lattice (original) (raw)
A sublattice of L is a subposet of L which is a lattice, that is, which is closed under the operations ∧ and ∨ as defined in L.
Conspicuously absent from the above list of properties is distributivity (http://planetmath.org/DistributiveLattice). While many nice lattices, such as face lattices of polytopes, are distributive, there are also important classes of lattices, such as partition lattices (http://planetmath.org/PartitionLattice), that are usually not distributive.
Lattices, like posets, can be visualized by diagrams called Hasse diagrams. Below are two diagrams, both posets. The one on the left is a lattice, while the one on the right is not:
| \entrymodifiers=[o]\xymatrix@!=1pt&&∙\ar@-[ld]\ar@-[rd]&&&∙\ar@-[ld]\ar@-[rd]&&∙\ar@-[ld]\ar@-[rd]&∙\ar@-[rd]&&∙\ar@-[ld]\ar@-[rd]&&∙\ar@-[ld]&∙\ar@-[rd]&&∙\ar@-[ld]&&&∙&& \entrymodifiers=[o]\xymatrix@!=1pt&&∙\ar@-[ld]\ar@-[rd]&&&∙\ar@-[ld]\ar@-[rd]\ar@-[d]&&∙\ar@-[ld]\ar@-[rd]\ar@-[d]&∙\ar@-[rd]&\ar@-[d]&\ar@-[ld]\ar@-[rd]&\ar@-[d]&∙\ar@-[ld]&∙\ar@-[rd]&&∙\ar@-[ld]&&&∙&& |
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The vertices of a lattice diagram can also be labelled, so the lattice diagram looks like
| \xymatrix@!=1pt&&a\ar@-[ld]\ar@-[rd]&&&b\ar@-[ld]\ar@-[rd]&&c\ar@-[ld]\ar@-[rd]&d\ar@-[rd]&&e\ar@-[ld]\ar@-[rd]&&f\ar@-[ld]&g\ar@-[rd]&&h\ar@-[ld]&&&i&& |
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Remark. Alternatively, a lattice can be defined as an algebraic system. Please see the link below for details.