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Unique $a$-closure for some $\ell$-groups of rational valued functions
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 409-421
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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 $C(X)$.
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 $C(X)$.
Classification : 06F20, 06F25, 20F60, 54C30, 54F65
Keywords: archimedean lattice-ordered group; $a$-closure; rational-valued functions; zero-dimensional space 
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Hager, Anthony W.; Kimber, Chawne M.; McGovern, Warren W. Unique $a$-closure for some $\ell$-groups of rational valued functions. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 2, pp. 409-421. http://geodesic.mathdoc.fr/item/CMJ_2005_55_2_a10/

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