### Some Remarks on the Tensor Product of Function Rings.

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For a Tychonoff space $X$, $C\left(X\right)$ is the lattice-ordered group ($l$-group) of real-valued continuous functions on $X$, and ${C}^{*}\left(X\right)$ is the sub-$l$-group of bounded functions. A property that $X$ might have is (AP) whenever $G$ is a divisible sub-$l$-group of ${C}^{*}\left(X\right)$, containing the constant function 1, and separating points from closed sets in $X$, then any function in $C\left(X\right)$ can be approximated uniformly over $X$ by functions which are locally in $G$. The vector lattice version of the Stone-Weierstrass Theorem is more-or-less equivalent...

The $\sigma $-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\left\{{b}_{n}\right\}$ of positive elements of $B$, there is a sequence $\left\{{\lambda}_{n}\right\}$ of positive reals, and $b\in B$, with ${\lambda}_{n}{b}_{n}\le b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when “$\sigma $” obtains for a Riesz space of continuous real-valued functions $C\left(X\right)$. A basic result is: For discrete $X$, $C\left(X\right)$ has $\sigma $ iff the cardinal $\left|X\right|<\U0001d51f$, Rothberger’s bounding number. Consequences and...

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:...

$\mathrm{\mathbf{S}\mathbf{p}\mathbf{F}\mathbf{i}}$ is the category of spaces with filters: an object is a pair $(X,\mathcal{F})$, $X$ a compact Hausdorff space and $\mathcal{F}$ a filter of dense open subsets of $X$. A morphism $f\phantom{\rule{0.222222em}{0ex}}(Y,\mathcal{G})\to (X,\mathcal{F})$ is a continuous function $f\phantom{\rule{0.222222em}{0ex}}Y\to X$ for which ${f}^{-1}\left(F\right)\in \mathcal{G}$ whenever $F\in \mathcal{F}$. 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 kernel functor” $W\stackrel{k}{\to}LFrm$ from the category $W$ of archimedean lattice-ordered groups with distinguished weak unit onto LFrm, of Lindelöf completely regular frames, preserves and reflects monics. In $W$, monics are one-to-one, but not necessarily so in LFrm. An embedding $\varphi \in W$ for which $k\varphi $ 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 $W$-objects every embedding of which is KI are characterized;...

For Tychonoff $X$ and $\alpha $ an infinite cardinal, let $\alpha defX:=$ the minimum number of $\alpha $ cozero-sets of the Čech-Stone compactification which intersect to $X$ (generalizing $\mathbb{R}$-defect), and let $rtX:={min}_{\alpha}max(\alpha ,\alpha defX)$. Give $C\left(X\right)$ the compact-open topology. It is shown that $\tau C\left(X\right)\le n\chi C\left(X\right)\le rtX=max\left(L\right(X),L(X)defX)$, where: $\tau $ is tightness; $n\chi $ is the network character; $L\left(X\right)$ is the Lindel"of number. For example, it follows that, for $X$ Čech-complete, $\tau C\left(X\right)=L\left(X\right)$. The (apparently new) cardinal functions $n\chi C$ and $rt$ are compared with several others.

Usually, an abelian $\ell $-group, even an archimedean $\ell $-group, has a relatively large infinity of distinct $a$-closures. Here, we find a reasonably large class with unique and perfectly describable $a$-closure, the class of archimedean $\ell $-groups with weak unit which are “$\mathbb{Q}$-convex”. ($\mathbb{Q}$ is the group of rationals.) Any $C(X,\mathbb{Q})$ is $\mathbb{Q}$-convex and its unique $a$-closure is the Alexandroff algebra of functions on $X$ defined from the clopen sets; this is sometimes $C\left(X\right)$.

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