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Topologies corresponding to metrics on \(\ell\)-groups. (English) Zbl 0578.06013
Let G be a lattice ordered group and let H be an abelian lattice ordered group. A mapping \(v: G\to H\) is said to be an \(\ell p\)-valuation on G if \(v(x)+v(y)=v(x\vee y)+v(x\wedge y)\) for all \(x,y\in G\). Put \(d(x,y)=v(x\vee y)-v(x\wedge y)\), \(\bar v(x)=v(x)-v(0)\). Then d is called an \(\ell\)-metric on G. The author investigates conditions (expressed in terms of the \(\ell\)-metric d) for \(\bar v\) to be a group homomorphism. Further there are studied two topologies on G defined by means of d (under the assumption that d is compatible with the group operation). In the first of these topologies, the complete system of neighbourhoods of 0 is \(\{U_ h\}_{0\neq h\in H}\), where \(U_ h=\{g\in G:\) \(\bar v(| g|)<| h|\}\); in the second one the corresponding system is \(\{U^ h\}_{0\neq h\in H}\), where \(U^ h=\{g\in G:\) \(\check v{\bar{\;}}(| g|)\) non\(\geq | h| \}\).
Reviewer: J.Jakubík

06F15 Ordered groups
06F30 Ordered topological structures
Full Text: EuDML
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