In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). The word homomorphism comes from the ancient Greek language: ὁμός (homos) meaning same and μορφή (morphe) meaning form or shape.ince inverses exist in G and H, one can show that the identity of G maps to the identity of H and that inverses are preserved. The algebraic structure to be preserved may include more than one operation, and a homomorphism is required to preserve each operation.