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closed sublattice (Definition)

A sublattice $K$ of a complete lattice $L$ is a closed sublattice of $L$ iff $K$ contains the meet and the join of any its nonempty subset.

Examples:

Any complete sublattice is a closed sublattice.

$[0;1]$ is a closed sublattice of $(-\infty;\infty)$ .

The set of rational numbers is not a closed sublattice of the set of real numbers.



"closed sublattice" is owned by porton.
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See Also: sublattice
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Cross-references: real numbers, rational numbers, complete, subset, join, meet, contains, iff, complete lattice, sublattice

This is version 2 of closed sublattice, born on 2009-06-09, modified 2009-06-13.
Object id is 11815, canonical name is ClosedSublattice.
Accessed 317 times total.

Classification:
AMS MSC: 06B23 (Order, lattices, ordered algebraic structures :: Lattices :: Complete lattices, completions)
Pending Errata and Addenda
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