Quantale-Valued Uniformizations of Quantale-Valued Generalizations of Approach Groups

T. M. G. Ahsanullah and Gunther Jäger ()
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T. M. G. Ahsanullah: Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
Gunther Jäger: School of Mechanical Engineering, University of Applied Sciences Stralsund, Stralsund 18435, Germany

New Mathematics and Natural Computation (NMNC), 2019, vol. 15, issue 03, 517-538

Abstract: We introduce the categories of quantale-valued approach uniform spaces and quantale-valued uniform gauge spaces, and prove that they are topological categories. We first show that the category of quantale-valued uniform gauge spaces is a full bireflective subcategory of the category of quantale-valued approach uniform spaces and; second, we prove that only under strong restrictions on the quantale these two categories are isomorphic. Besides presenting embeddings of the category of quantale-valued metric spaces into the categories of quantale-valued approach uniform spaces as well as quantale-valued uniform gauge spaces, we show that every quantale-valued approach system group and quantale-valued gauge group has a natural underlying quantale-valued approach uniform space, respectively, a quantale-valued uniform gauge space.

Keywords: Approach space; approach group; quantale-valued metric space; quantale-valued approach system group; quantale-valued gauge group; quantale-valued approach uniform space; quantale-valued uniform gauge space; adjoint; category theory (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1142/S1793005719500303

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