Mathematical theory of saturation
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Original Wikitext content:
The set of n-tuples of integers Z^n contained in the n-dimensional [[http://en.wikipedia.org/wiki/Vector_space|real vector space]] R^n is often called the [[http://en.wikipedia.org/wiki/Integer_lattice|integer lattice]], or grid lattice. Two lattice points consisting of n-tuples of integers can be added coordinatewise, making Z^n a [[http://en.wikipedia.org/wiki/Free_abelian_group|free abelian group]] of rank n. Its subgroups, which are also sublattices, have the property of //saturation// if for any element a of Z^n, 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 linear combination with rational coefficients q1*v1 + ... + qk*vk of elements of vk belongs to Z^n, then it belongs to V.
Original HTML content:
<html><head><title>Saturation</title></head><body>The set of n-tuples of integers Z^n contained in the n-dimensional <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Vector_space" rel="nofollow">real vector space</a> R^n is often called the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Integer_lattice" rel="nofollow">integer lattice</a>, or grid lattice. Two lattice points consisting of n-tuples of integers can be added coordinatewise, making Z^n a <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Free_abelian_group" rel="nofollow">free abelian group</a> of rank n. Its subgroups, which are also sublattices, have the property of <em>saturation</em> if for any element a of Z^n, 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 linear combination with rational coefficients q1*v1 + ... + qk*vk of elements of vk belongs to Z^n, then it belongs to V.</body></html>