I'm an undergraduate student, and I'm facing problems with entropy for the first time.
The word "adiathermal" is used here, as suggested in the comments, in order to describe a process occurring without any flow of heat and matter. "Adiabatic" was the word used at first. (Use the word you prefer, the meaning is what matters)
I know that if I consider an irreversible adiathermal process from A to B, the change in entropy $\Delta S_{AB}$ is more than $0$, and that if I consider a reversible adiathermal process starting from A, its path will not pass through B (and its difference of entropy will be $\Delta S_{AP} = 0$, where $P$ is its ending point).
My question is: when I have an irreversible $x$ process (where $x =$ isothermal, isobaric, etc...) from A to B, is there any reversible $x$ process going from state A to state B? I.e. Irreversible isothermal process from A to B, is there a reversible isothermal process from A to B?
As explained before, I have found a counterexample for $x = $ adiathermal, so if I want to evaluate $\Delta S_{AB}$ I need to find another reversible path from A to B (i.e. rev. isobaric + rev. isothermal).
I think such $x$ process does not exist, otherwise we would just use Clausius integral ($\int_{A}^{B} \frac{\delta Q}{T}$) without noticing any difference between a reversible and irreversible process. Furthermore, I think there should be some deeper concept that I'm missing here.