Marlow Anderson

The Lattice of Convex l -Subgroups of a Lattice-Ordered Group

In this chapter we shall examine in some detail the lattice C(G) of convex l-subgroups of a lattice-ordered group G. Our emphasis is on determining how much information is available from a strictly lattice-theoretic consideration of C(G). Many important classes of l-groups are described in these terms, but it is impossible to so describe varieties of l-groups; in section 5.4 we shall consider examples which reveal this limitation to our approach.

Turning Lights Out with Linear Algebra

(1998). Turning Lights Out with Linear Algebra. Mathematics Magazine: Vol. 71, No. 4, pp. 300-303.

A representation theorem for distributive

In 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.

l

A small variety of representable lattice-ordered groups is constructed, which contains all of the representable covers of the abelian variety.

-monoids

Abstract We provide a large class of coherent domains whose rings of formal power series are not coherent, by proving that if R is a pseudo-Bezout domain and R [[ X ]] is coherent, then R is completely integrally closed; this generalizes a theorem of Jondrup and Smalls for valuation domains. We also obtain a large class of completely integrally closed pseudo-Bezout domains R for which R [[ X ]] is not coherent; in particular, if R is a rank one valuation domain whose group of divisibility is a proper dense subgroup of the reals, then R [[ X ]] is not coherent.

A variety of lattice-ordered groups containing all representable covers of the abelian variety

This paper discusses the interplay between the theories of abelian l-groups and Bezout domains via the group of divisibility This interplay depends on a one-to-one correspondence between onto l-homomorphisms and overrings. A method for interpreting l-embeddings in the context of Bezout domains is conjectured.

Coherence of power series rings over pseudo-Bezout domains

Abstract History of mathematics courses based on original source materials are becoming increasingly common. However, are they more suitable for particular types of students? Here we compare two such upper-level courses, with a similar structure and using the same textbook, taken by liberal arts students at Colorado College and mathematics specialists at Oxford University.

Lattice-Ordered Groups of Divisibility: An Expository Introduction

The question of the existence of a lateral completion for an arbitrary l-group has been an important and fruitful one in the theory of l-groups. Conrad’s important paper [69] laid the groundwork for the problem, and proved existence and uniqueness for representable l-groups and l-groups with zero radical. Byrd and Lloyd [69] applied Conrad’s methods to the completely distributive case (for a definition and results concerning complete distributivity, the reader should consult Chapter 10). It was Bernau [74b],[75a] who provided the first proof for arbitrary l-groups. His construction, like that he made for the orthocompletion [66b] , involves a very technical brute force construction of the elements of the positive cone of the lateral completion in terms of the original l-group. This construction has since been simplified by McCleary [81], but it remains exceedingly complicated.

Mathematics Emerging: from Colorado to Oxford

The idea of constructing a completion for a lattice-ordered group arises in two rather different contexts. On the one hand, it is natural to inquire as to the relationship .between an l-group and the l-group inside of which it might be represented by one of the representation theorems discussed in Chapters 2 through 5. Can the target l-groups for these representation theorems be described algebraically in terms of the l-group to be represented? One might naturally expect such results to be phrased in terms of the larger l-group being complete in some sense.

The Lateral Completion

Recall example 1.1.15: Let T be a totally ordered set and A(T) the l-group of orderpreserving permutations of T, where α ∈ A(T) is positive if tα ≥ t for all t ∈ T. Birkhoff [B] asked what l-groups can be constructed in this manner. Holland [63] gave a partial answer to this question and in the process provided a new perspective from which l-groups can be studied. (Bigard, Keimel and Wolfenstein [BKW] refer to this as “l’ecole americaine”.) Holland’s main result is that every l-group is l-isomorphic to an l-subgroup of the l-group of order-preserving permutations of some totally ordered set. In this chapter we will derive Holland’s theorem, along with some immediate consequences. For a more complete study of groups viewed in this manner see [G]. The reader should note that after Holland’s theorem is proved we will employ multiplicative notation extensively.