The π-Full Tight Riesz Orders on A(Ω)

1981 ◽  
Vol 24 (2) ◽  
pp. 137-151
Author(s):  
Gary Davis ◽  
Stephen H. McCleary
Keyword(s):  
Permutation Group ◽  
Ordered Set ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Totally Ordered Set

Let G be a lattice-ordered group (l-group), and let t, u∈ G+. We write tπu if t ∧ g = 1 is equivalent to u ∧ g = 1, and say that a tight Riesz order T on G is π-full if t ∈ T and t π U imply u∈T. We study the set of π-full tight Riesz orders on an l-permutation group (G, Ω), Ω a totally ordered set.

scholarly journals Compatible tight Riesz orders on ordered permutation groups

1976 ◽  
Vol 21 (3) ◽  
pp. 317-333 ◽  
Author(s):  
G. Davis ◽  
E. Loci
Keyword(s):  
Ordered Set ◽  
Permutation Groups ◽  
Real Field ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Open Problems ◽  
First Order ◽  
Ordered Field ◽  
Totally Ordered Set ◽  
Point Lattice

AbstractThe pointwise order makes the group A(Ω) of order-preserving permutations of a totally-ordered set Ω a lattice-ordered group. We give some criteria for determining the compatible tight Riesz orders on A(Ω) in the case of Ω being a totally-ordered field, and then obtain various adjunctionshellip one between tight Riesz orders on A(Ω) and certain ideals of the fixed point lattice Φ(Ω), and a second between maximal tangents and certain filters of Φ(Ω). We also establish a correspondence between tight Riesz orders and first-order properties. Finally, we make use of our results to say what we can in the case of the automorphisms of the real field, and to pose several open problems.

scholarly journals Automorphism groups of totally ordered sets: A retrospective survey

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
V. Bludov ◽  
M. Droste ◽  
A. Glass
Keyword(s):  
Automorphism Groups ◽  
Ordered Set ◽  
Lattice Ordered Group ◽  
Ordered Sets ◽  
Ordered Group ◽  
Retrospective Survey ◽  
Important Theorem ◽  
Totally Ordered Set ◽  
The Many

AbstractIn 1963, W. Charles Holland proved that every lattice-ordered group can be embedded in the lattice-ordered group of all order-preserving permutations of a totally ordered set. In this article we examine the context and proof of this result and survey some of the many consequences of the ideas involved in this important theorem.

scholarly journals The lateral completion of a completely distributive lattice-ordered group (revisited)

1981 ◽  
Vol 31 (1) ◽  
pp. 114-128 ◽  
Author(s):  
Gary Davis ◽  
Stephen H. McCleary
Keyword(s):  
Distributive Lattice ◽  
Permutation Group ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Completely Distributive ◽  
Completely Distributive Lattice ◽  
The Given ◽  
Ordered Permutation Group

AbstractThe lateral completion of a completely distributive lattice-ordered permutation group is investigated via various completions, obtained by adjoining permutations which match some elements of the given group in various ways. This makes known results on the lateral completion of a completely distributive lattice-ordered group both transparent and easy.

scholarly journals Highly transitive representations of free groups and free products

1991 ◽  
Vol 43 (1) ◽  
pp. 19-36 ◽  
Author(s):  
A.M.W. Glass ◽  
Stephen H. McCleary
Keyword(s):  
Positive Integer ◽  
Permutation Group ◽  
Free Product ◽  
Free Group ◽  
Infinite Order ◽  
Transitive Group ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Free Products ◽  
Rational Line

A permutation group is highly transitive if it is n–transitive for every positive integer n. A group G of order-preserving permutations of the rational line Q is highly order-transitive if for every α1 < … < αn and β1 < … < βn in Q there exists g ∈ G such that αig = βi, i = 1, …, n. The free group Fn(2 ≤ η ≤ אo) can be faithfully represented as a highly order-transitive group of order-preserving permutations of Q, and also (reproving a theorem of McDonough) as a highly transitive group on the natural numbers N. If G and H are nontrivial countable groups having faithful representations as groups of order-preserving permutations of Q, then their free product G * H has such a representation which in addition is highly order-transitive. If G and H are nontrivial finite or countable groups and if H has an element of infinite order, then G * H can be faithfully represented as a highly transitive group on N. Some of the representations of Fη on Q can be extended to faithful representations of the free lattice-ordered group Lη.

ON RIBBON PRESENTATIONS OF RIBBON KNOTS

1994 ◽  
Vol 03 (02) ◽  
pp. 223-231
Author(s):  
TOMOYUKI YASUDA
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Totally Ordered Set

A ribbon n-knot Kn is constructed by attaching m bands to m + 1n-spheres in the Euclidean (n + 2)-space. There are many way of attaching them; as a result, Kn has many presentations which are called ribbon presentations. In this note, we will induce a notion to classify ribbon presentations for ribbon n-knots of m-fusions (m ≥ 1, n ≥ 2), and show that such classes form a totally ordered set in the case of m = 2 and a partially ordered set in the case of m ≥ 1.

scholarly journals On the structure of the totally ordered set of unimodal cycles

2001 ◽  
Vol 25 (5) ◽  
pp. 323-329 ◽  
Author(s):  
Irene Mulvey
Keyword(s):  
Ordered Set ◽  
Totally Ordered Set

We continue the study of a class of unimodal cycles where each cycle in the class is forced by every unimodal cycle not in the class. For every order, we identify the cycle in the class of that order, which is maximal with respect to the forcing relation.

scholarly journals Minimal vector lattice covers

1971 ◽  
Vol 5 (3) ◽  
pp. 331-335 ◽  
Author(s):  
Roger D. Bleier
Keyword(s):  
Vector Lattice ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Vector Lattices ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

A Representation Theorem for Distributive l-Monoids

1984 ◽  
Vol 27 (2) ◽  
pp. 238-240 ◽  
Author(s):  
Marlow Anderson ◽  
C. C. Edwards
Keyword(s):  
Group Theory ◽  
Representation Theorem ◽  
Ordered Set ◽  
Totally Ordered Set

AbstractIn this note the Holland representation theorem for l-groups is extended to l-monoids by the following theorem: an l-monoid is distributive if and only if it may be embedded into the l-monoid of order-preserving functions on some totally ordered set. A corollary of this representation theorem is that a monoid is right orderable if and only if it is a subsemigroup of a distributive l-monoid; this result is analogous to the group theory case.

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