Abstract
This paper considers metrics valued in abelian ℓ-groups and their induced topologies. In addition to a metric into an ℓ-group, one needs a filter in the positive cone to determine which balls are neighborhoods of their center. As a key special case, we discuss a topology on a lattice ordered abelian group from the metric d G and the positive filter consisting of the weak units of G; in the case of \({\mathbb R^{n}}\) , this is the Euclidean topology. We also show that there are many Nachbin convex topologies on an ℓ-group which are not induced by any positive filter of the ℓ-group.
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Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, 3, Clarendon Press, (1994), 1–168. Corrected and expanded version available at http://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf
Birkhoff, G.: Lattice Theory. Amer. Math. Soc. Colloquium Publications, vol. XXV (1967)
Darnel M.: The Theory of Lattice-Ordered Groups. Marcel Dekker, New York (1995)
Hewitt E.: Rings of real-valued continuous functions I. Trans. Amer. Math. Soc. 64, 45–99 (1948)
Holland W.C.: Intrinsic metrics for lattice ordered groups. Algebra Universalis 19, 142–150 (1984)
Jasem M.: Intrinsic metric preserving maps on partially ordered groups. Algebra Universalis 36, 135–140 (1996)
Kalman J. A.: Triangle Inequality in l-groups. Proc. Amer Math. Soc. 11, 395 (1960)
Kopperman R.: All topologies come from generalized metrics. Am. Math. Monthly 95, 89–97 (1988)
Kopperman R., Matthews S., Pajoohesh H.: Partial metrizability in value quantales. Applied General Topology 5, 115–127 (2004)
Matthews, S.: Partial metric topology. In: Itzkowitz, G., et al., (eds.),College, 1992. Annals of the New York Academy of Sciences, vol. 728, pp. 183–197. New York Academy of Sciences, New York (1994)
Nachbin, L.: Topology and Order. Van Nostrand Mathematical Studies #4, Princeton (1965)
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Presented by J. Martinez.
In memory of Mel Henriksen
The first author wishes to acknowledge the support of this research by PSC-CUNY grants #69487-00 38 and #61743-00 39.
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Kopperman, R., Pajoohesh, H. & Richmond, T. Topologies arising from metrics valued in abelian ℓ-groups. Algebra Univers. 65, 315–330 (2011). https://doi.org/10.1007/s00012-011-0132-5
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DOI: https://doi.org/10.1007/s00012-011-0132-5