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.

A free abstract mathematical object is generated by $n$ elements in a ``free manner,'' 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 $f:G\to Y$ 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) $g:X\to Y$ 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, Gadget, Mirror Image, Rank