EQUIVALENT AUTOMATIC STRUCTURES AND THEIR BOUNDARIES

1992 ◽  
Vol 02 (04) ◽  
pp. 443-469 ◽  
Author(s):  
WALTER D. NEUMANN ◽  
MICHAEL SHAPIRO
Keyword(s):  
Equivalence Class ◽  
Abelian Groups ◽  
Automatic Structures ◽  
Automatic Structure

Two (synchronous, asynchronous, or non-deterministic asynchronous) automatic structures on a group G are “equivalent” if their union is a non-deterministic asynchronous automatic structure. We discuss this relation, giving a classification of structures up to equivalence for abelian groups and partial results in some other cases. We also discuss a “boundary” of an asynchronous automatic structure. We show that it is an invariant of the equivalence class of the structure, and describe other properties. We describe a “rehabilitated boundary” which yields Sn−1 for any automatic structure on ℤn.

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

Some results on the classification of expansive automorphisms of compact abelian groups

1996 ◽  
Vol 16 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Fabio Fagnani
Keyword(s):  
Finite Group ◽  
Topological Entropy ◽  
Abelian Groups ◽  
Topological Classification ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Abelian Groups ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractIn this paper we study expansive automorphisms of compact 0-dimensional abelian groups. Our main result is the complete algebraic and topological classification of the transitive expansive automorpisms for which the maximal order of the elements isp2for a primep. This yields a classification of the transitive expansive automorphisms with topological entropy logp2. Finally, we prove a necessary and sufficient condition for an expansive automorphism to be conjugated, topologically and algebraically, to a shift over a finite group.

A Classification Of n-Abelian Groups

1969 ◽  
Vol 21 ◽  
pp. 1238-1244 ◽  
Author(s):  
J. L. Alperin
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Recent Result ◽  
Abelian Groups ◽  
The Way

The concept of an abelian group is central to group theory. For that reason many generalizations have been considered and exploited. One, in particular, is the idea of an n-abelian group (6). If n is an integer and n > 1, then a group G is n-abelian if, and only if,(xy)n = xnynfor all elements x and y of G. Thus, a group is 2-abelian if, and only if, it is abelian, while non-abelian n-abelian groups do exist for every n > 2.Many results pertaining to the way in which groups can be constructed from abelian groups can be generalized to theorems on n-abelian groups (1; 2). Moreover, the case of n = p, a prime, is useful in the study of finite p-groups (3; 4; 5). Moreover, a recent result of Weichsel (9) gives a description of all p-abelian finite p-groups.

DETERMINISTIC AND NON-DETERMINISTIC ASYNCHRONOUS AUTOMATIC STRUCTURES

1992 ◽  
Vol 02 (03) ◽  
pp. 297-305 ◽  
Author(s):  
MICHAEL SHAPIRO
Keyword(s):  
Equivalence Relation ◽  
Automatic Structures ◽  
Fellow Traveller ◽  
Automatic Structure ◽  
Definition Of ◽  
Fellow Traveller Property

Following the definition of asynchronous automatic structures in [3], we define non-deterministic asynchronous automatic structures and characterize these in terms of the asynchronous fellow traveller property. We show that any group with a non-deterministic asynchronous automatic structure has an asynchronous automatic structure. Non-deterministic asynchronous automatic structures are a labor saving method of showing that a group has an asynchronous automatic structure. They also allow one to define an equivalence relation on the class of non-deterministic asynchronous automatic structures which descends to the subclasses of deterministic asynchronous automatic structures and synchronous automatic structures.

scholarly journals Homogeneity of inverse semigroups

2018 ◽  
Vol 28 (05) ◽  
pp. 837-875 ◽  
Author(s):  
Thomas Quinn-Gregson
Keyword(s):  
Inverse Semigroup ◽  
Finite Groups ◽  
Abelian Groups ◽  
Semigroup Theory ◽  
Inverse Semigroups ◽  
Finitely Generated ◽  
Homogeneous Groups ◽  
Inverse Subsemigroups

An inverse semigroup [Formula: see text] is a semigroup in which every element has a unique inverse in the sense of semigroup theory, that is, if [Formula: see text] then there exists a unique [Formula: see text] such that [Formula: see text] and [Formula: see text]. We say that a countable inverse semigroup [Formula: see text] is a homogeneous (inverse) semigroup if any isomorphism between finitely generated (inverse) subsemigroups of [Formula: see text] extends to an automorphism of [Formula: see text]. In this paper, we consider both these concepts of homogeneity for inverse semigroups, and show when they are equivalent. We also obtain certain classifications of homogeneous inverse semigroups, in particular periodic commutative inverse semigroups. Our results may be seen as extending both the classification of homogeneous semilattices and the classification of certain classes of homogeneous groups, such as homogeneous abelian groups and homogeneous finite groups.

Automatic Structure-Sensitive Searching in X-Ray Powder Diffraction

1990 ◽  
Vol 5 (4) ◽  
pp. 181-185 ◽  
Author(s):  
S.D. Kirik ◽  
S.A. Kovyazin ◽  
A.M. Fedotov
Keyword(s):  
Crystal Structures ◽  
Powder Diffraction ◽  
Search System ◽  
X Ray ◽  
Automatic Structure ◽  
Structure Sensitive ◽  
Unit Cells ◽  
Low Symmetry

AbstractThe resemblance between powder patterns because of similarity of crystal structures is well known and widely used. This phenomenon facilitates the determination of unit cells and is frequently used to predict crystal structures of new substances. At present the matching of diffraction analogues is done mainly by hand. Some approaches have been considered in this paper for applying a computer to the problem. Four numerical criteria for resemblance of powder patterns are suggested. Powder patterns are matched with patterns in a database by making use of a computer program based on these criteria.The procedure results in a short list of powder patterns to be examined by the expert. The efficiency of the program is illustrated by examples of calculations for substances of both high and low symmetry. The search system may find an important application in X-ray powder diffraction analysis for the identification of solid solutions, of substances documented under unusual conditions, of structure analogues and for classification of patterns in a database.

scholarly journals Algebras of acyclic cluster type: Tree type and type Ã

2013 ◽  
Vol 211 ◽  
pp. 1-50 ◽  
Author(s):  
Claire Amiot ◽  
Steffen Oppermann
Keyword(s):  
Equivalence Class ◽  
Complete Classification ◽  
Global Dimension ◽  
Path Algebra ◽  
Derived Category ◽  
Derived Equivalence ◽  
Cluster Category ◽  
Cluster Type ◽  
Tree Type

AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

scholarly journals Antiautomorphisms and Biantiautomorphisms of Some Finite Abelian Groups

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 294
Author(s):  
Daniel López-Aguayo ◽  
Servando López Aguayo
Keyword(s):  
Lower Bound ◽  
Abelian Groups ◽  
Additive Group ◽  
Exact Formula ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Partial Classification ◽  
Open Questions ◽  
Odd Order

We extend the concepts of antimorphism and antiautomorphism of the additive group of integers modulo n, given by Gaitanas Konstantinos, to abelian groups. We give a lower bound for the number of antiautomorphisms of cyclic groups of odd order and give an exact formula for the number of linear antiautomorphisms of cyclic groups of odd order. Finally, we give a partial classification of the finite abelian groups which admit antiautomorphisms and state some open questions.

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