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bounded lattice (Definition)

A lattice $ L$ is said to be bounded from below if there is an element $ 0\in L$ such that $ 0\leq x$ for all $ x\in L$. Dually, $ L$ is bounded from above if there exists an element $ 1\in L$ such that $ x\leq1$ for all $ x\in L$. A bounded lattice is one that is bounded both from above and below.

For example, any finite lattice $ L$ is bounded, as $ \bigvee L$ and $ \bigwedge L$, being join and meet of finitely many elements, exist. $ \bigvee L=1$ and $ \bigwedge L=0$.

Remarks. Let $ L$ be a bounded lattice with 0 and $ 1$ as described above.

  • $ 0\land x=0$ and $ 0\lor x=x$ for all $ x\in L$.
  • $ 1\land x=x$ and $ 1\lor x=1$ for all $ x\in L$.
  • As a result, 0 and $ 1$, if they exist, are necessarily unique. For if there is another such a pair $ 0^{\prime}$ and $ 1^{\prime}$, then $ 0=0\land 0^{\prime}=0^{\prime}\land 0=0^{\prime}$. Similarly $ 1=1^{\prime}$.
  • 0 is called the bottom of $ L$ and $ 1$ is called the top of $ L$.
  • $ L$ is a lattice interval and can be written as $ [0,1]$.

Remark. More generally, a poset $ P$ is said to be bounded if it has both a greatest element $ 1$ and a least element 0.



"bounded lattice" is owned by CWoo.
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Also defines:  top, bottom, bounded poset
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Cross-references: least element, greatest element, poset, lattice interval, meet, join, bounded, finite, lattice
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This is version 10 of bounded lattice, born on 2005-02-16, modified 2007-04-29.
Object id is 6755, canonical name is BoundedLattice.
Accessed 4609 times total.

Classification:
AMS MSC06B05 (Order, lattices, ordered algebraic structures :: Lattices :: Structure theory)
 06A06 (Order, lattices, ordered algebraic structures :: Ordered sets :: Partial order, general)

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