Greatest / least element of a lattice

rindphi
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Greatest / least element of a lattice

Beitrag von rindphi » 25. Nov 2012 17:10

Hello all,

I have a question concercing the definition of lattices (slide set 1, slide 15):
There, it is stated that the least element of a lattice is the least upper bound of the empty set (and, similarly, the greatest element is the greatest lower bound of the empty set). Now, regarding the definitions on the previous slides, the necessary preconditions are trivially satisfied since there are no elements in the empty set. But this would also, like I understand it, be the case for every other element in the lattice. So I do not get the sense behind this definition of greatest / least elements.

Is there an intuitive way of explaining the definition? Or did I understand something wrong?

Thanks,
Best Regards,
Dominic

aderhold
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Re: Greatest / least element of a lattice

Beitrag von aderhold » 27. Nov 2012 17:35

As you correctly observed, each element of a lattice is an upper bound of the empty set. Thus the least upper bound of the empty set is the least element of the lattice. That's the intuition behind the definition.

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