# Is there a name for this function with properties…

Let $V$ be a vector space over an algebraic structure $\mathbb{A}$, and suppose we have a binary operation $\star:V^2\to V$. Consider a function $f:V\to \mathbb{A}$ with the property that $$f(x\star y)=f(x)f(y),$$ for all $x,y\in V$.

Q1: Is there a name for such a function $f$ given that $\mathbb{A}$ is not necessarily a field, for example? I know there are norms with this property, but since $\mathbb{A}$ is not necessarily a field then I don't think I can call it a norm (see my related question here) Q2: Additionally is there a name of a system to take into account all the above conditions?

• $f$ is an algebra homomorphism, where $(V,\star)$ is an algebra, and $\mathbb{A}$ is an algebra. Example: $V=M_n(\mathbb{H})$, with $A\star B=AB$ and $\mathbb{A}=\mathbb{H}$ the quaternion algebra, and $f=\det$. – Dietrich Burde Aug 13 '15 at 11:16
• Thanks for clarifying Dietrich, and giving an example of such a system. – Pixel Aug 13 '15 at 12:47