Flats of a linear space

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Substructures
Let be a linear space. A subset is called a flat of if for all with

Of course, the intersection of flats is another flat, hence each subset is contained in a smallest flat <E>. In particular, the set of all flats, ordered by inclusion, forms a complete lattice.

The rank of a linear space is defined to be

If has finite rank, we define the dimension of as


Contributed by Hauke Klein
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