When referring to a planar object, "free" means that the object is regarded as capable of being picked up out of the plane and flipped over. As a result, mirror images are equivalent for free objects.

The word "free" is also used in technical senses to refer to a free group, free semigroup, free tree, free variable, etc.

In algebraic topology, a free abstract mathematical object is generated by n elements in a "free manner" ("freely"), i.e., such that the n elements satisfy no nontrivial relations among themselves. To make this more formal, an algebraic gadget X is freely generated by a subset G if, for any function f:G->Y where Y is any other algebraic gadget, there exists a unique homomorphism (which has different meanings depending on what kind of gadgets you're dealing with) g:X->Y such that g restricted to G is f.

If the algebraic gadgets are vector spaces, then G freely generates X iff G is a basis for X. If the algebraic gadgets are Abelian groups, then G freely generates X iff X is a direct sum of the integers, with G consisting of the standard basis.

See also

Fixed, Free Group, Free Variable, Freely, Gadget, Mirror Image, Rank

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Cite this as:

Weisstein, Eric W. "Free." From MathWorld--A Wolfram Web Resource.

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