We define a set \mathbb{F}^{s}, which is the set of unit functions that maps from any set S to \mathbb{F}. closeness of addition \begin{equation} (f+g)(x) = f(x)+g(x), \forall f,g \in \mathbb{F}^{S}, x \in S \end{equation} closeness of scalar multiplication \begin{equation} (\lambda f)(x)=\lambda f(x), \forall \lambda \in \mathbb{F}, f \in \mathbb{F}^{S}, x \in S \end{equation} commutativity inherits \mathbb{F} (for the codomain of functions f and g) associativity inherits \mathbb{F} for codomain or is just \mathbb{F} for scalar distribution inherits distribution in \mathbb{F} on the codomain again additive identity \begin{equation} 0(x) = 0 \end{equation} additive inverse \begin{equation} (-f)(x) = -f(x) \end{equation} multiplicative identity 1 hee hee