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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

##### MSC:
 06F15 Ordered groups 06F30 Ordered topological structures
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##### References:
  G. BIRKHOFF: Lattice Theory. New York 1973. · Zbl 0126.03801  L. FUCHS: Partially Ordered Algebraic Systems (Russian translation). Moskva 1965.  B. ŠMARDA: Topologies in tl-groups. Arch. Math., Brno, 1967, 69-81. · Zbl 0224.06013  B. ŠMARDA: Some types of tl-groups. Publ. Fac. Sci. Univ. J. E. Purkyně, Brno, 507, 1969, 341-352.  B. ŠMARDA: Connectivity in tl-groups. Arch. Math., Brno, 1976, 1-7. · Zbl 0373.22003  R. N. BALL: Topological lattice ordered groups. Pacific. J. Math., 83, 1979, 1-26. · Zbl 0434.06016
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