Distinguished extensions of a lattice-ordered group.

*(English)*Zbl 0842.06012Author’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 \(G\)”.

Reviewer: J.Jakubík (Košice)

##### Keywords:

category of \(\ell\)-groups with \(\ell\)-homomorphisms; lattice-ordered group; lexicographic product with a totally ordered group; maximal essential extension; maximal distinguished extension; projectable hull
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