Abstract
We introduce a special type of order-preserving maps between quasiordered sets, the so-called cut-stable maps. These form the largest morphism class such that the corresponding category of quasiordered sets contains the category of complete lattices and complete homomorphisms as a full reflective subcategory, the reflector being given by the Dedekind-MacNeille completion (alias normal completion or completion by cuts). Suitable restriction of the object class leads to the category of separated quasiordered sets and its full reflective subcategory of completely distributive lattices. Similar reflections are obtained for continuous lattices, algebraic lattices, etc.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.Abian (1968) On definitions of cuts and completion of partially ordered sets, Z. Math. Logik Grundl. der Math. 14, 299–309.
B.Banaschewski (1956) Hüllensysteme und Erweiterung von Quasi-Ordnungen, Z. Math. Logik Grundl. der Math. 2, 117–130.
B.Banaschewski and G.Bruns (1967) Categorical characterization of the MacNeille completion, Arch. Math. (Basel) 43, 369–377.
G. Birkhoff (1973) Lattice Theory, Amer. Math. Soc. Coll. Publ. 25, 3rd ed., Providence, R.I.
A.Bishop (1978) A universal mapping characterization of the completion by cuts. Algebra Universalis 8, 349–353.
T. S.Blyth and M. S.Janowitz (1972) Residuation Theory, Pergamon Press, Oxford.
G.Bruns (1962) Darstellungen und Erweiterungen geordneter Mengen I, II, J. Reine Angew. Math. 209, 167–200, and 210, 1–23.
P.Crawley and R. P.Dilworth (1973) Algebraic Theory of Lattices, Prentice-Hall, Inc., Englewood Cliffs, N.J.
R.Dedekind (1872) deStetigkeit und irrationale Zahlen, 7th ed. (1969), Vieweg, Braunschweig.
M. Erné (1980) Verallgemeinerungen der Verbandstheorie, I, II, Preprint No. 109 and Habilitations-schrift, Institut für Mathematik, Universität Hannover.
M.Erné (1980) Separation axioms for interval topologies, Proc. Amer. Math. Soc. 79, 185–190.
M.Erné (1981) A completion-invariant extension of the concept of continuous lattices. In: B.Banaschewski and R.-E.Hoffmann (eds.), Continuous Lattices, Proc. Bremen 1979, Lecture Notes in Math. 871, Springer-Verlag, Berlin-Heidelberg-New York, 43–60.
M. Erné (1981) Scott convergence and Scott topology in partially ordered sets, II. In: Continuous Lattices, Proc. Bremen 1979 (see [12]), 61–96.
M.Erné (1982) Distributivgesetze und Dedekindsche Schnitte, Abh. Braunschweig. Wiss. Ges. 33, 117–145.
M.Erné (1982) deEinführung in die Ordnungstheorie, Bibl. Inst. Wissenschaftsverlag, Mannheim.
M.Erné (1983) Adjunctions and standard constructions for partially ordered sets. In: G.Eigenthaler et al. (eds.), Contributions to General Algebra 2, Proc. Klagenfurt Conf. 1982. Hölder-Pichler-Tempski, Wien, 77–106.
M.Erné (1986) Order extensions as adjoint functors, Quaestiones Math. 9, 149–206.
M.Erné (1987) Compact generation in partially ordered sets, J. Austral. Math. Soc. 42, 69–83.
M. Erné (1988) The Dedekind-MacNeille completion as a reflector. Preprint No. 1183, Techn. Hochschule Darmstadt.
M. Erné (1988) Bigeneration and principal separation in partially ordered sets, Preprint No. 1185, Techn. Hochschule Darmstadt; (1991) Order 8, 197–221.
M. Erné (1989) Distributive laws for concept lattices, Preprint No. 1230, Techn. Hochschule Darmstadt.
O.Frink (1954) Ideals in partially ordered sets, Amer. Math. Monthly 61, 223–234.
G.Gierz, K. H.Hofmann, K.Keimel, J. D.Lawson, M.Mislove, and D. S.Scott (1980) A Compendium of Continuous Lattices, Springer-Verlag, Berlin-Heidelberg-New York.
M.Kolibiar (1962) Bemerkungen über Intervalltopologie in halbgeordneten Mengen. In: General Topology and Its Relations to Modern Analysis and Algebra, Proc. Sympos. Prague 1961, Academic Press, New York, 252–253.
H. M.MacNeille (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416–460.
G. N.Raney (1953) A subdirect union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4, 518–522.
J.Schmidt (1956) Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Menge, Arch. Math. (Basel) 7, 241–249.
D. P.Strauss (1968) Topological lattices, Proc. London Math. Soc. 18, 217–230.
R.Wille (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. In: I.Rival (ed.), Ordered Sets, Reidel, Dordrecht-Boston, 445–470.
Author information
Authors and Affiliations
Additional information
Communicated by K. Keimel
Rights and permissions
About this article
Cite this article
Erné, M. The Dedekind-MacNeille completion as a reflector. Order 8, 159–173 (1991). https://doi.org/10.1007/BF00383401
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00383401