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(Completely regular) locales generalize (Tychonoff) spaces; indeed, the passage from a locale to its spatial sublocale is a well understood coreflection. But a locale also possesses an equally important pointless sublocale, and with morphisms suitably restricted, the passage from a locale to its pointless sublocale is also a coreflection. Our main theorem is that every locale can be uniquely represented as a subdirect product of its pointless and spatial parts, again with suitably restricted projections. We then exploit this representation by showing that any locale is determined by (what may be described as) the placement of its points in its pointless part., Comment: 48 pages

The truncation operation facilitates the articulation and analysis of several aspects of the structure of archimedean vector lattices; we investigate two such aspects in this article. We refer to archimedean vector lattices equipped with a truncation as \emph{truncs}. In the first part of the article we review the basic definitions, state the (pointed) Yosida Representation Theorem for truncs, and then prove a representation theorem which subsumes and extends the (pointfree) Madden Representation Theorem. The proof has the virtue of being much shorter than the one in the literature, but the real novelty of the theorem lies in the fact that the topological data dual to a given trunc $G$ is a (localic) compactification, i.e., a dense pointed frame surjection $q \colon M \to L$ out of a compact regular pointed frame $M$. The representation is an amalgam of the Yosida and Madden representations; the compact frame $M$ is sufficient to describe the behavior of the bounded part $G^*$ of $G$ in the sense that $\widetilde{G}^*$ separates the points of the compact Hausdorff pointed space $X$ dual to $M$, while the frame $L$ is just sufficient to capture the behavior of the unbounded part of $G$ in $\mathcal{R}_0 L$. The truncation operation lends itself to identifying those elements of a trunc which behave like characteristic functions, and in the second part of the article we characterize in several ways those truncs composed of linear combinations of such elements. Along the way, we show that the category of such truncs is equivalent to the category of pointed Boolean spaces, and to the category of generalized Boolean algebras. The short third part contains a characterization of the kernels of truncation homomorphisms in terms of pointwise closure. In it we correct an error in the literature.

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 $m\circ n\equiv m+n-2$, 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.

We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of $\mathcal{R}L$, the family of continuous real valued functions on a locale, or pointfree space. Our setting is the category $\mathbf{W}$ of archimedean lattice-ordered groups ($\ell$-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. A main result is the appropriate analog of the Nakano-Stone Theorem: a (completely regular) locale $L$ has the feature that $\mathcal{R}L$ is conditionally pointwise complete ($\sigma $-complete), i.e., every bounded (countable) family from $\mathcal{R}L$ has a pointwise supremum in $\mathcal{R}L$, iff $L $ is boolean (a $P$-locale). We adopt a maximally broad definition of unconditional pointwise completeness ($\sigma$-completeness): a divisible $\mathbf{W}$-object $G$ is pointwise complete ($\sigma$-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete ($\sigma$-complete) $\mathbf{W}$-objects are those of the form $\mathcal{R}L$ for $L$ a boolean locale ($P$-locale). Finally, we show that a $\mathbf{W}$-object $G$ is pointwise $\sigma$-complete iff it is epicomplete.

This is the second of three articles on the topic of truncation as an operation on divisible abelian lattice-ordered groups, or simply $\ell$-groups. This article uses the notation and terminology of the first article and assumes its results. In particular, we refer to an $\ell$-group with truncation as a truncated $\ell$-group, or simply a trunc, and denote the category of truncs with truncation morphisms by $\mathbf{AT}$. Here we develop the analog for $\mathbf{AT}$ of Madden's pointfree representation for $\mathbf{W}$, the category of archimedean $\ell$-groups with designated order unit. More explicitly, for every archimedean trunc $A$ there is a regular Lindel\"{o}f frame $L$ equipped with a designated point $\ast : L \rightarrow 2$, a subtrunc $\widehat{A}$ of $\mathcal{R}_{0}L$, the trunc of pointed frame maps $\mathcal{O}_{0}\mathbb{R}\rightarrow L$, and a trunc isomorphism $A\rightarrow\widehat{A}$. A pointed frame map is just a frame map between frames which commutes with their designated points, and $\mathcal{O}_{0}\mathbb{R}$ stands for the pointed frame which is the topology $\mathcal{O}\mathbb{R}$ of the real numbers equipped with the frame map of the insertion $0 \to \mathbb{R}$. $\left( L,\ast\right) $ is unique up to pointed frame isomorphism with respect to its properties. Finally, we reprove an important result from the first article, namely that $\mathbf{W}$ is a non-full monoreflective subcategory of $\mathbf{AT}$.

We use a landmark result in the theory of Riesz spaces - Freudenthal's 1936 Spectral Theorem - to canonically represent any Archimedean lattice-ordered group $G$ with a strong unit as a (non-separating) lattice-group of real valued continuous functions on an appropriate $G$-indexed zero-dimensional compactification $w_GZ_G$ of its space $Z_G$ of \emph{minimal} prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of $G$ on its space $X_G$ of \emph{maximal} ideals, and the well-known continuous surjection of $Z_G$ onto $X_G$. We then establish our main result by showing that the inclusion-minimal extension of this representation of $G$ that separates the points of $Z_G$ - namely, the sublattice subgroup of ${\rm C}\,(Z_G)$ generated by the image of $G$ along with all characteristic functions of clopen (closed and open) subsets of $Z_G$ which are determined by elements of $G$ - is precisely the classical projectable hull of $G$. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit., Comment: 19 pages. Major revision. Our previous version contains a mistake: the clopens on the minimal spec of G given by zerosets of principal polars are assumed to be all clopens of the space. We are deeply grateful to an anonymous referee for a counterexample to our previous main statement. Please see the paper for details. Also, minor changes to some proofs, and added examples and references

We prove that the category of unital hyperarchimedean vector lattices is equivalent to the category of Boolean algebras. The key result needed to establish the equivalence is that, via the Yosida representation, such a vector lattice is naturally isomorphic to the vector lattice of all locally constant real-valued continuous functions on a Boolean (=compact Hausdorff totally disconnected) space. We give two applications of our main result., Comment: 15 pages. Submitted paper