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The \(C\)-topology on lattice-ordered groups. (English) Zbl 1194.06011

Summary: Let \(A\) be a lattice-ordered group. I. Gusić [Proc. Am. Math. Soc. 126, No. 9, 2593–2597 (1998; Zbl 0943.06009)] showed that \(A\) can be equipped with a \(C\)-topology which makes \(A\) a topological group. We give a generalization of Gusić’s theorem, and reveal the very nature of a “\(C\)-group” of Gusić. Moreover, we show that the \(C\)-topological groups are topological lattice-ordered groups, and prove that every Archimedean lattice-ordered vector space is a \(T_{2}\) topological lattice-ordered vector space under the \(C\)-topology. An easy example shows that a \(C\)-group need not be \(T_{2}\). A further example demonstrates that a \(T_{2}\) topological Archimedean lattice-ordered group need not be \(C\)-Archimedean, either.

MSC:

06F15 Ordered groups
06F30 Ordered topological structures
22A99 Topological and differentiable algebraic systems
46A40 Ordered topological linear spaces, vector lattices

Citations:

Zbl 0943.06009

Software:

QPL
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References:

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