Richard N. Ball

Convergence and Cauchy structures on lattice ordered groups

This paper employs the machinery of convergence and Cauchy struc- tures in the task of obtaining completion results for lattice ordered groups. §§ 1 and 2 concern /-convergence and /-Cauchy structures in general. §4 takes up the order convergence structure; the resulting completion is shown to be the Dedekind- MacNeille completion. §5 concerns the polar convergence structure; the corre- sponding completion has the property of lateral completeness, among others. A simple theory of subset types routinizes the adjoining of suprema in §3. This procedure, nevertheless, is shown to be sufficiently general to prove the existence and uniqueness of both the Dedekind-MacNeille completion in §4 and the lateral completion in §5. A proof of the existence and uniqueness of a proper class of similar completions comes free. The principal new hull obtained by the techniques of adjoining suprema is the type •?) hull, strictly larger than the lateral completion in general. As is nearly always true in the field of lattice ordered groups, this research follows a path first trod by Conrad (10) and Holland (14). The contributions of Papangelou ((19), (20)) and Kenny (15) have also been important. But the novelty of the present approach lies in the systematic application of convergence and Cauchy structure techniques, significantly more powerful than topological and uniformity techniques. These new techniques have been developed recently by Kent and others ((16), (22)). The author owes a more personal debt of gratitude to professors Gary Davis, G. Otis Kenny, and Darrell Kent for many stimulating conversations on these topics. A collection x) all x G X}. LG(X) and UG(X) are often written more simply L(X) and U(X). The order closure of X, written ocl(X), is defined inductively. X0 = X,

On the localic Yosida representation of an archimedean lattice ordered group with weak order unit

W is the category of archimedean lattice ordered groups with weak order unit, and F is the category of frames. C:F→W is the functor which assigns to a given frame F the W-object CF=HomF(OR,F), where OR designates the frame of open sets of the real numbers R, and assigns to the frame homomorphism f:F→L the W-homomorphism Cf:CF→CL defined by Cf(g)=fg for all g ϵ CF. On the other hand, Y:W → F is the functor which assigns to a given W-object G the frame YG of W-kernels of G, and assigns to the W-homomorphism u:G→H the frame homomorphism Yu:YG→YH whose value at K ϵ YG is the W-kernel of H generated by u(K). We prove in Theorem 2.3.2 that Y is left adjoint to C, and in Theorem 2.4.3 that the adjunction restricts to an equivalence between the full subcategory RL of regular Lindelof frames and the full subcategory C of W-objects of the form CF for some frame F. The equivalence was first established by Madden and Vermeer. The unit μ=(μG) is the reflection of W in C, herein termed the (localic) Yosida representation, while the counit λ=(λF) is the (frame counterpart of the) reflection of locales into regular Lindelof locales. Finally, we note that the sense in which the Yosida representation is unique is precisely that (μG, YG) is the C-universal map for G.

Colored graphs without colorful cycles

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation mon ≡ m + n − 2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized.We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having 2, 4, and 5 vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality.

Normal subgroups of doubly transitive automorphism groups of chains

We characterize the structure of the normal subgroup lattice of 2-transitive automorphism groups A(Q) of infinite chains (Q, <) by the structure of the Dedekind completion (Q, <) of the chain (Q7 <). As a consequence we obtain various group-theoretical results on the normal subgroups of A(Q), including that any proper subnormal subgroup of A(Q) is indeed normal and contained in a maximal proper normal subgroup of A(Q), and that A(Q) has precisely 5 normal subgroups if and only if the coterminality of the chain (Q, < ) is countable.

Configurations in coproducts of Priestley spaces

Abstract Let P be a configuration, i.e., a finite poset with top element. Let

Characterization of Epimorphisms in Archimedean Lattice-Ordered Groups and Vector Lattices

\hbox{\textsf{Forb}}(P)

Distinguished extensions of a lattice-ordered group

be the class of bounded distributive lattices L whose Priestley space ℘(L) contains no copy of P. We show that the following are equivalent:

Completions of l-Groups

\hbox{\textsf{Forb}}(P)

Dualities in full homomorphisms

is first-order definable, i.e., there is a set of first-order sentences in the language of bounded lattice theory whose satisfaction characterizes membership in

The relative uniform density of the continuous functions in the Baire functions, and of a divisible Archimedean ℓ-group in any epicompletion

{\hbox{\textsf{Forb}}}(P)