A [[Homomorphism]] is a [[Morphism|structure-preserving]] [[Function|map]] $f$ between two algabraic structures $A \to B$ such that it preserves the operations over those structures $*$, $\cdot$. If the $\mu_{A}$ and $\mu_{B}$ are some [[Binary Operation]] $*$ and $\cdot$, this can be expressed simply as: $\huge f(a*b) = f(a) \cdot f(b) $ If they are not binary operations, then you can express the same property more explicitly like so: $\huge \begin{align} f(\mu_{A}(a_{1}, a_{2}, \dots, a_{k})) &=\mu_{B}(f(a_{1}), f(a_{2}), \dots, f(a_{k})) \end{align}$ Note the [[Set|sets]] $A$ and $B$ must be of the same type, eg. both must be [[Vector Space|vector spaces]] or [[Field|fields]], etc. >[!example] >The [[Exponential]] [[Function]] $x \mapsto e^{x}$ is a [[Homomorphism]] between the [[Group]] $(\R,+)$ to the [[Group]] $(\R, \cdot)$. >$\begin{align} >e^{x+y} =& e^{x}e^{y}\\ >x+y &= z \\ >e^{x} \cdot e^{z} &= e^{z} >\end{align} $ ### Examples - [[Homomorphism]] of [[Vector Space|Vector Space]] are called [[Linear Transformation|Linear Maps]]