PlanetMath: irredundant

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irredundant

(Definition)

Let be a lattice. A finite join

$\displaystyle a_1\vee a_2\vee \cdots \vee a_n$

of elements in

is said to be irredundant if one can not delete an element from from the join without resulting in a smaller join. In other words,

$\displaystyle \bigvee \lbrace a_j \mid j\ne i\rbrace < a_1\vee a_2\vee \cdots \vee a_n$

for all $i=1,\ldots, n$ . This definition can be extended to the case where the join is taken over an infinite number of elements, provided that the join exists. If the join is not irredundant, it is redundant

Irredundant meets are dually defined.

Example. In the lattice of all subsets (ordered by inclusion) of $\mathbb{Z}$ , the set of all integers, the join

$\displaystyle \mathbb{Z}= \bigvee \lbrace p\mathbb{Z} \mid p$ is prime $\displaystyle \rbrace$

is irredudant. Another irredundant join representation of $\mathbb{Z}$ is just the join of all atoms, the singletons consisting of the individual elements of $\mathbb{Z}$ . However,

$\displaystyle \mathbb{Z}=\bigvee \lbrace n\mathbb{Z} \mid n$ is any positive integer $\displaystyle \rbrace$

is redundant, since $n\mathbb{Z}$ can be removed whenever

is a composite number. The join of all doubletons is also redundant, for $\lbrace a,b\rbrace \le \lbrace a,c\rbrace \vee \lbrace c,b\rbrace$ , for any $c\notin \lbrace a,b\rbrace$ .

"irredundant" is owned by CWoo.

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Also defines:

redundant

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Cross-references: composite number, singletons, atoms, representation, integers, inclusion, subsets, meets, number, infinite, join, finite, lattice
There are 13 references to this entry.

This is version 2 of irredundant, born on 2008-06-30, modified 2008-06-30.
Object id is 10730, canonical name is Irredundant.
Accessed 430 times total.

Classification:

AMS MSC:

06B05 (Order, lattices, ordered algebraic structures :: Lattices :: Structure theory)

Pending Errata and Addenda

None.

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