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Each join-completion of a partially ordered set in the solution of a universal problem

Published online by Cambridge University Press:  09 April 2009

Jürgen Schmidt
Affiliation:
Department of Mathematics, University of Houston, Houston, Texas 77004, U. S. A.
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The main result of this paper is the theorem in the title. Only special cases of it seem to be known so far. As an application, we obtain a result on the unique extension of Galois connexions. As a matter of fact, it is only by the use of Galois connexions that we obtain the main result, in its present generality.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Aumann, G., Reellee Fanktionen. (Berlin 1954).CrossRefGoogle Scholar
[2]Aumann, G., ‘Bemerkung über Galois-Verbindungen’, S. Ber. math. nat. Kl. Bayer. Akad. Wiss. 1955, 281284.Google Scholar
[3]Banaschewski, B., ‘Hüllensysteme und Erweiterungen von Quasi-Ordnungen’, Z. math Logik Grand. Math. 2 (1956), 117130.CrossRefGoogle Scholar
[4]Bruns, G., ‘Darstellungen und Erweiterungen geordneter Mengen’, J. reine angew. Math. 209 (1962), 167200.CrossRefGoogle Scholar
[5]Doctor, H. P., Extensions of a partially ordered Set. Thesis (Hamilton 1967.)Google Scholar
[6]Everett, C. J., ‘Closure operators and Galois theory in lattices’, Trans. Amer. Math. Soc. 55 (1944), 514535.CrossRefGoogle Scholar
[7]Ore, O., ‘Galois connexions’, Trans. Amer. Math. Soc. 55 (1944), 493513.CrossRefGoogle Scholar
[8]Pickert, G., ‘Bemerkungen über Galois-Verbindungen, Arch. Math. 3 (1952), 285289.CrossRefGoogle Scholar
[9]Raney, G. N., ‘Tight Galois connections and complete distributivity’, Trans. Amer. Math. Soc. 97 (1960), 723730.CrossRefGoogle Scholar
[10]Schmidt, J., ‘Abgeschlossenheits — und Homomorphiebegriffe in der Ordungstheorie’, Wiss. Z. Humboldt-Univ. Berlin, math. nat. R. 3. (1953/1954), 223225.Google Scholar
[11]Schmidt, J., ‘Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Menge’, Arch. Math. 7, (1956), 241249.CrossRefGoogle Scholar
[12]Schmidt, J., ‘Universal and internal properties of some extensions of partially ordered sets’, J. reine angew. Math. 253 (1972), 2842.Google Scholar
[13]Schmidt, J., ‘Universal and internal properties of some completions of k-join-semilattices and k-join-distributive partially ordered sets’, J. reine angew. Math. 255 (1972), 822.Google Scholar
[14]Wright, F. B., ‘Polarity and duality’, Pacif. J. Math. 10 (1960), 723730.CrossRefGoogle Scholar