zbMATH — the first resource for mathematics

Bigeneration in complete lattices and principal separation in ordered sets. (English) Zbl 0738.06005
Summary: By a recent observation of Monjardet and Wille, a finite distributive lattice is generated by its doubly irreducible elements iff the poset of all join-irreducible elements has a distributive MacNeille completion. This fact is generalized in several directions, by dropping the finiteness condition and considering various types of bigeneration via arbitrary meets and certain distinguished joins. This leads to a deeper investigation of so-called \({\mathcal Z}\)-generators resp. \({\mathcal Z}\)- subbases, translating well-known notions of topology to order theory. A strong relationship is established between bigeneration by (minimal) \({\mathcal Z}\)-generators and so-called principal separation, which is defined in order-theoretical terms but may be regarded as a strong topological separation axiom. For suitable \({\mathcal Z}\), the complete lattices with a smallest join-dense \({\mathcal Z}\)-subbasis consisting of \({\mathcal Z}\)-primes are the \({\mathcal Z}\)-completions of principally separated posets.

06B23 Complete lattices, completions
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
06B15 Representation theory of lattices
06D05 Structure and representation theory of distributive lattices
PDF BibTeX Cite
Full Text: DOI
[1] A.Abian (1968) On definitions of cuts and completion of partially ordered sets, Z. Math. Logik Grundl. der Math. 14, 229-309. · 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] N.Bourbaki (1949/50) Sur le théorème de Zorn, Arch. Math. (Basel) 2, 434-437. · Zbl 0045.32902
[4] G.Bruns (1962) Darstellungen und Erweiterungen geordneter Mengen I, II, J. Reine Angew. Math. 209, 167-200; ibid. 210, 1-23. · Zbl 0148.01202
[5] J. R.Büchi (1952) Representations of complete lattices by sets, Portugal. Math. 11, 151-167. · Zbl 0048.02202
[6] J.Dugundji (1966) Topology. Allyn & Bacon, Boston 1966.
[7] M. Erné (1980) Verallgemeinerungen der Verbandstheorie I: Halbgeordnete Mengen und das Prinzip der Vervollständigungsinvarianz. Techn. Report 109, University of Hannover.
[8] M. Erné (1979/80) Verallgemeinerungen der Verbandstheorie II: m-Ideale in halbgeordneten Mengen und Hüllendräumen. Habilitationsschrift, University of Hannover.
[9] M.Erné (1983) On the existence of decompositions in lattices, Alg. Univ. 16, 338-343. · Zbl 0516.06004
[10] M. Erné (1981) Homomorphisms of ?-generated and ?-distributive posets. Techn. Report 125, University of Hannover.
[11] M.Erné (1981) Scott convergence and Scott topology in partially ordered sets II, In Continuous Lattices, Proc. Bremen 1979. Lecture Notes in Math. 871; Springer-Verlag, Berlin-Heidelberg-New York.
[12] M.Erné (1983) Adjunctions and standard constructions for partially ordered sets. Contributions to general algebra 2, Proc. Klagenfurt Conf. 1982. Hölder-Pichler-Tempsky, Wien, 77-106.
[13] M.Erné (1986) Order extensions as adjoint functors, Quaest. Math. 9, 149-206. · Zbl 0602.06002
[14] M. Erné (1988) The Dedekind-MacNeille completion as a reflector. This journal (to appear). · Zbl 0738.06004
[15] M.Erné and H.Gatzke (1985) Convergence and continuity in partially ordered sets and semilattices, In Continuous Lattices and Their Applications, Proc. Bremen 1982. M. Dekker, New York, 9-40. · Zbl 0591.54029
[16] O.Frink (1954) Ideals in partially ordered sets, Amer. Math. Monthly 61, 223-233. · Zbl 0055.25901
[17] 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. · Zbl 0452.06001
[18] K. H.Hofmann and M. W.Mislove (1981) Local compactness and continuous lattices, In Continuous Lattices, Proc. Bremen 1979, Lecture Notes in Math. 871, Springer-Verlag, Berlin. · Zbl 0464.06005
[19] D. C.Kent (1966) On the order topology in a lattice, Illinois J. Math. 10, 90-96. · Zbl 0131.20401
[20] H. M.MacNeille (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416-460. · Zbl 0017.33904
[21] J.Meseguer (1983) Order completion monads, Alg. Univ. 16, 63-82. · Zbl 0522.18005
[22] B.Monjardet and R.Wille (1989) On finite lattices generated by their doubly irreducible elements. Discr. Math. 73, 163-164. · Zbl 0663.06008
[23] H. A.Priestley (1986) Ordered sets and duality for distributive lattices, In Proc. Conf. Ens. Ordonnés Appl., Lyon, 1982 (ed. M.Pouzet and D.Richard), North Holland Publ. Co., Amsterdam. · Zbl 0557.06007
[24] J.Rosický (1972) On a characterization of the lattice of m-ideals of an ordered set, Arch. Math. (Brno) 8, 137-142.
[25] J.Schmidt (1956) Zur Kennzeichnung der Dedekin-MacNeilleschen Hülle einer geordneten Menge, Arch. Math. (Basel) 7, 241-249. · Zbl 0073.03801
[26] M. H.Stone (1936) The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40, 37-111. · Zbl 0014.34002
[27] S.Weck (1981) Scott convergence and Scott topology in partially ordered sets I, In Continuous Lattices, Proc. Bremen 1979. Lecture Notes in Math. 871; Springer-Verlag, Berlin-Heidelberg-New York. · Zbl 0482.54022
[28] J. B.Wright, E. G.Wagner, and J. W.Thatcher (1978) A uniform approach to inductive posets and inductive closure, Theoret. Comput. Sci. 7, 57-77. · Zbl 0732.06001
[29] E.Zermelo (1908) Neuer Beweis für die Wohlordnung, Math. Ann. 65, 181-198. · JFM 39.0097.03
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.