PlanetMath: modular inequality

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modular inequality

(Theorem)

In any 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, $x\lor (y\land z)\leq(x\lor y)\land z$ . $\qedsymbol$

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Also defines:

modular inequality

Keywords:

lattice modular

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Cross-references: imply, self-dual, lattice
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This is version 5 of modular inequality, born on 2007-04-17, modified 2007-04-17.
Object id is 9210, canonical name is ModularInequality.
Accessed 1000 times total.

Classification:

AMS MSC:

06C05 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Modular lattices, Desarguesian lattices)

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