Some Remarks on the Tensor Product of Function Rings.
Local/global uniform approximation of real-valued continuous functions
For a Tychonoff space , is the lattice-ordered group (-group) of real-valued continuous functions on , and is the sub--group of bounded functions. A property that might have is (AP) whenever is a divisible sub--group of , containing the constant function 1, and separating points from closed sets in , then any function in can be approximated uniformly over by functions which are locally in . The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...
The -property in
The -property of a Riesz space (real vector lattice) is: For each sequence of positive elements of , there is a sequence of positive reals, and , with for each . This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “” obtains for a Riesz space of continuous real-valued functions . A basic result is: For discrete , has iff the cardinal , Rothberger’s bounding number. Consequences and...
Real-valued functions on Alexandroff (zero-set) spaces
Functorial rings of quotients 2: the ring of hyper-fractions.
The Projection Mapping and Other Continuous Functions on a Product Space.
Algebraic Classes of Abelian Torsion-Free and Lattice-Ordered Groups
Functorial approximation to the lateral completion in archimedean lattice-ordered groups with weak unit
Majorizing-injectivity in abelian lattice-ordered groups
Extremal units in an archimedean Riesz space
Some examples of hyperarchimedean lattice-ordered groups
All ℓ-groups shall be abelian. An a-extension of an ℓ-group is an extension preserving the lattice of ideals; an ℓ-group with no proper a-extension is called a-closed. A hyperarchimedean ℓ-group is one for which each quotient is archimedean. This paper examines hyperarchimedean ℓ-groups with unit and their a-extensions by means of the Yosida representation, focussing on several previously open problems. Paul Conrad asked in 1965: If G is a-closed and M is an ideal, is G/M a-closed? And in 1972:...
Network character and tightness of the compact-open topology
For Tychonoff and an infinite cardinal, let the minimum number of cozero-sets of the Čech-Stone compactification which intersect to (generalizing -defect), and let . Give the compact-open topology. It is shown that , where: is tightness; is the network character; is the Lindel"of number. For example, it follows that, for Čech-complete, . The (apparently new) cardinal functions and are compared with several others.
Frame monomorphisms and a feature of the -group of Baire functions on a topological space
“The kernel functor” from the category of archimedean lattice-ordered groups with distinguished weak unit onto LFrm, of Lindelöf completely regular frames, preserves and reflects monics. In , monics are one-to-one, but not necessarily so in LFrm. An embedding for which is one-to-one is termed kernel-injective, or KI; these are the topic of this paper. The situation is contrasted with kernel-surjective and -preserving (KS and KP). The -objects every embedding of which is KI are characterized;...
The commuting of coreflectors in uniform spaces with completion
Monomorphisms in spaces with Lindelöf filters
is the category of spaces with filters: an object is a pair , a compact Hausdorff space and a filter of dense open subsets of . A morphism is a continuous function for which whenever . This category arises naturally from considerations in ordered algebra, e.g., Boolean algebra, lattice-ordered groups and rings, and from considerations in general topology, e.g., the theory of the absolute and other covers, locales, and frames, though we shall specifically address only...
The category of compactifications and its coreflections
We define “the category of compactifications”, which is denoted , and consider its family of coreflections, denoted . We show that is a complete lattice with bottom the identity and top an interpretation of the Čech–Stone . A implies the assignment to each locally compact, noncompact a compactification minimum for membership in the “object-range” of . We describe the minimum proper compactifications of locally compact, noncompact spaces, show that these generate the atoms...
Inversion in a class of lattice-ordered algebras
Unique -closure for some -groups of rational valued functions
Usually, an abelian -group, even an archimedean -group, has a relatively large infinity of distinct -closures. Here, we find a reasonably large class with unique and perfectly describable -closure, the class of archimedean -groups with weak unit which are “-convex”. ( is the group of rationals.) Any is -convex and its unique -closure is the Alexandroff algebra of functions on defined from the clopen sets; this is sometimes .
Extensions of Zero-sets and of Real-valued Functions.
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