# modular inequality

In any lattice (http://planetmath.org/lattice) the self-dual modular inequality is true: if $x\leq z$ then $x\lor(y\land z)\leq(x\lor y)\land z$.

###### Proof.

$x\leq x\lor y$ and we are given that $x\leq z$, so $x\leq(x\lor y)\land z$. Also, $y\land z\leq y\leq x\lor y$ and $y\land z\leq z$ imply that $y\land z\leq(x\lor y)\land z$. Therefore, $x\lor(y\land z)\leq(x\lor y)\land z$. ∎

Title modular inequality ModularInequality 2014-02-01 1:48:21 2014-02-01 1:48:21 ixionid (16766) ixionid (16766) 10 ixionid (16766) Theorem msc 06C05 ModularLattice DistributiveInequalities modular inequality