Dedekind–MacNeille completion (English Wikipedia)

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Bishop (1978); Schröder (2003), Theorem 5.3.8, p. 121. Bishop, Alan A. (1978), "A universal mapping characterization of the completion by cuts", Algebra Universalis, 8 (3): 349–353, doi:10.1007/bf02485405, MR 0469839, S2CID 121624631. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
MacNeille (1937), Lemma 11.8, p. 444; Davey & Priestley (2002), Lemma 3.9(i), p. 166. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001.
This is the definition originally used by MacNeille (1937), for instance. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904.
MacNeille (1937). MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904.
Ganter & Kuznetsov (1998). Ganter, Bernhard; Kuznetsov, Sergei O. (1998), "Stepwise construction of the Dedekind-MacNeille completion", Proc. 6th Int. Conf. Conceptual Structures: Theory, Tools and Applications (ICCS98), Lecture Notes in Computer Science, vol. 1453, Springer-Verlag, pp. 295–302, doi:10.1007/BFb0054922, MR 1673860, Zbl 0928.06004.
Schmidt (1956). Schmidt, Jürgen (1956), "Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Hülle", Archiv der Mathematik (in German), 7: 241–249, doi:10.1007/BF01900297, MR 0084484
Cotlar (1944); Funayama (1944). Cotlar, Mischa (1944), "A method of construction of structures and its application to topological spaces and abstract arithmetic", Univ. Nac. Tucumán. Revista A., 4: 105–157, MR 0014062. Funayama, Nenosuke (1944), "On the completion by cuts of distributive lattices", Proceedings of the Imperial Academy, Tokyo, 20: 1–2, doi:10.3792/pia/1195573210, MR 0014063, Zbl 0063.01484.
This result is frequently attributed to an unpublished 1961 Harvard University honors thesis by K. A. Baker, "Dimension, join-independence and breadth in partially ordered sets". It was published by Novák (1969). Novák, Vítězslav (1969), "Über eine Eigenschaft der Dedekind-MacNeilleschen Hülle", Mathematische Annalen, 179: 337–342, doi:10.1007/BF01350778, MR 0240010, S2CID 120963245.
Banaschewski & Bruns (1967). Banaschewski, B.; Bruns, G. (1967), "Categorical characterization of the MacNeille completion", Archiv der Mathematik, 18 (4): 369–377, doi:10.1007/BF01898828, MR 0221984, S2CID 121216988.
Jourdan, Rampon & Jard (1994). Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude (1994), "Computing on-line the lattice of maximal antichains of posets" (PDF), Order, 11 (3): 197–210, doi:10.1007/BF02115811, MR 1308475, S2CID 120755660, Zbl 0814.06004.
For the equivalence between algorithms for antichains in partial orders and for independent sets in comparability graphs, see Cameron (1985), p. 251. Cameron, Kathie (1985), "Antichain sequences", Order, 2 (3): 249–255, doi:10.1007/BF00333130, MR 0824698

books.google.com

Davey & Priestley (2002, p. 166); Schröder (2003, p. 119). Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
Roman (2007). Roman, Steven (2007), Advanced Linear Algebra, Graduate Texts in Mathematics, vol. 135 (3rd ed.), Springer, pp. 10–11, ISBN 978-0-387-72831-5
Schröder (2003), definition 5.3.1, p. 119. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
O'Leary (2015). O'Leary, Michael L. (2015), A First Course in Mathematical Logic and Set Theory, John Wiley & Sons, p. 276, ISBN 978-0-470-90588-3
Bishop (1978); Schröder (2003), Theorem 5.3.8, p. 121. Bishop, Alan A. (1978), "A universal mapping characterization of the completion by cuts", Algebra Universalis, 8 (3): 349–353, doi:10.1007/bf02485405, MR 0469839, S2CID 121624631. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
MacNeille (1937), Lemma 11.8, p. 444; Davey & Priestley (2002), Lemma 3.9(i), p. 166. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001.
Davey & Priestley (2002), Example 7.44(1), p. 168; Schröder (2003), Example 5.3.3(2), p. 120. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
Davey & Priestley (2002), Example 7.44(2), p. 168. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001.
Schröder (2003), Proposition 5.3.7, p. 121. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
Birkhoff (1995), Theorem 27, p. 130. Birkhoff, Garrett (1995), "VI.9 Completion by Cuts", Lattice Theory, Colloquium Publications, vol. 25 (3rd ed.), American Mathematical Society, pp. 126–128, ISBN 978-0-8218-1025-5.
Gabbay, Shehtman & Skvortsov (2009). Gabbay, Dov M.; Shehtman, Valentin; Skvortsov, Dimitrij (2009), "3.4.12 The Dedekind–MacNeille completion of a residuated lattice", Quantification in Nonclassical Logic, Volume 1, Studies in logic and the foundations of mathematics, vol. 153, Elsevier, pp. 177–178, ISBN 978-0-444-52012-8, Zbl 1211.03002.
Birkhoff (1995). Birkhoff, Garrett (1995), "VI.9 Completion by Cuts", Lattice Theory, Colloquium Publications, vol. 25 (3rd ed.), American Mathematical Society, pp. 126–128, ISBN 978-0-8218-1025-5.

doi.org

Bishop (1978); Schröder (2003), Theorem 5.3.8, p. 121. Bishop, Alan A. (1978), "A universal mapping characterization of the completion by cuts", Algebra Universalis, 8 (3): 349–353, doi:10.1007/bf02485405, MR 0469839, S2CID 121624631. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
MacNeille (1937), Lemma 11.8, p. 444; Davey & Priestley (2002), Lemma 3.9(i), p. 166. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001.
This is the definition originally used by MacNeille (1937), for instance. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904.
MacNeille (1937). MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904.
Ganter & Kuznetsov (1998). Ganter, Bernhard; Kuznetsov, Sergei O. (1998), "Stepwise construction of the Dedekind-MacNeille completion", Proc. 6th Int. Conf. Conceptual Structures: Theory, Tools and Applications (ICCS98), Lecture Notes in Computer Science, vol. 1453, Springer-Verlag, pp. 295–302, doi:10.1007/BFb0054922, MR 1673860, Zbl 0928.06004.
Schmidt (1956). Schmidt, Jürgen (1956), "Zur Kennzeichnung der Dedekind-MacNeilleschen Hülle einer geordneten Hülle", Archiv der Mathematik (in German), 7: 241–249, doi:10.1007/BF01900297, MR 0084484
Cotlar (1944); Funayama (1944). Cotlar, Mischa (1944), "A method of construction of structures and its application to topological spaces and abstract arithmetic", Univ. Nac. Tucumán. Revista A., 4: 105–157, MR 0014062. Funayama, Nenosuke (1944), "On the completion by cuts of distributive lattices", Proceedings of the Imperial Academy, Tokyo, 20: 1–2, doi:10.3792/pia/1195573210, MR 0014063, Zbl 0063.01484.
This result is frequently attributed to an unpublished 1961 Harvard University honors thesis by K. A. Baker, "Dimension, join-independence and breadth in partially ordered sets". It was published by Novák (1969). Novák, Vítězslav (1969), "Über eine Eigenschaft der Dedekind-MacNeilleschen Hülle", Mathematische Annalen, 179: 337–342, doi:10.1007/BF01350778, MR 0240010, S2CID 120963245.
Banaschewski & Bruns (1967). Banaschewski, B.; Bruns, G. (1967), "Categorical characterization of the MacNeille completion", Archiv der Mathematik, 18 (4): 369–377, doi:10.1007/BF01898828, MR 0221984, S2CID 121216988.
Jourdan, Rampon & Jard (1994). Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude (1994), "Computing on-line the lattice of maximal antichains of posets" (PDF), Order, 11 (3): 197–210, doi:10.1007/BF02115811, MR 1308475, S2CID 120755660, Zbl 0814.06004.
For the equivalence between algorithms for antichains in partial orders and for independent sets in comparability graphs, see Cameron (1985), p. 251. Cameron, Kathie (1985), "Antichain sequences", Order, 2 (3): 249–255, doi:10.1007/BF00333130, MR 0824698
Nourine & Raynaud (2002). Nourine, Lhouari; Raynaud, Olivier (2002), "A fast incremental algorithm for building lattices", Journal of Experimental and Theoretical Artificial Intelligence, 14 (2): 217–227, doi:10.1080/09528130210164152, S2CID 38160433, Zbl 1022.68027.

inria.fr

hal.inria.fr

Jourdan, Rampon & Jard (1994). Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude (1994), "Computing on-line the lattice of maximal antichains of posets" (PDF), Order, 11 (3): 197–210, doi:10.1007/BF02115811, MR 1308475, S2CID 120755660, Zbl 0814.06004.

jstor.org

MacNeille (1937), Lemma 11.8, p. 444; Davey & Priestley (2002), Lemma 3.9(i), p. 166. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001.
This is the definition originally used by MacNeille (1937), for instance. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904.
MacNeille (1937). MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904.

semanticscholar.org

api.semanticscholar.org

Bishop (1978); Schröder (2003), Theorem 5.3.8, p. 121. Bishop, Alan A. (1978), "A universal mapping characterization of the completion by cuts", Algebra Universalis, 8 (3): 349–353, doi:10.1007/bf02485405, MR 0469839, S2CID 121624631. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
This result is frequently attributed to an unpublished 1961 Harvard University honors thesis by K. A. Baker, "Dimension, join-independence and breadth in partially ordered sets". It was published by Novák (1969). Novák, Vítězslav (1969), "Über eine Eigenschaft der Dedekind-MacNeilleschen Hülle", Mathematische Annalen, 179: 337–342, doi:10.1007/BF01350778, MR 0240010, S2CID 120963245.
Banaschewski & Bruns (1967). Banaschewski, B.; Bruns, G. (1967), "Categorical characterization of the MacNeille completion", Archiv der Mathematik, 18 (4): 369–377, doi:10.1007/BF01898828, MR 0221984, S2CID 121216988.
Jourdan, Rampon & Jard (1994). Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude (1994), "Computing on-line the lattice of maximal antichains of posets" (PDF), Order, 11 (3): 197–210, doi:10.1007/BF02115811, MR 1308475, S2CID 120755660, Zbl 0814.06004.
Nourine & Raynaud (2002). Nourine, Lhouari; Raynaud, Olivier (2002), "A fast incremental algorithm for building lattices", Journal of Experimental and Theoretical Artificial Intelligence, 14 (2): 217–227, doi:10.1080/09528130210164152, S2CID 38160433, Zbl 1022.68027.

zbmath.org

Davey & Priestley (2002, p. 166); Schröder (2003, p. 119). Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
MacNeille (1937), Lemma 11.8, p. 444; Davey & Priestley (2002), Lemma 3.9(i), p. 166. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001.
This is the definition originally used by MacNeille (1937), for instance. MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904.
MacNeille (1937). MacNeille, H. M. (1937), "Partially ordered sets", Transactions of the American Mathematical Society, 42 (3): 416–460, doi:10.2307/1989739, JFM 63.0833.04, JSTOR 1989739, MR 1501929, Zbl 0017.33904.
Davey & Priestley (2002), Example 7.44(1), p. 168; Schröder (2003), Example 5.3.3(2), p. 120. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001. Schröder, Bernd S. W. (2003), "5.3 Embeddings/The Dedekind/MacNeille Completion", Ordered Sets: An Introduction, Birkhäuser, pp. 119–122, ISBN 978-1-4612-6591-7.
Davey & Priestley (2002), Example 7.44(2), p. 168. Davey, B. A.; Priestley, Hilary A. (2002), "7.38 The Dedekind–MacNeille completion", Introduction to Lattices and Order (2nd ed.), Cambridge University Press, p. 166, ISBN 978-0-521-78451-1, Zbl 1002.06001.
Ganter & Kuznetsov (1998). Ganter, Bernhard; Kuznetsov, Sergei O. (1998), "Stepwise construction of the Dedekind-MacNeille completion", Proc. 6th Int. Conf. Conceptual Structures: Theory, Tools and Applications (ICCS98), Lecture Notes in Computer Science, vol. 1453, Springer-Verlag, pp. 295–302, doi:10.1007/BFb0054922, MR 1673860, Zbl 0928.06004.
Gabbay, Shehtman & Skvortsov (2009). Gabbay, Dov M.; Shehtman, Valentin; Skvortsov, Dimitrij (2009), "3.4.12 The Dedekind–MacNeille completion of a residuated lattice", Quantification in Nonclassical Logic, Volume 1, Studies in logic and the foundations of mathematics, vol. 153, Elsevier, pp. 177–178, ISBN 978-0-444-52012-8, Zbl 1211.03002.
Cotlar (1944); Funayama (1944). Cotlar, Mischa (1944), "A method of construction of structures and its application to topological spaces and abstract arithmetic", Univ. Nac. Tucumán. Revista A., 4: 105–157, MR 0014062. Funayama, Nenosuke (1944), "On the completion by cuts of distributive lattices", Proceedings of the Imperial Academy, Tokyo, 20: 1–2, doi:10.3792/pia/1195573210, MR 0014063, Zbl 0063.01484.
Jourdan, Rampon & Jard (1994). Jourdan, Guy-Vincent; Rampon, Jean-Xavier; Jard, Claude (1994), "Computing on-line the lattice of maximal antichains of posets" (PDF), Order, 11 (3): 197–210, doi:10.1007/BF02115811, MR 1308475, S2CID 120755660, Zbl 0814.06004.
Nourine & Raynaud (2002). Nourine, Lhouari; Raynaud, Olivier (2002), "A fast incremental algorithm for building lattices", Journal of Experimental and Theoretical Artificial Intelligence, 14 (2): 217–227, doi:10.1080/09528130210164152, S2CID 38160433, Zbl 1022.68027.