Does anyone know a good book about lattices (as subgroups of a vector space $V$)?
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These notes of mine on geometry of numbers begin with a section on lattices in Euclidean space. However they are a work in progress and certainly not yet fully satisfactory. Of the references I myself have been consulting for this material, the one I have found most helpful with regard to basic material on lattices is C.L. Siegel's Lectures on the Geometry of Numbers.
Check the following:
Roman,Steven: "Lattices and Ordered Sets", Springer
Blyth, T.S.: "Lattices and Ordered Algebraic Structures" , Springer
Roggenkamp, Klaus - Huber-Dyson, Verena : "Lattices over Orders" , Springer