A bijective monoid homomorphism is called a monoid isomorphism. Any bijective ring homomorphism is a ring isomorphism. A "K"-algebra isomorphism is a bijective "K"-algebra homomorphism. Two monoids are said to be isomorphic if there is a monoid isomorphism between them. An endomorphism that is also an isomorphism is an automorphism. An automorphism is an endomorphism that is also an isomorphism. This is the first isomorphism theorem for affine spaces. The function field is invariant under isomorphism and birational equivalence of varieties. Condition 3 is equivalent to the fact that this bundle homomorphism is an isomorphism. An isomorphism to itself is called an automorphism. An automorphism is an isomorphism from a structure to itself. These can therefore serve as isomorphism invariants of graphs. Representations of functors are unique up to a unique isomorphism. The Riesz representation theorem states that this mapping is an antilinear isomorphism. The term "isomorphism" is mainly used for algebraic structures.