nonmodular sublattice

Any nonmodular lattice $L$ contains the lattice $N_{5}$ (shown below) as a sublattice.

\xymatrix{&x\lor y\ar@{-}[ld]\ar@{-}[rdrd]&&\\ (x\lor y)\land z\ar@{-}[dd]&&&\\ &&&y\ar@{-}[ldld]\\ x\lor(y\land z)\ar@{-}[rd]&&&\\ &y\land z&&}

Proof.

Since $L$ is not modular, by definition it contains elements $x$ , $y$ and $z$ such that $x\leq z$ and $x\lor(y\land z)<(x\lor y)\land z$ . Then the sublattice formed by $y$ , $x\lor y$ , $y\land z$ , $(x\lor y)\land z$ and $x\lor(y\land z)$ is isomorphic to $N_{5}$ . This is because $y\land z\leq x\lor(y\land z)<(x\lor y)\land z\leq x\lor y$ while $[x\lor(y\land z)]\lor y=x\lor y$ by absorption and similarly $y\land[(x\lor y)\land z]=y\land z$ . Moreover, $x\lor(y\land z)$ covers $y\land z$ since $y\land z=x\lor(y\land z)$ would imply $x\leq y\land z\leq y$ , whence $x\lor y=y$ and $(x\lor y)\land z=y\land z=x\lor(y\land z)$ contrary to our hypothesis. By the same method, $(x\lor y)\land z=x\lor y$ leads to a contradiction of nonmodularity, so $x\lor y$ covers $(x\lor y)\land z$ . ∎

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	Theorem
Classification	msc 06C05
Related topic	ModularLattice