# Definition Given a [[map]] $f: X \rightarrow Y$ and the [[set|sets]] $X$ and $Y$ are endowed with some additional algebraic structure. If the map $f$ preserves such algebraic structures, then it is called a *homomorphism*. A homomorphism that is also [[Types of Maps|onto]] (i.e, a surjection) is said to be a homomorphism from $X$ *onto* $Y$. A [[Types of Maps|bijective]] homomorphism is an [[Isomorphism|isomorphism]]. [[group homomorphism|Group homomorphisms]] are special cases of homomorphisms that preserve the group structure.