×

zbMATH — the first resource for mathematics

The Dedekind-MacNeille completion as a reflector. (English) Zbl 0738.06004
Summary: 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.

MSC:
06B23 Complete lattices, completions
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
06A06 Partial orders, general
PDF BibTeX Cite
Full Text: DOI
References:
[1] A.Abian (1968) On definitions of cuts and completion of partially ordered sets, Z. Math. Logik Grundl. der Math. 14, 299-309. · Zbl 0169.30805
[2] B.Banaschewski (1956) Hüllensysteme und Erweiterung von Quasi-Ordnungen, Z. Math. Logik Grundl. der Math. 2, 117-130. · Zbl 0073.26904
[3] B.Banaschewski and G.Bruns (1967) Categorical characterization of the MacNeille completion, Arch. Math. (Basel) 43, 369-377. · Zbl 0157.34101
[4] G. Birkhoff (1973) Lattice Theory, Amer. Math. Soc. Coll. Publ. 25, 3rd ed., Providence, R.I.
[5] A.Bishop (1978) A universal mapping characterization of the completion by cuts. Algebra Universalis 8, 349-353. · Zbl 0385.06005
[6] T. S.Blyth and M. S.Janowitz (1972) Residuation Theory, Pergamon Press, Oxford.
[7] G.Bruns (1962) Darstellungen und Erweiterungen geordneter Mengen I, II, J. Reine Angew. Math. 209, 167-200, and 210, 1-23. · Zbl 0148.01202
[8] P.Crawley and R. P.Dilworth (1973) Algebraic Theory of Lattices, Prentice-Hall, Inc., Englewood Cliffs, N.J. · Zbl 0494.06001
[9] R.Dedekind (1872) deStetigkeit und irrationale Zahlen, 7th ed. (1969), Vieweg, Braunschweig.
[10] M. Erné (1980) Verallgemeinerungen der Verbandstheorie, I, II, Preprint No. 109 and Habilitations-schrift, Institut für Mathematik, Universität Hannover.
[11] M.Erné (1980) Separation axioms for interval topologies, Proc. Amer. Math. Soc. 79, 185-190. · Zbl 0398.54017
[12] 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.
[13] M. Erné (1981) Scott convergence and Scott topology in partially ordered sets, II. In: Continuous Lattices, Proc. Bremen 1979 (see [12]), 61-96.
[14] M.Erné (1982) Distributivgesetze und Dedekindsche Schnitte, Abh. Braunschweig. Wiss. Ges. 33, 117-145. · Zbl 0526.06005
[15] M.Erné (1982) deEinführung in die Ordnungstheorie, Bibl. Inst. Wissenschaftsverlag, Mannheim.
[16] 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. · Zbl 0533.06001
[17] M.Erné (1986) Order extensions as adjoint functors, Quaestiones Math. 9, 149-206. · Zbl 0602.06002
[18] M.Erné (1987) Compact generation in partially ordered sets, J. Austral. Math. Soc. 42, 69-83. · Zbl 0614.06004
[19] M. Erné (1988) The Dedekind-MacNeille completion as a reflector. Preprint No. 1183, Techn. Hochschule Darmstadt. · Zbl 0738.06004
[20] M. Erné (1988) Bigeneration and principal separation in partially ordered sets, Preprint No. 1185, Techn. Hochschule Darmstadt; (1991) Order 8, 197-221. · Zbl 0738.06005
[21] M. Erné (1989) Distributive laws for concept lattices, Preprint No. 1230, Techn. Hochschule Darmstadt. · Zbl 0795.06006
[22] O.Frink (1954) Ideals in partially ordered sets, Amer. Math. Monthly 61, 223-234. · Zbl 0055.25901
[23] 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. · Zbl 0452.06001
[24] 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.
[25] H. M.MacNeille (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416-460. · Zbl 0017.33904
[26] G. N.Raney (1953) A subdirect union representation for completely distributive complete lattices, Proc. Amer. Math. Soc. 4, 518-522. · Zbl 0053.35201
[27] J.Schmidt (1956) Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Menge, Arch. Math. (Basel) 7, 241-249. · Zbl 0073.03801
[28] D. P.Strauss (1968) Topological lattices, Proc. London Math. Soc. 18, 217-230. · Zbl 0153.33404
[29] R.Wille (1982) Restructuring lattice theory: an approach based on hierarchies of concepts. In: I.Rival (ed.), Ordered Sets, Reidel, Dordrecht-Boston, 445-470. · Zbl 0491.06008
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.