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Distinguished extensions of a lattice-ordered group. (English) Zbl 0842.06012
Author’s abstract: “Any lattice-ordered group (l-group for short) is essentially extended by its lexicographic product with a totally ordered group. That is, an l-homomorphism (i.e., a group and lattice homomorphism) on the extension which is injective on the l-group must be injective on the extension as well. Thus no l-group has a maximal essential extension in the category IGp of l-groups with l-homomorphisms. However, an l-group is a distributive lattice, and so it has a maximal essential extension in the category $$\mathbf D$$ of distributive lattices with lattice homomorphisms. A distinguished extension of one l-group by another is one which is essential in $$\mathbf D$$. We characterize such extensions, and show that every l-group $$G$$ has a maximal distinguished extension $$E(G)$$ which is unique up to an l-isomorphism over $$G$$. $$E(G)$$ contains most other known completions in which $$G$$ is order dense, and has most l-group completeness properties as a result. Finally, we show that if $$G$$ is projectable then $$E(G)$$ is the $$\alpha$$-completion of the projectable hull of $$G$$”.

##### MSC:
 06F15 Ordered groups 06D05 Structure and representation theory of distributive lattices
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