The amalgamation property for 𝐺-metric spaces

Hung, H. H.

doi:10.1090/S0002-9939-1973-0314008-1

The amalgamation property for $G$-metric spaces
HTML articles powered by AMS MathViewer

by H. H. Hung

Proc. Amer. Math. Soc. 37 (1973), 53-58

DOI: https://doi.org/10.1090/S0002-9939-1973-0314008-1

PDF | Request permission

Abstract:

Let G be a (totally) ordered (abelian) group. A G-metric space $(X,\rho )$ consists of a nonempty set X and a G-metric $\rho :X \times X \to G$ (satisfying the usual axioms of a metric, with G replacing the ordered group of real numbers). That the amalgamation property holds for the class of all metric spaces is attributed, by Morley and Vaught, to Sierpiński. The following theorem is proved. Theorem. The class of all G-metric spaces has the amalgamation property if, and only if, G is either the ordered group of the integers or the ordered group of the reals.

References

L. W. Cohen and Casper Goffman, The topology of ordered Abelian groups, Trans. Amer. Math. Soc. 67 (1949), 310–319. MR 32648, DOI 10.1090/S0002-9947-1949-0032648-5
L. W. Cohen and Casper Goffman, On the metrization of uniform space, Proc. Amer. Math. Soc. 1 (1950), 750–753. MR 38644, DOI 10.1090/S0002-9939-1950-0038644-X
Roland Fraïssé, Sur l’extension aux relations de quelques propriétés des ordres, Ann. Sci. École Norm. Sup. (3) 71 (1954), 363–388 (French). MR 69239, DOI 10.24033/asens.1027
Felix Hausdorff, Grundzüge der Mengenlehre, Chelsea Publishing Co., New York, 1949 (German). MR 31025
J. M. Howie, Embedding theorems with amalgamation for semigroups, Proc. London Math. Soc. (3) 12 (1962), 511–534. MR 138696, DOI 10.1112/plms/s3-12.1.511
Bjarni Jónsson, Universal relational systems, Math. Scand. 4 (1956), 193–208. MR 96608, DOI 10.7146/math.scand.a-10468
B. Jónsson, Homogeneous universal relational systems, Math. Scand. 8 (1960), 137–142. MR 125021, DOI 10.7146/math.scand.a-10601
Bjarni Jónsson, Algebraic extensions of relational systems, Math. Scand. 11 (1962), 179–205. MR 154833, DOI 10.7146/math.scand.a-10665
Bjarni Jónsson, Extensions of relational structures, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 146–157. MR 202601

The amalgamation property in varieties of modular lattices

18

On semigroups

Michael Morley and Robert Vaught, Homogeneous universal models, Math. Scand. 11 (1962), 37–57. MR 150032, DOI 10.7146/math.scand.a-10648
Keith R. Pierce, Amalgamations of lattice ordered groups, Trans. Amer. Math. Soc. 172 (1972), 249–260. MR 325488, DOI 10.1090/S0002-9947-1972-0325488-3
Waclaw Sierpiński, Sur un espace métrique séparable universel, Fund. Math. 33 (1945), 115–122 (French). MR 15451, DOI 10.4064/fm-33-1-115-122
Waclaw Sierpiński, Sur les espaces métriques universels, Fund. Math. 33 (1945), 123–136 (French). MR 15452, DOI 10.4064/fm-33-1-123-136
Roman Sikorski, Remarks on some topological spaces of high power, Fund. Math. 37 (1950), 125–136. MR 40643, DOI 10.4064/fm-37-1-125-136

Sur un espace métrique universal

180

Sur un espace métrique universel

51