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, 18435 Stralsund, Germany

New Mathematics and Natural Computation (NMNC), 2019, vol. 15, issue 01, 1-30

Abstract: The motive behind this paper is to generalize the concept of approach groups. In so doing, we introduce various categories, specifically, the categories of quantale-valued convergence groups, quantale-valued approach groups, quantale-valued gauge groups and quantale-valued approach system groups, and study their functorial relationships including the fact that the category of quantale-valued gauge groups as well as the category of quantale-valued approach system groups is topological over the category of groups. Besides obtaining a variety of categorical connections, we note that if the quantale is linearly ordered satisfying certain conditions, then the categories of quantale-valued approach system groups and quantale-valued gauge groups are isomorphic. Finally, we look into the embeddings of the category of quantale-valued metric groups into the categories of quantale-valued generalizations of approach groups including commutativity of some diagrams.

Keywords: Quantale; approach space; approach system; gauge space; quantale-valued metric space; approach group (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1142/S1793005719500017

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