×

A natural Galois connection between generalized norms and metrics. (English) Zbl 1507.06008

Summary: Having in mind a well-known connection between norms and metrics on vector spaces, for an additively written group \(X\), we establish a natural Galois connection between functions of \(X\) to \(\mathbb{R}\) and \(X^2\) to \(\mathbb{R}\).

MSC:

06A15 Galois correspondences, closure operators (in relation to ordered sets)
20A99 Foundations
54E25 Semimetric spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. Publ., 25, Providence, RI, 1967.; · Zbl 0153.02501
[2] T. S. Blyth, M. F. Janowitz, Residuation Theory, Pergamon Press, Oxford, 1972.; · Zbl 0301.06001
[3] S. Buglyó, Á. Száz, A more important Galois connection between distance functions and inequality relations, Sci. Ser. A Math. Sci. (N.S.), 18 (2009), 17-38.; · Zbl 1221.54031
[4] B. A. Davey, H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, Cambridge, 2002.; · Zbl 1002.06001
[5] K. Denecke, M. Erné, S. L. Wismath (Eds.) Galois Connections and Applications, Kluwer Academic Publisher, Dordrecht, 2004.; · Zbl 1050.06001
[6] P. Fischer, Gy. Muszély, On some new generalizations of the functional equation of Cauchy, Canadian Math. Bull., 10 (1967), 197-205.; · Zbl 0157.22303
[7] B. Ganter, R. Wille, Formal Concept Analysis, Springer-Verlag, Berlin, 1999.; · Zbl 0909.06001
[8] R. Ger, Fischer-Muszély additivity on Abelian groups, Comment Math., Tomus Specialis in Honorem Juliani Musielak (2004), 83-96.; · Zbl 1066.39024
[9] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott, A Compendium of Continuous Lattices, Springer-Verlag, Berlin, 1980.; · Zbl 0452.06001
[10] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Pan´stwowe Wydawnictwo Naukowe, Warszawa, 1985.;
[11] S. Kurepa, On P. Volkmann’s paper, Glasnik Mat., 22 (1987), 371-374.; · Zbl 0651.46031
[12] J. Lambek, Some Galois connections in elementary number theory, J. Number Theory, 47 (1994), 371-377.; · Zbl 0806.11009
[13] I. Makai, Über invertierbare Lösungen der additive Cauchy-Functionalgleichung, Publ. Math. Debrecen, 16 (1969), 239-243.; · Zbl 0202.42901
[14] Gy. Maksa, P. Volkmann, Characterizations of group homomorphisms having values in an inner product space, Publ. Math. Debrecen, 56 (2000), 197-200.; · Zbl 0991.39015
[15] P. Schöpf, Solutions of || f (ξ + η) || = || f (ξ) + f (η) ||, Math. Pannon.8 (1997), 117-127.; · Zbl 0881.39012
[16] Á. Száz, Preseminormed spaces, Publ. Math. Debrecen, 30 (1983), 217-224.; · Zbl 0553.46003
[17] Á. Száz, A Galois connection between distance functions and inequality relations, Math. Bohem., 127 (2002), 437-448.; · Zbl 1003.54012
[18] Á. Száz, Galois-type connections on power sets and their applications to relators, Tech. Rep., Inst. Math., Univ. Debrecen, 2005/2, 38 pp.;
[19] Á. Száz, Supremum properties of Galois-type connections, Comment. Math. Univ. Carolin., 47 (2006), 569-583.; · Zbl 1150.06300
[20] Á. Száz, An instructive treatment of convergence, closure and orthogonality in semi-inner product spaces, Tech. Rep., Inst. Math., Univ. Debrecen, 2006/2, 29 pp.;
[21] Á. Száz, Galois type connections and closure operations on preordered sets, Acta Math. Univ. Comen., 78 (2009), 1-21.; · Zbl 1199.06005
[22] Á. Száz, A common generalization of the postman, radial, and river metrics, Rostock. Math. Kolloq., 67 (2012), 89-125.; · Zbl 1285.54022
[23] Á. Száz, A particular Galois connection between relations and set functions, Acta Univ. Sapientiae, Math., 6 (2014), 73-91.; · Zbl 1317.06008
[24] Á. Száz, Galois and Pataki connections on generalized ordered sets, Tech. Rep., Inst. Math., Univ. Debrecen, 2014/3, 27 pp.; · Zbl 1453.06004
[25] Á. Száz, Remarks and Problems at the Conference on Inequalities and Applications, Hajdu´szoboszló, Hungary, 2014, Tech. Rep., Inst. Math., Univ. Debrecen, 2014/5, 12 pp.;
[26] Á. Száz, The closure-interior Galois connection and its applications to relational inclusions and equations, Tech. Rep., Inst. Math., Univ. Debrecen, 2015/2, 40 pp.;
[27] Á. Száz, Generalization of a theorem of Maksa and Volkmann on additive functions, Tech. Rep., Inst. Math., Univ. Debrecen, 1016/5, 6 pp. (An improved and enlarged version is available from the author.); · Zbl 1452.39008
[28] Á. Száz, Remarks and problems at the Conference on Inequalities and Applications, Hajdu´szoboszló, Hungary, 2016, Tech. Rep., Inst. Math., Univ. Debrecen, 2016/9, 34 pp.;
[29] Jacek Tabor, Józef Tabor, 19. Remark (Solution of the 7. Problem posed by K. Nikodem.), Aequationes Math., 61 (2001), 307-309.; · Zbl 0973.46012
[30] Jacek Tabor, Stability of the Fisher-Muszély functional equation, Publ. Math. Debrecen, 62 (2003), 205-211.; · Zbl 1026.39011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.