scholarly journals Groups of automorphisms of linearly ordered sets

1976 ◽  
Vol 15 (1) ◽  
pp. 13-32 ◽  
Author(s):  
J.L. Hickman
Keyword(s):  
Simple Group ◽  
Direct Product ◽  
Cyclic Group ◽  
Ordered Set ◽  
Embedding Theorems ◽  
Ordered Sets ◽  
Infinite Cyclic Group ◽  
Groups Of Automorphisms ◽  
Linearly Ordered Set

I show that a group of order-automorphisms of a linearly ordered set can be expressed as an unrestricted direct product in which each factor is either the infinite cyclic group or else a group of order-automorphisms of a densely ordered set. From this a couple of simple group embedding theorems can be derived. The technique used to obtain the main result of this paper was motivated by the Erdös-Hajnal inductive classification of scattered sets.

scholarly journals Corrigenda On a classification of the function fields of algebraic tori: (Nagoya Math. J. 56 (1975), 85–104)

1980 ◽  
Vol 79 ◽  
pp. 187-190 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Dihedral Group ◽  
Quaternion Group ◽  
Algebraic Tori ◽  
Root Of Unity ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

Topologies of Lattice Products

1966 ◽  
Vol 18 ◽  
pp. 1004-1014 ◽  
Author(s):  
Richard A. Alo ◽  
Orrin Frink
Keyword(s):  
Direct Product ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Order Relation ◽  
Direct Products ◽  
Ordered Sets ◽  
Partially Ordered

A number of different ways of defining topologies in a lattice or partially ordered set in terms of the order relation are known. Three of these methods have proved to be useful and convenient for lattices of special types, namely the ideal topology, the interval topology, and the new interval topology of Garrett Birkhoff. In another paper (2) we have shown that these three topologies are equivalent for chains (totally ordered sets), where they reduce to the usual intrinsic topology of the chain.Since many important lattices are either direct products of chains or sublattices of such products, it is natural to ask what relationships exist between the various order topologies of a direct product of lattices and those of the lattices themselves.

Automorphism groups of models of Peano arithmetic

2002 ◽  
Vol 67 (4) ◽  
pp. 1249-1264 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Topological Group ◽  
Automorphism Group ◽  
Closed Subgroup ◽  
Automorphism Groups ◽  
Ordered Set ◽  
Peano Arithmetic ◽  
Ordered Sets ◽  
Torsion Free ◽  
Linearly Ordered Set

Which groups are isomorphic to automorphism groups of models of Peano Arithmetic? It will be shown here that any group that has half a chance of being isomorphic to the automorphism group of some model of Peano Arithmetic actually is.For any structure , let Aut() be its automorphism group. There are groups which are not isomorphic to any model = (N, +, ·, 0, 1, ≤) of PA. For example, it is clear that Aut(N), being a subgroup of Aut((, <)), must be torsion-free. However, as will be proved in this paper, if (A, <) is a linearly ordered set and G is a subgroup of Aut((A, <)), then there are models of PA such that Aut() ≅ G.If is a structure, then its automorphism group can be considered as a topological group by letting the stabilizers of finite subsets of A be the basic open subgroups. If ′ is an expansion of , then Aut(′) is a closed subgroup of Aut(). Conversely, for any closed subgroup G ≤ Aut() there is an expansion ′ of such that Aut(′) = G. Thus, if is a model of PA, then Aut() is not only a subgroup of Aut((N, <)), but it is even a closed subgroup of Aut((N, ′)).There is a characterization, due to Cohn [2] and to Conrad [3], of those groups G which are isomorphic to closed subgroups of automorphism groups of linearly ordered sets.

scholarly journals Generalized Fibonacci groups H(r,n,s) that are connected labelled oriented graph groups

2019 ◽  
Vol 22 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Gerald Williams
Keyword(s):  
Cyclic Group ◽  
Betti Numbers ◽  
Oriented Graph ◽  
Complete Classification ◽  
Circulant Matrices ◽  
Fundamental Groups ◽  
Infinite Cyclic Group ◽  
Torus Knot ◽  
Graph Groups

Abstract The class of connected Labelled Oriented Graph (LOG) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in {S^{4}} , and so contains all knot groups. We investigate when Campbell and Robertson’s generalized Fibonacci groups {H(r,n,s)} are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups {H(r,n,s)} that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that {H(r,n,s)} is a 2-generator knot group if and only if it is a torus knot group.

ON THE TOTAL COMPONENT OF THE PARTIAL SCHUR MULTIPLIER

2016 ◽  
Vol 100 (3) ◽  
pp. 374-402 ◽  
Author(s):  
H. G. G. DE LIMA ◽  
H. PINEDO
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Closed Field ◽  
Schur Multiplier ◽  
Dihedral Groups ◽  
Infinite Cyclic Group ◽  
Algebraically Closed Field ◽  
Cyclic Groups

In this paper we determine the structure of the total component of the Schur multiplier over an algebraically closed field of some relevant families of groups, such as dihedral groups, dicyclic groups, the infinite cyclic group and the direct product of two finite cyclic groups.

LINEAR ORDERS REALIZED BY C.E. EQUIVALENCE RELATIONS

2016 ◽  
Vol 81 (2) ◽  
pp. 463-482 ◽  
Author(s):  
EKATERINA FOKINA ◽  
BAKHADYR KHOUSSAINOV ◽  
PAVEL SEMUKHIN ◽  
DANIEL TURETSKY
Keyword(s):  
Ordered Set ◽  
Equivalence Relations ◽  
Ordered Sets ◽  
Linear Orders ◽  
Algebraic Properties ◽  
Image Position ◽  
Quotient Set ◽  
The Relationship ◽  
Induced Structure ◽  
Linearly Ordered Set

AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$(or equivalently, the relationE)realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure ($\omega /E$; ⊴) is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.

scholarly journals Topological rigidity for FJ by the infinite cyclic group

2018 ◽  
Vol 10 (02) ◽  
pp. 421-445
Author(s):  
Kun Wang
Keyword(s):  
Direct Product ◽  
Free Group ◽  
Cyclic Group ◽  
Product Formula ◽  
Addition Formula ◽  
Infinite Cyclic Group ◽  
Torsion Free Group ◽  
Rigidity Results ◽  
Borel Conjecture ◽  
Torsion Free

We call a group FJ if it satisfies the [Formula: see text]- and [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. We show that if [Formula: see text] is FJ, then the simple Borel conjecture (in dimensions [Formula: see text]) holds for every group of the form [Formula: see text]. If in addition [Formula: see text], which is true for all known torsion-free FJ groups, then the bordism Borel conjecture (in dimensions [Formula: see text]) holds for [Formula: see text]. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion-free group [Formula: see text] satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text], then any semi-direct product [Formula: see text] also satisfies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in [Formula: see text]. Our result is indeed more general and implies the [Formula: see text]-theoretic Farrell–Jones conjecture with coefficients in additive categories is closed under extensions of torsion-free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.

scholarly journals On the endomorphism semigroup of an ordered set

1995 ◽  
Vol 37 (2) ◽  
pp. 173-178 ◽  
Author(s):  
T. S. Blyth
Keyword(s):  
Maximal Element ◽  
Ordered Set ◽  
Minimal Element ◽  
Hasse Diagram ◽  
Endomorphism Semigroup ◽  
Ordered Sets ◽  
Maximal Elements ◽  
Minimal Elements ◽  
Image Position

M. E. Adams and Matthew Gould [1] have obtained a remarkable classification of ordered sets P for which the monoid End P of endomorphisms (i.e. isotone maps) is regular, in the sense that for every f є End P there exists g є End P such that fgf = f. They show that the class of such ordered sets consists precisely of(a) all antichains;(b) all quasi-complete chains;(c) all complete bipartite ordered sets (i.e. given non-zero cardinals α β an ordered set Kα,β of height 1 having α minimal elements and β maximal elements, every minimal element being less than every maximal element);(d) for a non-zero cardinal α the lattice Mα consisting of a smallest element 0, a biggest element 1, and α atoms;(e) for non-zero cardinals α, β the ordered set Nα,β of height 1 having α minimal elements and β maximal elements in which there is a unique minimal element α0 below all maximal elements and a unique maximal element β0 above all minimal elements (and no further ordering);(f) the six-element crown C6 with Hasse diagramA similar characterisation, which coincides with the above for sets of height at most 2 but differs for chains, was obtained by A. Ya. Aizenshtat [2].

Shuffling of Linear Orders

1995 ◽  
Vol 38 (2) ◽  
pp. 223-229
Author(s):  
John Lindsay Orr
Keyword(s):  
Ordered Set ◽  
Linear Orders ◽  
Real Numbers ◽  
The Real ◽  
Linearly Ordered Set

AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.

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