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. $