# distributive inequalities

Let $L$ be a lattice. Then for $a,b,c\in L$, we have the following inequalities:

1. 1.

$a\vee(b\wedge c)\leq(a\vee b)\wedge(a\vee c)$,

2. 2.

$(a\wedge b)\vee(a\wedge c)\leq a\wedge(b\vee c)$.

###### Proof.

Since $a\leq a\vee b$ and $a\leq a\vee c$, $a\leq(a\vee b)\wedge(a\vee c)$. Similarly, $b\wedge c\leq b\leq a\vee b$ and $b\wedge c\leq c\leq a\vee c$ imply $b\wedge c\leq(a\vee b)\wedge(a\vee c)$. Together, we have $a\vee(b\wedge c)\leq(a\vee b)\wedge(a\vee c)$.

The second inequality is the dual of the first one. ∎

The two inequalities above are called the distributive inequalities.

A lattice $L$ is a distributive lattice if one of the following inequalities holds:

1. 1.

$(a\vee b)\wedge(a\vee c)\leq a\vee(b\wedge c)$,

2. 2.

$a\wedge(b\vee c)\leq(a\wedge b)\vee(a\wedge c)$.

###### Proof.

By the distributive inequalities, all we need to show is that 1. implies 2. (that 2. implies 1. is just the dual statement). So suppose 1. holds. Then

 $\displaystyle(a\wedge b)\vee(a\wedge c)$ $\displaystyle\geq((a\wedge b)\vee a)\wedge((a\wedge b)\vee c)$ $\displaystyle\qquad\mbox{by assumption}$ $\displaystyle=a\wedge((a\wedge b)\vee c)$ $\displaystyle\qquad\mbox{by absorption}$ $\displaystyle\geq a\wedge((c\vee a)\wedge(c\vee b))$ $\displaystyle\qquad\mbox{by assumption}$ $\displaystyle=(a\wedge(c\vee a))\wedge(c\vee b)$ $\displaystyle\qquad\mbox{meet associativity}$ $\displaystyle=a\wedge(c\vee b).$ $\displaystyle\qquad\mbox{by absorption}$

Title distributive inequalities DistributiveInequalities 2013-03-22 16:37:48 2013-03-22 16:37:48 CWoo (3771) CWoo (3771) 5 CWoo (3771) Derivation msc 06D99 ModularInequality