Mathematical theory of saturation: Difference between revisions

{{Expert|Saturation, torsion, and contorsion}}

The set of ''n''-tuples of integers <math>\mathbb{Z}^n</math> such that two ''n''-tuples can be added coordinatewise is the {{w|free abelian group}} of rank ''n''. Its subgroups have the property of '''saturation''' if for any element ''a'' of <math>\mathbb{Z}^n</math>, if an integer multiple ''m''·''a'' of ''a'' belongs to a sublattice ''V'', then ''a'' already belongs to ''V''. Another way to put it is that if some {{w|linear combination}} with rational coefficients {{nowrap| ''q''₁''v''₁ + … + ''q''_''k''''v''_''k'' }} of elements of ''V'' belongs to <span><math>\mathbb{Z}^n</math></span>, then it belongs to ''V''. For the latter definition we consider <math>\mathbb{Z}^n</math> to be contained in the ''n''-dimensional real {{w|vector space}} <math>\mathbb{R}^n</math>, in which case <math>\mathbb{Z}^n</math> is often called the {{w|integer lattice}}, or grid lattice.