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.
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










If the algebraic gadgets are vector spaces, then








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