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.