Right Topological Semigroup

Quasiminimal distal function on a semitopological semigroup is introduced. The concept of right topological semigroup compactification is employed to study the algebra of quasiminimal distal functions. The universal mapping property of the quasiminimal distal compactification is obtained.

scholarly journals Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

scholarly journals The minimally displaced set of an irreducible automorphism of $$F_N$$ is co-compact

2021 ◽  
Vol 116 (4) ◽  
pp. 369-383
Author(s):  
Stefano Francaviglia ◽  
Armando Martino ◽  
Dionysios Syrigos
Keyword(s):  
Growth Rate ◽  
Free Group ◽  
Exponential Growth ◽  
Single Point ◽  
Irreducible Automorphism

AbstractWe study the minimally displaced set of irreducible automorphisms of a free group. Our main result is the co-compactness of the minimally displaced set of an irreducible automorphism with exponential growth $$\phi $$ ϕ , under the action of the centraliser $$C(\phi )$$ C ( ϕ ) . As a corollary, we get that the same holds for the action of $$ <\phi>$$ < ϕ > on $$Min(\phi )$$ M i n ( ϕ ) . Finally, we prove that the minimally displaced set of an irreducible automorphism of growth rate one consists of a single point.

Subgroups induced by certain ideals of free group rings

1983 ◽  
Vol 11 (22) ◽  
pp. 2519-2525 ◽  
Author(s):  
Chander Kanta Gupta
Keyword(s):  
Free Group ◽  
Group Rings

scholarly journals Graphical splittings of Artin kernels

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Enrique Miguel Barquinero ◽  
Lorenzo Ruffoni ◽  
Kaidi Ye
Keyword(s):  
Free Group ◽  
Artin Group ◽  
Artin Groups ◽  
Graphs Of Groups ◽  
Underlying Graph ◽  
Block Graphs ◽  
Right Angled Artin Groups

Abstract We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.

scholarly journals The dynamics and geometry of free group endomorphisms

2021 ◽  
Vol 384 ◽  
pp. 107714
Author(s):  
Jean Pierre Mutanguha
Keyword(s):  
Free Group
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