# show that the set of functions from $\mathbb{Z}[X^{n}]$ to $A$ is generated by the characteristic functions

Let $X$ be a quandle and $A$ be an abelian group (for simplicity assume that $A$ is a finite cyclic group $\mathbb{Z}_{n}$ or the infinite cyclic group $\mathbb{Z}$). I need to show that the set of functions from $\mathbb{Z}[X^{n}]$ to $A$ is generated by the characteristic functions denoted $\chi_{x}$ where $x \in X^{n}.$

The characteristic function is defined by $\chi_{x}(y) = \begin{cases} 1 & \text{if$x = y$,} \\ 0 & \text{otherwise.} \end{cases}$

A quandle $X$ is a non-empty set with a binary operation $(a,b)\mapsto a\ast b$ such that:

(Q1) $\forall a \in X , a\ast a=a.$

(Q2) $\forall a,b \in X, \exists !c\in X, a=(c\ast b).$

(Q3) $\forall a, b, c\in X, (a\ast b)\ast (b\ast c)=(a\ast b) \ast c.$

• What sense of “generated” do you mean — that is, what kind of algebraic structure are you considering the set of functions $\mathbb{Z}[X^n] \to A$ as carrying? (This also sounds a little bit like a homework exercise, and not research-level, in which case it would be off-topic here and should be asked at math.stackexchange instead; but that’s not quite clear until we know what structure you’re considering on the function set. – Peter LeFanu Lumsdaine Jul 10 '17 at 18:01
• What is $X^n$? The direct product? if so, why ask the question for $X^n$: if it holds for every quandle, it should work for $X^n$. In any case, the question is unclear at this point (what is $\mathbb{Z}[X^n]$? "generated" in which sense?) – YCor Jul 10 '17 at 21:54