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Algebraic closures of \(\ell\)-groups of continuous functions. (English) Zbl 0616.06017

Rings of continuous functions, Pap. Spec. Sess. Annu. Meet. Am. Math. Soc., Cincinnati/Ohio 1982, Lect. Notes Pure Appl. Math. 95, 165-193 (1985).
[For the entire collection see Zbl 0559.00006.]
We consider the category \({\mathcal W}\) of Archimedean lattice-ordered groups with distinguished nonnegative weak order unit, with unit-preserving \(\ell\)-homomorphisms. Each object of \({\mathcal W}\) is an ”\(\ell\)-group of continuous functions” via the Yosida representation, which says that A is isomorphic to an \(\ell\)-group of continuous, extended-real valued, almost-finite functions on a certain compact Hausdorff space Y(A). Pursuing a familiar theme in general algebra, we describe certain extensions of an object A as obtained by closing A under certain collections of implicit operations.

MSC:

06F15 Ordered groups
54H15 Transformation groups and semigroups (topological aspects)
54C35 Function spaces in general topology

Citations:

Zbl 0559.00006