PlanetMath: modular lattice

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modular lattice

(Definition)

A lattice is said to be modular if $x \lor (y \land z) = (x \lor y) \land z$ for all $x,y,z\in L$ such that $x \leq z$ . In fact it is sufficient to show that $x \lor (y \land z) \ge (x \lor y) \land z$ for all $x,y,z\in L$ such that $x \leq z$ , as the reverse inequality holds in all lattices (see modular inequality).

There are a number of other equivalent conditions for a lattice to be modular:

$(x\land y)\lor (x\land z)=x\land (y\lor (x\land z))$ for all $x,y,z\in L$ .
$(x\lor y)\land (x\lor z)=x\lor (y\land (x\lor z))$ for all $x,y,z\in L$ .
For all $x,y,z\in L$ , if then either $x\land y<z\land y$ or $x\lor y<z\lor y$ .

The following are examples of modular lattices.

All distributive lattices.
The lattice of normal subgroups of any group.
The lattice of submodules of any module. (See modular law.)

A finite lattice is modular if and only if it is graded and its rank function $\rho$ satisfies $\rho(x)+\rho(y)=\rho(x\land y)+\rho(x\lor y)$ for all $x,y\in L$ .

"modular lattice" is owned by yark. [ full author list (2) | owner history (1) ]

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Other names:

Dedekind lattice

Also defines:

modular

Attachments:: normal subgroup lattice is modular (Derivation) by CWoo
nonmodular sublattice (Theorem) by ixionid
modular inequality (Theorem) by ixionid

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Cross-references: rank function, finite, modular law, submodules, group, normal subgroups, equivalent, number, modular inequality, inequality, sufficient, lattice
There are 29 references to this entry.

This is version 14 of modular lattice, born on 2002-02-24, modified 2007-04-20.
Object id is 2598, canonical name is ModularLattice.
Accessed 6460 times total.

Classification:

AMS MSC:

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

Pending Errata and Addenda

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