Conrad frames

A Conrad frame is a frame which is isomorphic to the frame C(G) of all convex ?-subgroups of some lattice-ordered group G. It has long been known that Conrad frames have the disjointification property. In this paper a number of properties are considered that strengthen the disjointification property; they are referred to as the Conrad conditions. A particularly strong form of the disjointification property, the C-frame condition, is studied in detail. The class of lattice-ordered groups G for which C(G) is a C-frame is shown to coincide with the class of pairwise splitting ?-groups. The arguments are mostly frame-theoretic and Choice-free, until one tackles the question of whether C-frames are Conrad frames. They are, but the proof is decidedly not point-free. This proof actually does more: it shows that every algebraic frame with the FIP and disjointification can be coherently embedded in a C-frame. When the discussion is restricted to normal-valued lattice-ordered groups, one is able to produce examples of coherent frames having disjointification, which are not Conrad frames.     

Yosida frames

A Yosida frame is an algebraic frame in which every compact element is a meet of maximal elements. Yosida frames are used to abstractly characterize the frame of z-ideals of a ring of continuous functions C(X), when X is a compact Hausdorff space. An algebraic frame in which the meet of any two compact elements is compact is Yosida precisely when it is “finitely subfit”; that is, if and only if for each pair of compact elements a<b, there is a z (not necessarily compact) such that a∨z<1=b∨z. This is used to prove that if L is an algebraic frame in which the meet of any two compact elements is compact, and L has disjointification and dim(L)=1, then it is Yosida. It is shown that this result fails with almost any relaxation of the hypotheses. The paper closes with a number of examples, and a characterization of the Bézout domains in which the frame of semiprime ideals is Yosida frame.     

Remarks on the frame envelope of a σ-frame

The familiar equivalence between σ-frames and σ-coherent frames, given by the frame envelopes of σ-frames, is shown to induce an equivalence between stably continuous σ-frames and stably continuous frames. Similarly, the analogue of the former for σ-biframes is proved to provide an equivalence between compact regular σ-biframes and compact regular biframes. As an application we obtain the equivalence between stably continuous σ-frames and compact regular σ-biframes due to Matutu as an easy consequence of its frame counterpart established earlier by Banaschewski and Brümmer. This provides an affirmative answer to a question posed by Dana Scott.     

When an algebraic frame is regular

It is shown that an algebraic frame L is regular if and only if its compact elements are complemented. More generally, it is shown that each pseudocomplemented element is regular if and only if each , with c compact, is complemented. With a mild assumption on L, each , with c compact, is regular precisely when for any two minimal primes p and q of L. These results are then interpreted in various frames of subobjects of lattice-ordered groups and f-rings.     

Quotients and colimits of κ-quantales

Let κQnt be the category of κ-quantales, quantales closed under κ-joins in which the monoid identity is the largest element. (κ is an infinite regular cardinal.) Although the lack of lattice completeness in this setting would seem to mitigate against the techniques which lend themselves so readily to the calculation of frame quotients, we show how to easily compute κQnt quotients by applying generalizations of the frame techniques to suitable extensions of this category.The second major tool in the analysis is the free κ-quantale over a λ-quantale, κ?λ. Surprisingly, these can be characterized intrinsically, and the generating sub-κ-quantale can even be identified. The result that the λ-free κ-quantales coincide with the λ-coherent κ-quantales directly generalizes Madden?s corresponding result for κ-frames.These tools permit a direct and intuitive construction of κQnt colimits. We provide two applications: an intrinsic characterization of κQnt colimits, and of free (over sets) κ-quantales. The latter is a direct generalization of Whitman?s condition for distributive lattices.     

Ring ideals and the Stone-?ech compactification in pointfree topology

This paper shows that the compact completely regular coreflection in the category of frames is given by the frame of Jacobson radical ideals of the ring RL of real-valued continuous functions on L, as an alternative to its familiar representations in terms of (i) the l-ideals of RL as lattice-ordered ring or (ii) the ideals of the bounded part of RL which are closed in the usual uniform topology. Further, in analogy with this, the compact zero-dimensional coreflection will also be described in terms of ring ideals, this time of the ring ZL of integer-valued continuous functions on L.     

On the atomic conditions of lattice-ordered groups

We introduce large convex -subgroups to study the structure of the lattice-ordered groups G whose C(G), P(G) and (G) satisfy atomic conditions, where C(G), P(G) and (G) denote respectively the lattice of all convex -subgroups, the lattice of all polar subgroups and the root system of all regular subgroups of G. In particular, we construct a new torsion class defined as the class of -groups G for which all large prime subgroups are maximal. We prove that the class of hyperarchimedean -groups is properly contained within and that any -group within has the property that any chain of prime subgroups has length at most 2.Received October 7, 2003; accepted in final form June 11, 2004.     

On the separation of convex sets in some idempotent semimodules

Two linear maps are usually needed to separate disjoint convex subsets of an idempotent semimodule. In the context of Max-Plus convexity separation can be achieved by a single map if one considers linear maps with values in a linearly ordered semimodule, whose construction is given here, which is not the Max-Plus semiring R∪{-∞}.     

Convex vector lattices and l-algebras

For A an Archimedean Riesz space (=vector lattice) with distinguished positive weak unit e_A, we have the Yosida representation  as a Riesz space in D(X_A), the lattice of extended real valued functions on the space of e_A-maximal ideas. This note is about those A for which  is a convex subset of D(X_A); we call such A “convex”.Convex Riesz spaces arise from the general issue of embedding as a Riesz ideal, from consideration of uniform- and order-completeness, and from some problems involving comparison of maximal ideal spaces (which we won't discuss here; see [10]).The main results here are: (2.4) A is convex iff A is contained as a Riesz ideal in a uniformly complete Φ-algebra B with identity e_A. (3.1) Any A has a convex reflection (i.e., embeds into a convex B with a universal mapping property for Riesz homomorphisms; moreover, the embedding is epic and large).     

Tightness relative to some (co)reflections in topology

We address what might be termed the reverse reflection problem: given a monoreflection from a category A onto a subcategory B, when is a given object B ∈ B the reflection of a proper subobject? We start with a well known specific instance of this problem, namely the fact that a compact metric space is never the ?ech-Stone compactification of a proper subspace. We show that this holds also in the pointfree setting, i.e., that a compact metrizable locale is never the ?ech-Stone compactification of a proper sublocale. This is a stronger result than the classical one, but not because of an increase in scope; after all, assuming weak choice prin­ciples, every compact regular locale is the topology of a compact Hausdorff space. The increased strength derives from the conclusion, for in general a space has many more sublocales than subspaces. We then extend the analysis from metric locales to the broader class of perfectly normal locales, i.e., those whose frame of open sets consists entirely of cozero elements. We include a second proof of these results which is purely algebraic in character.

At the opposite extreme from these results, we show that an extremally disconnected locale is a compactification of each of its dense sublocales. Finally, we analyze the same phenomena, also in the pointfree setting, for the 0-dimensional compact reflec­tion and for the Lindelöf reflection.     

A proof of Coleman's conjecture

Almost thirty years ago Coleman made a conjecture that for any convex lattice polygon with v vertices, g (g?1) interior lattice points and b boundary lattice points we have b?2g-v+10. In this note we give a proof of the conjecture. We also aim to describe all convex lattice polygons for which the bound b=2g-v+10 is attained.     

Rings of continuous functions on σ-frames

This paper deals with the ?-rings RS of all real-valued continuous functions on a completely regular σ-frame. It shows that, in marked contrast with the situation for frames, any ?-ring homomorphism RS→RT results from a σ-frame homomorphism S→T. Further, it proves the analogue of this for integer-valued continuous functions and 0-dimensional σ-frames. In all, this demonstrates that the important classical difference between Alexandroff spaces and Tychonoff spaces with respect to the real-valued continuous functions carries over fully to the pointfree setting - indeed, it adds the integer-valued case which seems to be new in this context.     

The orderability of nonarchimedean spaces

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F_σ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if $\text{C}\text{?}\text{?C}$ is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.     

A constructive and functorial embedding of locally compact metric spaces into locales

The paper establishes, within constructive mathematics, a full and faithful functor M from the category of locally compact metric spaces and continuous functions into the category of formal topologies (or equivalently locales). The functor preserves finite products, and moreover satisfies f?g if, and only if, M(f)?M(g) for continuous . This makes it possible to transfer results between Bishop's constructive theory of metric spaces and constructive locale theory.     

Properties of P-sets and trapped compact convex sets

New properties of P-sets, which constitute a large class of convex compact sets in ? ⁿ that contains all convex polyhedra and strictly convex compact sets, are obtained. It is shown that the intersection of a P-set with an affine subspace is continuous in the Hausdorff metric. In this theorem, no assumption of interior nonemptiness is made, unlike in other known intersection continuity theorems for set-valued maps. It is also shown that if the graph of a set-valued map is a P-set, then this map is continuous on its entire effective set rather than only on the interior of this set. Properties of the so-called trapped sets are also studied; well-known Jung’s theorem on the existence of a minimal ball containing a given compact set in ? ⁿ is generalized. As is known, any compact set contains n + 1 (or fewer) points such that any translation by a nonzero vector takes at least one of them outside the minimal ball. This means that any compact set is trapped in the minimal ball. Compact sets trapped in any convex compact sets, rather than only in norm bodies, are considered. It is shown that, for any compact set A trapped in a P-set M ? ? ⁿ , there exists a set A ⁰ ? A trapped in M and containing at most 2n elements. An example of a convex compact set M ? ? ⁿ for which such a finite set A ⁰ ? A does not exist is given.     

Concerning some variants of C-embedding in pointfree topology

We study three types of quotient maps of frames which are closely related to C- and C^?-quotient maps. We call them C₁-, strong C₁-, and uplifting quotient maps. C₁-quotient maps are precisely those whose induced ring homomorphisms contract maximal ideals to maximal ideals. We show that every homomorphism onto a frame is a C₁-, a strong C₁-, or an uplifting quotient map iff the frame is pseudocompact, compact, or almost compact and normal, respectively. These quotient maps are used to characterize normality and also certain weaker forms of normality in a manner akin to the characterization of normal frames as those for which every closed quotient map is a C-quotient map. Under certain conditions, we show that the Stone extension of a quotient map is C₁-, strongly C₁- or uplifting if the map has the corresponding property.     

Realcompactness and certain types of subframes

If a frame satisfies a property which is some variant of realcompactness, then certain types of its subframes inherit the property. Conversely, there are instances where a frame has a subframe satisfying some such property only if the frame itself satisfies the property. We analyze these phenomena for the case of realcompactness, almost realcompactness, a-realcompactness, c-realcompactness and weak realcompactness vis-a-vis perfect subframes and flat subframes. Received September 19, 2006; accepted in final form March 10, 2007.     

A new family of convex weakly compact valued random variables in Banach space and applications to laws of large numbers

We consider a new family of convex weakly compact valued integrable random sets which is called an adapted array of convex weakly compact valued integrable random variables of type p (1?p?2). By this concept, more general laws of large numbers will be established. Some illustrative examples are provided.