Definition talk:Lattice (Group Theory)

This definition of (point) Lattice is not right!

A lattice can have any real vector in its basis. For instance, the Lattice whose basis is $\left\{(3.14, 0), (1, \log 2)\right\}$ is not a subgroup of $\Z^n$.

Moreover, the span of a lattice basis does not need to be equal to the whole vector space, since the lattice basis may have less vectors than the vector space basis. When this span is equal to the vector space, the rank of the lattice is equal to the dimension of the vector space, and then we say that the lattice is a full-rank lattice.

--Hilder.vitor (talk) 06:36, 20 January 2017 (EST)

How certain are you that what has been defined is not what you have defined? It seems that what you wish to define is a point lattice which may or may not be the same as the object that was originally defined by ScorpionSamurai as a lattice.
It has happened before that contributors have an understanding of two different objects, both called by the same name and used in different contexts, and have believed they refer to the same thing, and they don't. --prime mover (talk) 09:46, 20 January 2017 (EST)
Well, the old definition was a particular case of this one, namely an integer full-rank lattice. To see that, just take $m = n$ and all the $b_i$ belonging to $\Z^n$ in this new definition.
I just wrote point lattice here to distinguish from order lattices. But this is not the most common name. Usually, it is called simply by lattice. (See, for instance, this page of math world).
--Hilder.vitor (talk) 10:20, 20 January 2017 (EST)