I'm writing a paper about lattices in the complex plane, and while trying to explain the crystallographic restriction theorem, I realized that I never actually defined what a 'lattice' is. The ostensive definition which I have effectively been using is that a lattice [in the complex plane]
is a [countable] set $\Lambda\subset\mathbb{C}$
forms a finitely generated Abelian group under addition
However, these criteria do not exlude structures which are not usually thought of as 'lattices'. For example, certain quasicrystals and aperiodic tilings may satisfy these criteria (thus I cannot safely say that a 'lattice' may only have 2,3,4, or 6-fold rotational symmetry).
But when I look back through my references, I can't find anything to suggest that these objects are not examples of lattices - albeit not the usual ones.
The distinction is easy enough to make and somewhat novel - I could easily replace 'lattice' in places with 'periodic lattice' to make the problem go away.
In any case, is there a "the" definition of a lattice I should be using which makes this distinction?
A lattice is a discrete subgroup of $\mathbb{R}^n$ which is of rank $n$ (i.e. it is isomorphic to $\mathbb{Z}^n$ under addition).