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sav08:lattices [2012/04/01 21:04] evka |
sav08:lattices [2015/04/21 17:30] (current) |
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| **Definition:** A lattice is //distributive// iff | **Definition:** A lattice is //distributive// iff | ||
| - | \[ | + | \begin{equation*} |
| \begin{array}{l} | \begin{array}{l} | ||
| x \sqcap (y \sqcup z) = (x \sqcap y) \sqcup (x \sqcap z) \\ | x \sqcap (y \sqcup z) = (x \sqcap y) \sqcup (x \sqcap z) \\ | ||
| x \sqcup (y \sqcap z) = (x \sqcup y) \sqcap (x \sqcup z) | x \sqcup (y \sqcap z) = (x \sqcup y) \sqcap (x \sqcup z) | ||
| \end{array} | \end{array} | ||
| - | \] | + | \end{equation*} |
| **Example:** Lattice of all subsets of a set is distributive. Linear order is a distributive lattice. See examples of non-distributive lattices in [[wk>Distributive lattice]] and the characterization of non-distributive lattices. | **Example:** Lattice of all subsets of a set is distributive. Linear order is a distributive lattice. See examples of non-distributive lattices in [[wk>Distributive lattice]] and the characterization of non-distributive lattices. | ||