nonmodular sublattice

Any nonmodular latticeMathworldPlanetmath L contains the lattice N5 (shown below) as a sublattice.


Since L is not modular, by definition it contains elements x, y and z such that xz and x(yz)<(xy)z. Then the sublattice formed by y, xy, yz, (xy)z and x(yz) is isomorphic to N5. This is because yzx(yz)<(xy)zxy while [x(yz)]y=xy by absorption and similarly y[(xy)z]=yz. Moreover, x(yz) covers yz since yz=x(yz) would imply xyzy, whence xy=y and (xy)z=yz=x(yz) contrary to our hypothesisMathworldPlanetmathPlanetmath. By the same method, (xy)z=xy leads to a contradictionMathworldPlanetmathPlanetmath of nonmodularity, so xy covers (xy)z. ∎

Title nonmodular sublattice
Canonical name NonmodularSublattice
Date of creation 2013-03-22 16:55:25
Last modified on 2013-03-22 16:55:25
Owner ixionid (16766)
Last modified by ixionid (16766)
Numerical id 8
Author ixionid (16766)
Entry type TheoremMathworldPlanetmath
Classification msc 06C05
Related topic ModularLattice