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 elements in a ``free manner,'' 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 also Fixed, Gadget, Mirror Image, Rank