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modular inequality
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(Theorem)
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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$ . 
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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 1718 times total.
Classification:
| AMS MSC: | 06C05 (Order, lattices, ordered algebraic structures :: Modular lattices, complemented lattices :: Modular lattices, Desarguesian lattices) |
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Pending Errata and Addenda
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