I am reading about pushdown automata and I don't understand the definition of $\vdash$.
My book writes that $(q,aw,Z\alpha)\vdash(p,w,\beta\alpha)$ if $(p,\beta)\in\delta(q,a,Z)$
Can someone please explain to me this definition ?
I understand that the automata is at first in state $q$ and the reminder of the input is $aw$ (so it starts with $a$ so it fits to the second argument of $\delta)$,
but I have problems understanding whats happening with $Z\alpha$ and $\beta\alpha$, is the content of the stack $Z\alpha$ or is it just whats written at the top ? why go from $Z\alpha$ to $\beta\alpha$ ? and actually where is $\alpha$, is it in $\Gamma^{*}$?
another small issue I have: does $\Sigma\subset\Gamma$ ? it is not written in my book but I suspect so.
if it does, does $\Gamma=\Sigma\cup\{\dashv\}$? does $\dashv\not\in\Sigma$ by definition ?
I'm sorry for all this definition questions, it is not defined in my book.