- #1

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When X is a finite set and R is a relation....

If (X,R) is a lattice, then (X,R) is also a complete lattice.

Does this make sense? The question then is, why is is also automatically complete. I don't understand that.

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- Thread starter XodoX
- Start date

- #1

- 203

- 0

When X is a finite set and R is a relation....

If (X,R) is a lattice, then (X,R) is also a complete lattice.

Does this make sense? The question then is, why is is also automatically complete. I don't understand that.

- #2

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- 3,297

A lattice is an ordered set where every finite subset has a supremum and an infimum. A complete lattice is an ordered set where every subset has a supremum and an infimum.

Clearly, if the underlying set ##X## is finite, then every subset of ##X## is finite. Thus every subset has a supremum and an infimum since every finite subset has a supremum and an infimum.

- #3

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A lattice is an ordered set where every finite subset has a supremum and an infimum. A complete lattice is an ordered set where every subset has a supremum and an infimum.

Clearly, if the underlying set ##X## is finite, then every subset of ##X## is finite. Thus every subset has a supremum and an infimum since every finite subset has a supremum and an infimum.

Makes sense. Does this prove it, though ? I mean, is there any calculation to prove this, or you just basically say that's how it is.

- #4

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- #5

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Thank you! Didn't think of proof by induction.

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