Abstract
In [8], Vogt used so-called bialgebraic contexts to represent the lattice Sub(L) of all sublattices of a finite distributive lattice L as the substructure lattice of an appropriately defined finite (universal) algebra, based on Rival’s description (see [4] and [5]) by means of deleting suitable intervals from L. We show how to extend Vogt’s context in order to obtain a conceptually simpler description of Sub 01(L) – the lattice of all 0-1-preserving sublattices of L – by means of quasiorders and an associated total binary operation on J (L)2, the set of all pairs of non-zero join-irreducibles of L. Our approach is based on Birkhoff- resp. Priestley-duality, a standard reference is [1].
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Davey, B.A., Priestley, H.A.: Introduction to lattices and order, 2nd edn. Cambridge Univ. Press, Cambridge (2002)
Ganter, B., Wille, R.: Formal concept analysis. Springer, New York (1999)
Lengvárszky, Z., McNulty, G.: Covering in the lattice of subuniverses of a finite distributive lattice. J. Austral. Math. Soc. (Series A) 65, 333–353 (1998)
Rival, I.: Maximal sublattices of finite distributive lattices. Proc. Amer. Math. Soc. 37, 417–420 (1973)
Rival, I.: Maximal sublattices of finite distributive lattices, II. Proc. Amer. Math. Soc. 44, 263–268 (1974)
Schmid, J.: Quasiorders and Sublattices of Distributive Lattices. Order 19, 11–34 (2002)
Vogt, F.: Subgroup lattices of finite Abelian groups: structure and cardinality. In: Baker, K.A., Wille, R. (eds.) Lattice theory and its applications, pp. 241–259. Heldermann-Verlag, Berlin (1995)
Vogt, F.: Bialgebraic contexts for finite distributive lattices. Algebra Universalis 35, 151–165 (1996)
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© 2005 Springer-Verlag Berlin Heidelberg
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Schmid, J. (2005). Bialgebraic Contexts for Distributive Lattices – Revisited. In: Ganter, B., Godin, R. (eds) Formal Concept Analysis. ICFCA 2005. Lecture Notes in Computer Science(), vol 3403. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32262-7_28
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DOI: https://doi.org/10.1007/978-3-540-32262-7_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24525-4
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