Lattice/Complete lattice

  • Thread starter XodoX
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  • #1
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I'm not sure if I am using the right terms here, but:

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.
 

Answers and Replies

  • #2
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Yes, it makes sense and it is true.

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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Yes, it makes sense and it is true.

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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It depends on the definition you've used. Some books only require, for a poset to be a lattice, that every pair ##x,y## of elements have a supremum ##x\vee y:= \bigvee\{x,y\}## and infimum ##x\wedge y:= \bigwedge\{x,y\}##. If this is so in your course, you still need to prove (by induction) that every finite set in a lattice has a supremum and infimum.
 
  • #5
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Thank you! Didn't think of proof by induction.
 

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