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Topologies on abelian lattice ordered groups induced by a positive filter and completeness

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Abstract

We consider topologies on an abelian lattice ordered group that are determined by the absolute value and a positive filter. We show that the topological completions of these objects are also determined by the absolute value and a positive filter. We investigate the connection between the topological completion of such objects and the Dedekind–MacNeille completion of the underlying lattice ordered group. We consider the preservation of completeness for such topologies with respect to homomorphisms of lattice ordered groups. Finally, we show that topologies defined in terms of absolute value and a positive filter on the space C(X) of all real-valued continuous functions defined on a completely regular topological space X are always complete.

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Acknowledgements

The authors thank the referee(s) of this paper for their many suggestions, which have greatly improved this paper. In particular, Section 6 would not exist without their comments.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Queensborough Community College, New York, USA

    Francis Jordan

  2. Department of Mathematics, Medgar Evers College, New York, USA

    Homeira Pajoohesh

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  1. Francis Jordan
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  2. Homeira Pajoohesh
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Correspondence to Homeira Pajoohesh.

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Presented by W. Wm. McGovern.

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Jordan, F., Pajoohesh, H. Topologies on abelian lattice ordered groups induced by a positive filter and completeness. Algebra Univers. 79, 62 (2018). https://doi.org/10.1007/s00012-018-0543-7

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  • DOI: https://doi.org/10.1007/s00012-018-0543-7

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