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distributive_lattices [2010/07/29 18:30]
127.0.0.1 external edit
distributive_lattices [2018/08/01 11:42] (current)
jipsen
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,\wedge \rangle $ such that $\mathbf{L}$ has no sublattice isomorphic ,\wedge \rangle $ such that $\mathbf{L}$ has no sublattice isomorphic
to the diamond $\mathbf{M}_{3}$ or the pentagon $\mathbf{N}_{5}$ to the diamond $\mathbf{M}_{3}$ or the pentagon $\mathbf{N}_{5}$
 +
 +====Definition====
 +A \emph{distributive lattice} is a structure $\mathbf{L}=\langle L,\vee ,\wedge
 +\rangle $ of type $\langle 2,2\rangle $ such that
 +
 +$x\wedge(x\vee y)=x$ and
 +
 +$x\wedge(y\vee z)=(z\wedge x)\vee(y\wedge x)$.[(Sholander1951)]
 +
==Morphisms== ==Morphisms==
-Let $\mathbf{L}$ and $\mathbf{M}$ be distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:Larrow M$ that is a+Let $\mathbf{L}$ and $\mathbf{M}$ be distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ that is a
homomorphism: homomorphism:
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\emph{On the number of distributive lattices}, \emph{On the number of distributive lattices},
Electronic J. Combinatorics 9 (2002), no. 1, Research Paper 24, 23 pp. Electronic J. Combinatorics 9 (2002), no. 1, Research Paper 24, 23 pp.
 +)]
 +[(Sholander1951>
 +M. Sholander,
 +\emph{Postulates for distributive lattices}.
 +Canadian J. Math. 3, (1951). 28–30.
)] )]