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Free -- from Wolfram MathWorld

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology Alphabetical Index New in MathWorld Free

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

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

See alsoFixed

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Free Group

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Free Variable

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Freely

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Gadget

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Mirror Image

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Rank Explore with Wolfram|Alpha Cite this as:

Weisstein, Eric W. "Free." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Free.html

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