Todd Feil

Turning Lights Out with Linear Algebra

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

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

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

The probability of splitters in a list

Abstract A list a (1),…, a ( n ) splits if there exists j with a ( i ) ≤ a ( j ) if i j and a ( i ) ≥ a ( j ) if i > j . We show, for lists of distinct elements, that the probability of a list splitting is asymptotic to 2 n ( n being the length of the list).

The Lateral Completion

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.

Completions of Representable and Archimedean ℓ-groups

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.

Holland’s Embedding Theorem

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.

Finite-valued and Special-valued ℓ-groups

In this chapter we shall inquire more carefully into the root system of values of an l-group, with the aim of seeing how much information about the l-group itself is contained in this set. The Conrad-Harvey-Holland embedding theorem (Theorem 3.2) serves as a model for this work; we shall discover that many of the results in this chapter, which apply for all normal-valued l-groups, are actually analogues of the abelian case. In particular, we shall determine more precisely the sense in which Read’s representation theorem 7.1.9 for normal-valued l-groups is the correct analogue of the Conrad-Harvey-Holland theorem.

Bernau’s Representation for Archimedean ℓ-groups

In the next four chapters we shall prove the four fundamental representation theorems for lattice-ordered groups, in increasing order of generality. Consequently, we shall begin with the representation of archimedean l-groups, as continuous almost finite real-valued functions on a topological space. It was Bernau [65b] who first stated and proved this theorem in the l-group context, but because of its analytic flavor it should not be surprising that this result was anticipated in work on partially ordered vector spaces, done by Amemiya [53], Nakano [N], Vulich [V] and Pinsker [49], and others.

The Conrad-Harvey-Holland Representation

In 1907 H. Hahn [07] proved that every abelian totally ordered group could be represented as a group of real-valued functions on a totally ordered set. His proof is long and difficult, and is one of the first successful uses of the technique of transfinite induction. It was not until after the Second World War that simplified proofs of Hahn’s theorem appeared, by Conrad [53] and Clifford [54]. In 1963, Conrad, Harvey and Holland [63] were able to generalize Hahn’s theorem to the class of lattice-ordered groups, using the idea of Banaschewski’s proof [57] of the original theorem; their theorem states that any abelian l-group can be represented as a group of real-valued functions on a partially ordered set (in fact, on a root system). In this chapter we shall present Wolfenstein’s proof [66] (or see [C]) of the Conrad-Harvey-Holland theorem.

Representable and Normal-valued ℓ-groups

Clifford [40] and Lorenzen [49a] first observed that an abelian l-group could be represented as a subdirect product of totally ordered groups; soon thereafter Lorenzen [49b] provided a characterization of those l-groups admitting such a representation.