# Differences

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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: | ||

Line 104: | Line 113: | ||

\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. | ||

)] | )] |

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