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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. $

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  • $\begingroup$ 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. $\endgroup$ – Peter LeFanu Lumsdaine Jul 10 '17 at 18:01
  • $\begingroup$ 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?) $\endgroup$ – YCor Jul 10 '17 at 21:54

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