On the existence of ergodic automorphisms in ergodic ℤd-actions on compact groups

2009 ◽  
Vol 30 (6) ◽  
pp. 1803-1816 ◽  
Author(s):  
C. R. E. RAJA
Keyword(s):  
London Math ◽  
Abelian Groups ◽  
Chain Condition ◽  
Compact Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Metrizable Group ◽  
Compact Abelian Groups ◽  
Descending Chain Condition

AbstractLet K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

Varieties generated by 2-testable monoids

2012 ◽  
Vol 49 (3) ◽  
pp. 366-389 ◽  
Author(s):  
Edmond Lee
Keyword(s):  
Present Article ◽  
Chain Condition ◽  
Finite Basis ◽  
Finitely Based ◽  
Finitely Generated ◽  
Finite Basis Problem ◽  
Basis Problem ◽  
Descending Chain Condition ◽  
Ascending Chain Condition

The smallest monoid containing a 2-testable semigroup is defined to be a 2-testable monoid. The well-known Brandt monoid B21 of order six is an example of a 2-testable monoid. The finite basis problem for 2-testable monoids was recently addressed and solved. The present article continues with the investigation by describing all monoid varieties generated by 2-testable monoids. It is shown that there are 28 such varieties, all of which are finitely generated and precisely 19 of which are finitely based. As a comparison, the sub-variety lattice of the monoid variety generated by the monoid B21 is examined. This lattice has infinite width, satisfies neither the ascending chain condition nor the descending chain condition, and contains non-finitely generated varieties.

scholarly journals Locally compact products and coproducts in categories of topological groups

1977 ◽  
Vol 17 (3) ◽  
pp. 401-417 ◽  
Author(s):  
Karl Heinrich Hofmann ◽  
Sidney A. Morris
Keyword(s):  
Abelian Groups ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Torsion Free Group ◽  
Locally Compact Abelian Groups ◽  
Discrete Torsion ◽  
Compact Abelian Groups ◽  
Torsion Free

In the category of locally compact groups not all families of groups have a product. Precisely which families do have a product and a description of the product is a corollary of the main theorem proved here. In the category of locally compact abelian groups a family {Gj; j ∈ J} has a product if and only if all but a finite number of the Gj are of the form Kj × Dj, where Kj is a compact group and Dj is a discrete torsion free group. Dualizing identifies the families having coproducts in the category of locally compact abelian groups and so answers a question of Z. Semadeni.

scholarly journals A GENERATING SET FOR THE AUTOMORPHISM GROUP OF A GRAPH PRODUCT OF ABELIAN GROUPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250003 ◽  
Author(s):  
L. J. CORREDOR ◽  
M. A. GUTIERREZ
Keyword(s):  
Automorphism Group ◽  
London Math ◽  
Abelian Groups ◽  
Graph Products ◽  
Graph Product ◽  
Finitely Generated ◽  
Generating Set ◽  
Labeled Graph

We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut ⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

A Conjecture of Bachmuth and Mochizuki on Automorphisms of Soluble Groups

1976 ◽  
Vol 28 (6) ◽  
pp. 1302-1310 ◽  
Author(s):  
Brian Hartley
Keyword(s):  
Finite Index ◽  
Soluble Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Linear Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Group Of Automorphisms ◽  
Finitely Generated Group ◽  
Soluble Subgroup

In [1], Bachmuth and Mochizuki conjecture, by analogy with a celebrated result of Tits on linear groups [8], that a finitely generated group of automorphisms of a finitely generated soluble group either contains a soluble subgroup of finite index (which may of course be taken to be normal) or contains a non-abelian free subgroup. They point out that their conjecture holds for nilpotent-by-abelian groups and in some other cases.

scholarly journals Fourier duality in the Brascamp–Lieb inequality

2021 ◽  
pp. 1-23
Author(s):  
JONATHAN BENNETT ◽  
EUNHEE JEONG
Keyword(s):  
Abelian Groups ◽  
Discrete Analogue ◽  
Duality Principle ◽  
Natural Setting ◽  
Finitely Generated ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Duality Property ◽  
Compact Abelian Groups

Abstract It was observed recently in work of Bez, Buschenhenke, Cowling, Flock and the first author, that the euclidean Brascamp–Lieb inequality satisfies a natural and useful Fourier duality property. The purpose of this paper is to establish an appropriate discrete analogue of this. Our main result identifies the Brascamp–Lieb constants on (finitely-generated) discrete abelian groups with Brascamp–Lieb constants on their (Pontryagin) duals. As will become apparent, the natural setting for this duality principle is that of locally compact abelian groups, and this raises basic questions about Brascamp–Lieb constants formulated in this generality.

scholarly journals Automorphisms of compact groups

1989 ◽  
Vol 9 (4) ◽  
pp. 691-735 ◽  
Author(s):  
Bruce Kitchens ◽  
Klaus Schmidt
Keyword(s):  
Lie Group ◽  
Invariant Subgroup ◽  
Chain Condition ◽  
Periodic Points ◽  
Compact Lie Group ◽  
Higher Dimensional ◽  
Metrizable Group ◽  
Algebraic Description ◽  
Descending Chain Condition ◽  
Dense Set

AbstractWe study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs (X, Γ). If Γ acts expansively on X then (X, Γ) satisfies the descending chain condition, and (X, Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of GΓ, where G is a compact Lie group. Furthermore every such subgroup of GΓ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X. Furthermore, if X is a compact group and (X, Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs (X, Γ) satisfying the descending chain condition under the assumption that X is abelian.

scholarly journals Trees as commutative BCK-Algebras

1981 ◽  
Vol 23 (2) ◽  
pp. 181-190
Author(s):  
William H. Cornish
Keyword(s):  
Height Function ◽  
Chain Condition ◽  
New Method ◽  
Subdirectly Irreducible ◽  
Natural Numbers ◽  
Finitely Generated ◽  
Congruence Distributivity ◽  
Finite Height ◽  
Descending Chain Condition ◽  
Simple Algebras

A new method of constructing commutative BCK-algebras is given. It depends upon the notion of a valuation of a lower semilattice in a given commutative BCK-algebra. Any tree vith the descending chain condition has a valuation in the natural numbers, considered as a commutative BCK-algebra; the valuation is the height-function. Thus, any tree of finite height possesses a uniquely determined commutative BCK-structure. The finite trees with at most one atom and height at most n are precisely the finitely generated subdirectly irreducible (simple) algebras in the subvariety of commutative BCK-algebras which satisfy the identity (En): xyn = xyn+1. Due to congruence-distributivity, it is then possible to describe the associated lattice of subvarieties.

scholarly journals Noetherian and Artinian ordered groupoids—semigroups

2005 ◽  
Vol 2005 (13) ◽  
pp. 2041-2051
Author(s):  
Niovi Kehayopulu ◽  
Michael Tsingelis
Keyword(s):  
Chain Condition ◽  
Growth Conditions ◽  
Ideal Theory ◽  
Minimum Condition ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Maximal Ideals ◽  
Ordered Groupoid ◽  
Descending Chain Condition

Chain conditions, finiteness conditions, growth conditions, and other forms of finiteness, Noetherian rings and Artinian rings have been systematically studied for commutative rings and algebras since 1959. In pursuit of the deeper results of ideal theory in ordered groupoids (semigroups), it is necessary to study special classes of ordered groupoids (semigroups). Noetherian ordered groupoids (semigroups) which are about to be introduced are particularly versatile. These satisfy a certain finiteness condition, namely, that every ideal of the ordered groupoid (semigroup) is finitely generated. Our purpose is to introduce the concepts of Noetherian and Artinian ordered groupoids. An ordered groupoid is said to be Noetherian if every ideal of it is finitely generated. In this paper, we prove that an equivalent formulation of the Noetherian requirement is that the ideals of the ordered groupoid satisfy the so-called ascending chain condition. From this idea, we are led in a natural way to consider a number of results relevant to ordered groupoids with descending chain condition for ideals. We moreover prove that an ordered groupoid is Noetherian if and only if it satisfies the maximum condition for ideals and it is Artinian if and only if it satisfies the minimum condition for ideals. In addition, we prove that there is a homomorphismπof an ordered groupoid (semigroup)Shaving an idealIonto the Rees quotient ordered groupoid (semigroup)S/I. As a consequence, ifSis an ordered groupoid andIan ideal ofSsuch that bothIand the quotient groupoidS/Iare Noetherian (Artinian), then so isS. Finally, we give conditions under which the proper prime ideals of commutative Artinian ordered semigroups are maximal ideals.

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