The first definition has been standard at least since the 1930s and probably since Dedekind worked on lattice theory in the 19th century; though he may not
have used that name. -- JanHidders

Aren't finitely generated subgroups of**R**^{n} or **C**^{n} also called lattices? I wonder if they are related to the order-lattices. --AxelBoldt

The definition is given as:

- A
*least upperbound*of*V*is an element*x*in*L*such that- for all
*y*in*V*it holds that*y*<=*x*, and - for all
*z*in*L*it holds that if*z*<=*v*for all*v*in*V*then*x*<=*z*.

- for all
- A
*greatest lowerbound*of*V*is an element*x*in*S*such that- for all
*y*in*V*it holds that*x*<=*y*, and - for all
*z*in*L*it holds that if*v*<=*z*for all*v*in*V*then*z*<=*x*.

- for all

Isn't the first inequality in the second subbullet under both of the main bullets backwards? Shouldn't it be *v* <= *z* in the first case and *z* <= *v* in the second case, rather than vice versa?

*Yup.*

Aren't finitely generated subgroups of

Discrete subgroups, rather than finitely-generated subgroups, I think. E.g., <1,π> is a finitely generated subgroup of **R**, but it isn't a lattice. They aren't related to the type of lattice described in the current article. I was going to add a mention of them yesterday, but I couldn't think of anything much to write.

Zundark, 2001-08-20

I see. Maybe Minkowski's theorem about the number of lattice points in a convex set could be linked. --AxelBoldt The new material science definition seems to be the same as a discrete subgroup. --AxelBoldt

Yes. I think what we should do is to add the discrete subgroup definition, and then modify the materials science definition to mention that this is a special case of one of the mathematical definitions. --Zundark, 2001-08-21