Let $(X,\leq)$ and $(Y,\leq)$ be locally finite partially ordered sets with Möbius functions $\mu_X$ respectively $\mu_Y$, and let $\varphi:X\to Y$ be an order isomorphism. I want to show that \begin{equation*} \mu_Y(\varphi(x_1),\varphi(x_2)) = \mu_X(x_1,x_2) \end{equation*} for all $x_1,x_2\in X$, $x_1\leq x_2$.
In e.g. Carl G. Wagner's book, A First Course in Enumerative Combinatorics, p. 235, I find a proof that use's Philip Hall's theorem. I am not familiar with this theorem, but it appears to rely on the chain structure of posets. Instead of diving into this approach, I was wondering if the following approach is correct.
By definition, the Möbius function is the inverse of the zeta function with respect to convolution on the incidence algebra, i.e., \begin{equation*} \sum_{x_1\leq x_\leq x_2} \mu(x_1,x)\zeta(x,x_2) = (\mu * \zeta)(x_1,x_2) = \delta(x_1,x_2) \end{equation*} where \begin{equation*} \delta(x_1,x_2) = \left\{\begin{array}{rl} 1&\text{if }x_1=x_2, \\ 0&\text{otherwise} \end{array}\right. \end{equation*} is the multiplicative unit in the incidence algebra, and $\zeta(x_1,x_2) = 1$ for all $x_1,x_2\in X$, $x_1\leq x_2$.
Using subscripts $X$ and $Y$ to refer to the specific posets $X$ and $Y$, I now argue that by convoluting $\mu_Y(\varphi(\cdot),\varphi(\cdot))$ with $\delta_X$, then using the identity $\delta_X = \zeta_X * \mu_X$, followed by associativity of convolution and the fact that $\zeta_X = \zeta_Y(\varphi(\cdot),\varphi(\cdot))$, will yield the desired identity. That is, for any $x_1,x_2\in X$, $x_1\leq x_2$, we have \begin{aligned} \mu_Y(\varphi(x_1),\varphi(x_2)) &= (\mu_Y(\varphi(\cdot),\varphi(\cdot)) * \delta_X)(x_1,x_2) \\ &= \sum_{x_1\leq x\leq x_2} \mu_Y(\varphi(x_1),\varphi(x))\delta_X(x,x_2) \\ &= \sum_{x_1\leq x\leq x_2} \mu_Y(\varphi(x_1),\varphi(x))(\zeta_X * \mu_X)(x,x_2) \\ &= \sum_{x_1\leq x\leq y\leq x_2} \mu_Y(\varphi(x_1),\varphi(x)) \zeta_X(x,y) \mu_X(y,x_2) \\ &= \sum_{x_1\leq x\leq y\leq x_2} \mu_Y(\varphi(x_1),\varphi(x)) \zeta_Y(\varphi(x),\varphi(y)) \mu_X(y,x_2) \\ &= \sum_{x_1\leq y\leq x_2} \delta_Y(\varphi(x_1),\varphi(y)) \mu_X(y,x_2) \\ &= \mu_X(x_1,x_2) \end{aligned} Is this a valid proof of the above statement?