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Intervals in Lattices of κ-Meet-Closed Subsets

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Abstract

We study abstract properties of intervals in the complete lattice of all κ-meet-closed subsets (κ-subsemilattices) of a κ-(meet-)semilattice S, where κ is an arbitrary cardinal number. Any interval of that kind is an extremally detachable closure system (that is, for each closed set A and each point x outside A, deleting x from the closure of A∪{x} leaves a closed set). Such closure systems have many pleasant geometric and lattice-theoretical properties; for example, they are always weakly atomic, lower locally Boolean and lower semimodular, and each member has a decomposition into completely join-irreducible elements. For intervals of κ-subsemilattices, we describe the covering relation, the coatoms, the ∨-irreducible and the ∨-prime elements in terms of the underlying κ-semilattices. Although such intervals may fail to be lower continuous, they are strongly coatomic if and only if every element has an irredundant (and even a least) join-decomposition. We also characterize those intervals which are Boolean, distributive (equivalently: modular), or semimodular.

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References

  1. Adaricheva, K. V.: Semidistributive and coalgebraic lattices of subsemilattices, Algebra and Logic 27 (1988), 385–395.

    Article  Google Scholar 

  2. Alexandroff, P.: Diskrete Räume, Mat. Sb. 2 (1937), 501–518.

    Google Scholar 

  3. Birkhoff, G.: Lattice Theory, 3rd edn, Amer. Math. Soc. Publ. XX, Providence, RI, 1967.

  4. Crawley, P. and Dilworth, R. P.: Algebraic Theory of Lattices, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1973.

    Google Scholar 

  5. Diercks, V.: Zerlegungstheorie in vollständigen Verbänden, Diplomarbeit (Thesis), University of Hannover, 1982.

  6. Edelman, P. H. and Jamison, R. E.: The theory of convex geometries, Geom. Dedicata 19 (1985), 247–270.

    Article  Google Scholar 

  7. Erné, M.: Ordnungs- und Verbandstheorie, Fernuniversität Hagen, 1985.

  8. Erné, M.: Bigeneration in complete lattices and principal separation in ordered sets, Order 8 (1991), 197–221.

    Article  Google Scholar 

  9. Erné, M.: Algebraic ordered sets and their generalizations, In: I. Rosenberg and G. Sabidussi (eds), Algebras and Orders, Proc. Montreal, 1992, Kluwer Acad. Publ., Amsterdam, 1993.

    Google Scholar 

  10. Erné, M.: Convex geometries and anti-exchange properties, Preprint, University of Hannover, 2002.

  11. Erné, M., Šešelja, B. and Tepavčević, A.: Posets generated by irreducible elements, Order 20 (2003), 79–89.

    Article  Google Scholar 

  12. Gorbunov, V.: Canonical decompositions in complete lattices, Algebra and Logic 17 (1978), 323–332.

    Article  Google Scholar 

  13. Jamison-Waldner, R.: Copoints in antimatroids, In: Proc. 11th SE Conf. Comb., Graph Theory and Comput., Congressus Numerantium 29 (1980), 535–544.

  14. Semyonova, V. M.: Lattices with unique irreducible decompositions, Algebra and Logic 39 (2000), 54–60.

    Google Scholar 

  15. Semyonova, V. M.: Decompositions in complete lattices, Algebra and Logic 40 (2001), 384–390.

    Article  Google Scholar 

  16. Šešelja, B. and Tepavčević, A.: Collection of finite lattices generated by a poset, Order 17 (2000), 129–139.

    Article  Google Scholar 

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  1. Institute of Mathematics, University of Hannover, Germany

    Marcel Erné

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  1. Marcel Erné
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Correspondence to Marcel Erné.

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Mathematics Subject Classifications (2000)

Primary: 06A12; Secondary: 06B05, 06A23, 52A01.

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Erné, M. Intervals in Lattices of κ-Meet-Closed Subsets. Order 21, 137–153 (2004). https://doi.org/10.1007/s11083-004-3716-2

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